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✓ 11 results · all verified · 0 also independently AI-judged

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Murnaghan Stability via FI-Modules

1 · Prerequisites

2 · Summary

Objective. This page proves Murnaghan's stabilization theorem for Kronecker coefficients, develops the reduced coefficients through the lambda-ring and vertex-operator calculus, and computes the exact global stabilization threshold of a fixed Kronecker product. The route is the FI-module proof: the padded irreducible families (Each padded irreducible family is realised by a finitely generated FI-module) tensor together to a finitely generated FI-module, and finite generation is equivalent to uniform representation stability over characteristic zero (Finite generation is equivalent to uniform representation stability), so the padded Kronecker multiplicities $g_{α [n], β [n]}^{γ [n]}$ are eventually constant (Murnaghan stability for Kronecker coefficients). The limit is the reduced Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ (The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ ).

The second half makes the stable family computable. The Jacobi–Trudi completion of an arbitrary integer sequence (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ ) and the vertex-operator generating series $\sum_{n \in Z} s_{α [n]} t^{n} = σ [t X] s_{α} [t X - 1]$ (Vertex-operator generating series for the padded Schur functions) are combined into the compatibility of padding with the internal product (The Jacobi–Trudi completion is compatible with the internal product); comparing with the stable expansion identifies the generating polynomial with $\sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ}$ and yields Brion's formula (Brion's formula for the reduced Kronecker coefficients) in both its two-alphabet and symmetric three-alphabet forms. Brion's formula expands into four nonnegative coefficient families (The nonnegative expansion of a reduced Kronecker coefficient), which give Murnaghan's size inequalities (The support of the reduced Kronecker coefficients satisfies the size triangle inequalities), the top-degree Murnaghan–Littlewood rule (Murnaghan–Littlewood: top-degree reduced coefficients are Littlewood–Richardson coefficients) and the full expansion at every admissible rank (The Murnaghan expansion at every admissible rank). The signed straightening terms in that expansion genuinely cancel at small admissible ranks, and the page keeps that behaviour rather than replacing it by coefficientwise stability. Finally the support maximum $∣ γ ∣ + γ_{1} \leq ∣ α ∣ + ∣ β ∣ + α_{1} + β_{1}$ , attained at $γ = α + β$ with coefficient one, is shown to be the exact least global stabilization index $stab (α, β) = ∣ α ∣ + ∣ β ∣ + α_{1} + β_{1}$ (The exact global stabilization threshold for a fixed Kronecker product), with the one-empty and both-empty cases handled explicitly. No positivity or combinatorial rule for general Kronecker coefficients is asserted, and the global threshold is not claimed to be sharp for individual coefficients; the companion page murnaghan-stability-via-fi-modules-examples works the pair $α = β = (2)$ and exhibits both boundaries.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Specializations of symmetric functions and the series $σ$

Definition

Work in $Λ_{Q} : = Q \otimes_{Z} Λ$ , the $Q$ -scalar extension of the stable graded ring (The stable graded ring of symmetric functions). Its power sums $p_{1}, p_{2}, \dots$ (Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ) are algebraically independent generators: $Λ_{Q} = Q [p_{1}, p_{2}, \dots]$ , and for every $d$ the products $p_{λ}$ , $λ ⊢ d$ , form a $Q$ -basis of $Λ_{Q}^{d}$ (Power sums form a rational but not integral stable basis).

A specialization is a $Q$ -algebra homomorphism $f \mapsto f [A]$ from $Λ_{Q}$ to a commutative $Q$ -algebra. By the algebraic independence of the power sums, a specialization is determined by the sequence of values $(p_{k} [A])_{k \geq 1}$ , and conversely every sequence of elements of a commutative $Q$ -algebra extends uniquely to a specialization. For specializations $A, B$ , a scalar $c \in Q$ and a one-letter alphabet $t$ one defines the specializations $A + B$ , $- A$ , $A B$ , $c \cdot A$ and $t A$ by their power-sum values

$p_{k} [A + B] : = p_{k} [A] + p_{k} [B], p_{k} [- A] : = - p_{k} [A], p_{k} [A B] : = p_{k} [A] p_{k} [B],$

$p_{k} [c \cdot A] : = c p_{k} [A], p_{k} [t] : = t^{k} and hence p_{k} [t A] = t^{k} p_{k} [A] .$

Here $c \cdot A$ is a coefficient multiplicity, to be distinguished from the one-letter alphabet $t$ , for which $p_{k} [t] = t^{k}$ is a power and $p_{k} [t A] = t^{k} p_{k} [A]$ ; all occurrences of $t X$ , $t^{- 1}$ , $t A$ and $σ [t]$ on this page use this one-letter convention. The symbol $1$ denotes the one-letter alphabet with $p_{k} [1] = 1$ for all $k$ , and $- 1$ denotes its plethystic negative with $p_{k} [- 1] = - 1$ for all $k$ (Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ); it is not the ordinary one-letter alphabet whose letter is minus one, for which $p_{k}$ would equal $(- 1)^{k}$ . One writes $f [A + B]$ , $f [- A]$ , $f [A B]$ and $f [c \cdot A]$ for the images of $f$ under these specializations.

Define $X^{⊥}$ to be the specialization with $f [X^{⊥}] : = f^{⊥}$ for all $f$ , where $f^{⊥}$ is the adjoint of multiplication by $f$ with respect to the Hall inner product $⟨, ⟩_{H}$ (The Hall inner product on symmetric functions), characterized by $⟨ f^{⊥} g, h ⟩_{H} = ⟨ g, f h ⟩_{H}$ . It exists and is unique because the Hall form is nondegenerate on each finite-dimensional graded piece $Λ_{Q}^{d}$ ; the adjoints of commuting multiplication operators commute, so the operators $f^{⊥}$ span a commutative $Q$ -algebra and $X^{⊥}$ is a specialization in the sense above.

Define the series $σ [X] : = \sum_{n \geq 0} h_{n} [X],$ $h_{n}$ the complete homogeneous symmetric function (Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ). Here $σ$ is interpreted in the graded completion of $Λ_{Q}$ (or with a formal degree variable): it is a formal series, not an element of $Λ_{Q}$ , and every occurrence of $σ$ below is to be read degree by degree. Every Hall pairing of a series with a polynomial is the finite sum obtained by truncating to the degrees of the polynomial. No convergence of a scalar infinite sum is asserted, and no value of $σ$ at a specialization with infinitely many letters is interpreted analytically.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Threefold Schur coefficients $c_{a, b, c}^{δ}$

Definition

For partitions $δ, a, b, c$ with $∣ δ ∣ = ∣ a ∣ + ∣ b ∣ + ∣ c ∣$ , define the threefold Schur coefficient $c_{a, b, c}^{δ} : = [s_{δ}] s_{a} s_{b} s_{c},$ the coefficient of $s_{δ}$ in the ordinary product of the three Schur functions of $Λ$ (Stable Schur functions from bialternants); when $∣ δ ∣ \neq ∣ a ∣ + ∣ b ∣ + ∣ c ∣$ the coefficient is $0$ , the product being homogeneous of degree $∣ a ∣ + ∣ b ∣ + ∣ c ∣$ . Since $s_{a} s_{b} = \sum_{ϕ} c_{a, b}^{ϕ} s_{ϕ}$ with $c_{a, b}^{ϕ}$ the Littlewood–Richardson coefficients (The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients), substituting gives $c_{a, b, c}^{δ} = \sum_{ϕ} c_{a, b}^{ϕ} c_{ϕ, c}^{δ} = \sum_{ψ} c_{a, c}^{ψ} c_{ψ, b}^{δ},$ the sums over partitions $ϕ$ of $∣ a ∣ + ∣ b ∣$ and $ψ$ of $∣ a ∣ + ∣ c ∣$ ; hence each $c_{a, b, c}^{δ}$ is a nonnegative integer, and the two displayed expressions agree with the definition. These coefficients are symmetric in $a, b, c$ because the product $s_{a} s_{b} s_{c}$ is commutative, and equivalently $c_{a, b, c}^{δ} = ⟨ s_{a} s_{b} s_{c}, s_{δ} ⟩_{H},$ by the orthonormality of the Schur basis (Schur functions form an orthonormal integral basis). No fourfold or longer iterated coefficient is introduced here; iterating the two identities above gives every nested product of three factors, and the symmetry of the three subscripts is used throughout this page.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$

Definition

Padded sequences. For a partition $α = (α_{1}, \dots, α_{k})$ and an integer $n$ , with missing parts read as zero, the first-row padding is the sequence $α [n] : = (n - ∣ α ∣, α_{1}, \dots, α_{k})$ , a partition of $n$ exactly when $n \geq ∣ α ∣ + α_{1}$ (First-row padding of a partition, Padded Kronecker coefficient sequences). When this sequence is used as a partition label, omit trailing zero entries; in particular $\emptyset [0] = \emptyset$ and the missing first part is zero.

The Jacobi–Trudi completion. For an arbitrary finite sequence of integers $μ = (μ_{1}, \dots, μ_{m})$ , define $s_{μ} : = \det (h_{μ_{j} - j + i})_{1 \leq i, j \leq m},$ where $h_{0} = 1$ and $h_{k} = 0$ for $k < 0$ (Jacobi–Trudi and dual Jacobi–Trudi identities); for the empty sequence the empty determinant is $1$ . Every such determinant is $0$ or $\pm 1$ times a Schur function. Indeed, put $q_{j} : = μ_{j} + m - j$ . If $q_{j} = q_{k}$ for $j \neq k$ , the $j$ -th and $k$ -th columns of the matrix agree, so the determinant vanishes. Otherwise let $σ$ be the permutation sorting $q$ into decreasing order $q_{σ (1)} > \dots > q_{σ (m)}$ and set $ν_{j} : = q_{σ (j)} + j - m$ ; then $ν$ is weakly decreasing and, permuting columns with sign $sgn (σ)$ and using $h_{ν_{j} - j + i} = h_{q_{σ (j)} + i - m}$ , the determinant equals $sgn (σ) \det (h_{ν_{j} - j + i})$ . If $\min_{j} q_{j} < 0$ then $ν_{m} < 0$ and the last column of this determinant is zero, so the value is $0$ ; if $\min_{j} q_{j} \geq 0$ then $ν$ , after deleting trailing zero entries, is a partition of $∣ μ ∣$ with at most $m$ parts and the Jacobi–Trudi identity gives the value $sgn (σ) s_{ν}$ . In particular $s_{α [n]} \in Λ^{n}$ for every $n \geq 0$ , and $s_{α [n]} = 0$ for $n < 0$ : every term of the determinant expansion has total degree $n$ , so for $n < 0$ some factor has negative index and vanishes.

The operators $U_{n}$ and $V$ . The Schur functions form a $Z$ -basis of $Λ$ (Schur functions form an orthonormal integral basis), so there are unique $Z$ -linear endomorphisms $U_{n}$ ( $n \in Z$ ) and $V$ of $Λ$ determined on the Schur basis by $U_{n} (s_{λ}) : = s_{λ [n]}, V (s_{λ}) : = s_{λ + (1)},$ where $λ + (1) = (λ_{1} + 1, λ_{2}, \dots)$ . Thus $U_{n} (Λ) \subseteq Λ^{n}$ for $n \geq 0$ (and $U_{n} = 0$ for $n < 0$ ), while $V$ raises degree by one; in particular $V (1) = s_{(1)}$ . The operators are extended linearly, and $V^{k}$ denotes the $k$ -fold iterate of $V$ .

The stabilization index. The internal product $*$ on $Λ$ is the degree-preserving product of The internal (Kronecker) product on symmetric functions and Kronecker coefficients and the internal tensor product. For partitions $α, β$ , define $stab (α, β)$ to be the least integer $n_{0} \geq 0$ such that $s_{α [n_{0} + k]} * s_{β [n_{0} + k]} = V^{k} (s_{α [n_{0}]} * s_{β [n_{0}]}) for all k > 0,$ following the source. This definition records the source's convention only: it does not assert that such an $n_{0}$ exists, nor that a value of $n_{0}$ satisfying the display is least for any other property. Existence, least-ness and the exact value of $stab (α, β)$ are proved in The exact global stabilization threshold for a fixed Kronecker product.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Schur functions on sums and products of alphabets

Statement

Let $A, B$ be specializations (Specializations of symmetric functions and the series $σ$ ).

(a) For every partition $λ$ , $s_{λ} [A + B] = \sum_{μ, ν} c_{μ, ν}^{λ} s_{μ} [A] s_{ν} [B],$ the sum over pairs of partitions with $∣ μ ∣ + ∣ ν ∣ = ∣ λ ∣$ , where $c_{μ, ν}^{λ}$ is the Littlewood–Richardson coefficient (Littlewood--Richardson tableaux and coefficients).
(b) For every partition $λ$ , $s_{λ} [A B] = \sum_{μ, ν} g_{μ, ν}^{λ} s_{μ} [A] s_{ν} [B],$ the sum over pairs with $∣ μ ∣ = ∣ ν ∣ = ∣ λ ∣$ , where $g_{μ, ν}^{λ}$ is the ordinary Kronecker coefficient (Kronecker coefficients and the internal tensor product, The internal (Kronecker) product on symmetric functions).

Both sums are finite. No multiplicativity or bilinearity of $s_{λ}$ in the alphabet variables, and no unproved Schur-functor decomposition, is assumed.

Facts & Assumptions

Given: Specializations $A, B, U, V, W$ into commutative $Q$ -algebras, partitions $λ, μ, ν, ρ$ and integers $d, n \geq 0$ .

[F1]

Specializations are the algebra homomorphisms determined by arbitrary power-sum values, with $p_{k} [A + B] = p_{k} [A] + p_{k} [B]$ , $p_{k} [A B] = p_{k} [A] p_{k} [B]$ ; for a product $ρ$ one has $p_{ρ} [A + B] = \prod_{i} (p_{ρ_{i}} [A] + p_{ρ_{i}} [B])$ and $p_{ρ} [A B] = p_{ρ} [A] p_{ρ} [B]$ (Specializations of symmetric functions and the series $σ$ ).

[F2]

$Λ_{Q} = Q [p_{1}, p_{2}, \dots]$ and ${p_{λ} : λ ⊢ d}$ is a $Q$ -basis of $Λ_{Q}^{d}$ (Power sums form a rational but not integral stable basis).

[F3]

$h_{n} = \sum_{λ ⊢ n} p_{λ} / z_{λ}$ in $Λ_{Q}$ , where $z_{λ} = \prod_{i} i^{m_{i} (λ)} m_{i} (λ)!$ ; in a finite alphabet, $h_{k}$ is the sum of all monomials of total degree $k$ and $\sum_{k \geq 0} h_{k} t^{k} = \prod_{x} (1 - x t)^{- 1}$ (Complete homogeneous functions expand in power sums with cycle-distribution coefficients, Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

[F4]

Newton's identities: with $e_{0} = 1$ , $e_{k} = 0$ for $k > n$ , one has $k e_{k} = \sum_{i = 1}^{k} (- 1)^{i - 1} e_{k - i} p_{i}$ in $R [x_{1}, \dots, x_{n}]^{{Sym}_{n}}$ (Newton's identities: $k e_{k} = \sum_{i = 1}^{k} (- 1)^{i - 1} e_{k - i} p_{i}$ ).

[F5]

The fundamental theorem of symmetric polynomials: substitution $T_{k} \mapsto e_{k}$ is an isomorphism $R [T_{1}, \dots, T_{n}] \to R [x_{1}, \dots, x_{n}]^{{Sym}_{n}}$ (Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_{1}, \dots, e_{n}$ ).

[F6]

In the bidegree completion Bidegree completion of two symmetric-function rings, the Cauchy kernel satisfies $Ω (X, Y) = \sum_{λ} h_{λ} (X) m_{λ} (Y) = \sum_{λ} s_{λ} (X) s_{λ} (Y)$ , and in the rational completion $Ω (X, Y) = \sum_{λ} z_{λ}^{- 1} p_{λ} (X) p_{λ} (Y)$ ; each sum is by diagonal bidegree (Power-sum, complete, and Schur expansions of the Cauchy kernel).

[F7]

$s_{μ} s_{ν} = \sum_{λ} c_{μ ν}^{λ} s_{λ}$ in $Λ$ , and $s_{λ / μ} = \sum_{ν} c_{μ ν}^{λ} s_{ν}$ (The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients).

[F8]

The Schur functions form an orthonormal $Z$ -basis: $⟨ s_{λ}, s_{μ} ⟩_{H} = δ_{λ μ}$ (Schur functions form an orthonormal integral basis, The Hall inner product on symmetric functions), and $⟨ p_{ρ}, p_{σ} ⟩_{H} = δ_{ρ σ} z_{ρ}$ (Power sums are orthogonal for the Hall form).

[F9]

The internal product $*$ satisfies $s_{μ} * s_{ν} = \sum_{λ} g_{μ ν}^{λ} s_{λ}$ and $p_{ρ} * p_{σ} = δ_{ρ σ} z_{ρ} p_{ρ}$ (Kronecker coefficients and the internal tensor product, The internal (Kronecker) product on symmetric functions, Power sums diagonalize the internal product).

[F10]

The rank- $N$ projection $Λ^{d} \to A_{N}^{d}$ , $f \mapsto f (x_{1}, \dots, x_{N})$ , sends $m_{λ}$ to the orbit sum $m_{λ} (x_{1}, \dots, x_{N})$ for $ℓ (λ) \leq N$ (and to $0$ otherwise) and $s_{λ}$ to $s_{λ} (x_{1}, \dots, x_{N})$ ; for $N \geq d$ it carries the basis ${m_{λ} : λ ⊢ d}$ bijectively onto the basis ${m_{λ} (x) : λ ⊢ d}$ of the symmetric polynomials, hence is injective on $Λ^{d}$ (The monomial symmetric functions form the integral stable basis, Monomial symmetric polynomials form an $R$ -basis of the symmetric-polynomial ring, Stable Schur functions from bialternants, The stable graded ring of symmetric functions).

Proof

technique · universal polynomial identities checked on finite alphabets

1.1F4F5

Genericity. Fix $d \geq 0$ and $N \geq d$ . The substitution $σ_{d}$ from $Q [A_{1}, \dots, A_{d}; B_{1}, \dots, B_{d}]$ to $Q [x_{1}, \dots, x_{N}, y_{1}, \dots, y_{N}]$ with $A_{k} \mapsto p_{k} (x)$ , $B_{k} \mapsto p_{k} (y)$ is injective. Indeed, by [F4] over $Q$ the identities $k e_{k} = (- 1)^{k - 1} p_{k} + \sum_{i = 1}^{k - 1} (- 1)^{i - 1} e_{k - i} p_{i}$ express each $e_{k}$ recursively in $p_{1}, \dots, p_{k}$ and each $p_{k}$ recursively in $e_{1}, \dots, e_{k}$ for $k \leq N$ , so $Q [e_{1}, \dots, e_{N}] = Q [p_{1}, \dots, p_{N}]$ as subrings of $Q [x_{1}, \dots, x_{N}]$ ; by [F5] that ring is a free polynomial ring on $N$ generators, so $p_{1}, \dots, p_{N}$ , and hence $p_{1}, \dots, p_{d}$ , are algebraically independent. The same holds in the $y$ -variables, and the tensor product over $Q$ of two injective substitutions is injective.

2.1F1F3step 1.1

Multiplicativity. For all specializations $U, V$ and all $n \geq 0$ one has $h_{n} [U + V] = \sum_{i + j = n} h_{i} [U] h_{j} [V]$ ; consequently $σ [U + V] = σ [U] σ [V]$ coefficientwise. By [F3], $h_{n} = \sum_{λ ⊢ n} p_{λ} / z_{λ}$ involves only $p_{1}, \dots, p_{n}$ , so both sides of the claimed identity are the value at $(p_{1} [U], \dots, p_{n} [U], p_{1} [V], \dots, p_{n} [V])$ of the universal polynomial $D_{n} (A; B) : = \sum_{λ ⊢ n} \prod_{i} (A_{λ_{i}} + B_{λ_{i}}) / z_{λ} - \sum_{i + j = n} (\sum_{μ ⊢ i} p_{μ} (A) / z_{μ}) (\sum_{ν ⊢ j} p_{ν} (B) / z_{ν})$ , where $p_{μ} (A) = \prod_{i} A_{μ_{i}}$ . At honest alphabets $x_{1}, \dots, x_{n}$ and $y_{1}, \dots, y_{n}$ , the generating function $\sum_{k} h_{k} (ℓ) t^{k} = \prod_{ℓ} (1 - ℓ t)^{- 1}$ of [F3] gives $\sum_{k} h_{k} (x ⊔ y) t^{k} = \prod_{ℓ} (1 - ℓ t)^{- 1} = (\sum_{i} h_{i} (x) t^{i}) (\sum_{j} h_{j} (y) t^{j})$ , so $D_{n} (p (x); p (y)) = 0$ . By step 1.1 with $d = N = n$ the substitution is injective, so $D_{n} = 0$ , and the identity holds universally; since $σ = \sum_{n} h_{n}$ , this is multiplicativity of $σ$ , coefficient by coefficient finite.

2.2F1F2F3F6F10step 1.1

Cauchy identities for arbitrary specializations. For all specializations $X, Y$ and all $d \geq 0$ , $h_{d} [X Y] = \sum_{λ ⊢ d} s_{λ} [X] s_{λ} [Y] = \sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [X] p_{ρ} [Y]$ , equivalently $σ [X Y] = \sum_{λ} s_{λ} [X] s_{λ} [Y] = \sum_{λ} z_{λ}^{- 1} p_{λ} [X] p_{λ} [Y]$ coefficientwise. The second equality is [F3] applied to the alphabet $X Y$ : $h_{d} [X Y] = \sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [X Y] = \sum_{ρ} z_{ρ}^{- 1} p_{ρ} [X] p_{ρ} [Y]$ by [F1]. For the first, fix $d$ ; both $\sum_{λ ⊢ d} s_{λ} [X] s_{λ} [Y]$ and $\sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [X] p_{ρ} [Y]$ are values of universal polynomials in $(p_{k} [X], p_{k} [Y])_{k \leq d}$ , by [F2]. At honest alphabets $x_{1}, \dots, x_{N}$ , $y_{1}, \dots, y_{N}$ with $N \geq d$ , apply the projection $Λ^{d} (X) \otimes Λ^{d} (Y) \to A_{N}^{d} (x) \otimes A_{N}^{d} (y)$ of [F10] to the degree- $(d, d)$ component of the Cauchy identity [F6]: it maps the two sides to $\sum_{λ ⊢ d} s_{λ} (x) s_{λ} (y)$ and $\sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} (x) p_{ρ} (y)$ , which are therefore equal. By step 1.1 with $d = N$ the universal difference vanishes identically, proving the first equality for all $X, Y$ .

3.1F1F7F8step 2.1step 2.2

Coproduct (a). Apply step 2.2 to the pair of specializations $(U + V, Z)$ , where $Z$ denotes the tautological specialization into a fresh copy of $Λ$ ; the resulting chain of identities is $\sum_{λ} s_{λ} [U + V] s_{λ} [Z] = σ [(U + V) Z] = σ [U Z + V Z] = σ [U Z] σ [V Z] = \sum_{μ, ν} s_{μ} [U] s_{ν} [V] s_{μ} [Z] s_{ν} [Z]$ , where the second equality uses $p_{k} [(U + V) Z] = p_{k} [U Z] + p_{k} [V Z]$ from [F1], the third is step 2.1, and the last is step 2.2 applied twice. By [F7], $s_{μ} [Z] s_{ν} [Z] = \sum_{λ} c_{μ ν}^{λ} s_{λ} [Z]$ , so the right side equals $\sum_{λ} (\sum_{μ, ν} c_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]) s_{λ} [Z]$ , each coefficient being a finite sum. The $s_{λ} [Z]$ form a $Z$ -basis of $Λ^{d} (Z)$ by [F8] and [F10], so comparing coefficients of $s_{λ} [Z]$ gives $s_{λ} [U + V] = \sum_{μ, ν} c_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]$ for every $λ$ .

3.2F8F9step 2.2

Kernel identity. For $f, g \in Λ_{Q}^{d}$ and independent formal alphabet variables $U, V, W$ , with the Hall form of The Hall inner product on symmetric functions extended $Q$ -bilinearly to $Λ (U) \hat{\otimes} Λ (V)$ , one has $⟨ f [U] g [V] ∣ σ [U V W] ⟩_{U, V} = (f * g) [W]$ . By step 2.2 applied to the pair $(U V, W)$ , $σ [U V W] = \sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [U] p_{ρ} [V] p_{ρ} [W]$ in the relevant degree, and the pairing is a finite sum because only $ρ ⊢ d$ matches the bidegree $(d, d)$ of $f [U] g [V]$ . Writing $f = \sum_{ρ} a_{ρ} p_{ρ}$ and $g = \sum_{ρ} b_{ρ} p_{ρ}$ , so that $⟨ f, p_{ρ} ⟩_{H} = a_{ρ} z_{ρ}$ and $⟨ g, p_{ρ} ⟩_{H} = b_{ρ} z_{ρ}$ by [F8], the left side is $\sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [W] a_{ρ} b_{ρ} z_{ρ}^{2} = \sum_{ρ} a_{ρ} b_{ρ} z_{ρ} p_{ρ} [W]$ , which is $(f * g) [W]$ by the power-sum diagonalization [F9].

4.1F8F9step 2.2step 3.2

Plethysm (b). Keep $U, V, W$ independent formal alphabet variables. Apply step 2.2 to the pair $(U V, W)$ : $σ [U V W] = \sum_{λ} s_{λ} [U V] s_{λ} [W]$ . Pairing with $f [U] g [V]$ and using step 3.2 gives $\sum_{λ} ⟨ f [U] g [V], s_{λ} [U V] ⟩_{U, V} s_{λ} [W] = (f * g) [W] = \sum_{λ} ⟨ f * g, s_{λ} ⟩_{H} s_{λ} [W]$ , the last equality being the orthonormal expansion [F8]. Comparing coefficients of $s_{λ} [W]$ , legitimate exactly as in step 3.1, yields $⟨ f [U] g [V], s_{λ} [U V] ⟩_{U, V} = ⟨ f * g, s_{λ} ⟩_{H}$ . Taking $f = s_{μ}$ , $g = s_{ν}$ of degree $d$ gives $⟨ s_{μ} [U] s_{ν} [V], s_{λ} [U V] ⟩_{U, V} = g_{μ ν}^{λ}$ by [F9] and [F8]. The products $s_{μ} [U] s_{ν} [V]$ with $μ, ν ⊢ d$ are an orthonormal basis of the bidegree- $(d, d)$ piece of $Λ (U) \hat{\otimes} Λ (V)$ for the tensor Hall form, because tensor products of orthonormal families are orthonormal and each bidegree piece is finite dimensional; expanding $s_{λ} [U V]$ in that basis gives $s_{λ} [U V] = \sum_{μ, ν} g_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]$ . This is a finite identity in the independent power sums of $U, V$ , so applying the two specialization homomorphisms proves (b) for arbitrary $A, B$ .

5.1F1F7F9step 3.1step 4.1∎

Finiteness, empty case and choice. In (a), $c_{μ ν}^{λ} = 0$ unless $μ \subseteq λ$ and $∣ μ ∣ + ∣ ν ∣ = ∣ λ ∣$ , so only finitely many pairs contribute, and the identity is verified coefficient of $s_{λ} [Z]$ by coefficient of $s_{λ} [Z]$ . In (b), $g_{μ ν}^{λ} = 0$ unless $∣ μ ∣ = ∣ ν ∣ = ∣ λ ∣$ by [F9], so again only finitely many pairs contribute. Both identities hold for arbitrary specializations into arbitrary commutative $Q$ -algebras, and the proofs use only finite sums, finite bases and the substitution of step 1.1; no choice principle is used. The empty case is included: $s_{\emptyset} = 1$ and $c_{\emptyset, \emptyset}^{\emptyset} = g_{\emptyset, \emptyset}^{\emptyset} = 1$ , so both displays hold for $λ = \emptyset$ .

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The series $σ$ : product formula, Cauchy identity, adjunction and reproducing kernel

Statement

Let $σ [X] = \sum_{n \geq 0} h_{n} [X]$ be the series of Specializations of symmetric functions and the series $σ$ , and let $⟨, ⟩_{H}$ be the Hall form (The Hall inner product on symmetric functions).

$σ [X] = \prod_{x \in X} (1 - x)^{- 1}$ and $σ [- X] = \prod_{x \in X} (1 - x)$ for honest alphabets $X$ ; in particular $σ [t] = (1 - t)^{- 1}$ and $σ [- t] = 1 - t$ for the one-letter alphabet $t$ , as formal series in $t$ .
$σ [A + B] = σ [A] σ [B]$ for all specializations $A, B$ , and for the one-letter alphabet $t$ one has $σ [t A] = \sum_{n \geq 0} t^{n} h_{n} [A]$ .
Cauchy identity: $σ [X Y] = \sum_{λ} s_{λ} [X] s_{λ} [Y] = \sum_{λ} z_{λ}^{- 1} p_{λ} [X] p_{λ} [Y]$ for all specializations $X, Y$ , summed by degree.
For every specialization $A$ , the operator $σ [A X^{⊥}]$ is the adjoint of multiplication by $σ [A X]$ : $σ [A X^{⊥}] f [X] = f [X + A]$ for all $f \in Λ$ .
Reproducing kernel: $⟨ σ [A X] ∣ f [X] ⟩_{X} = f [A]$ for all $f \in Λ$ and all specializations $A$ .

Facts & Assumptions

Given: Specializations $A, B, U, X, Y$ into commutative $Q$ -algebras, a partition $λ$ , and integers $d, n \geq 0$ .

[F1]

Specializations are the algebra homomorphisms determined by arbitrary power-sum values, with $p_{k} [A + B] = p_{k} [A] + p_{k} [B]$ , $p_{k} [A B] = p_{k} [A] p_{k} [B]$ , $p_{k} [t A] = t^{k} p_{k} [A]$ , and $X^{⊥}$ the specialization $f [X^{⊥}] = f^{⊥}$ , where $f^{⊥}$ is the adjoint of multiplication by $f$ ; $σ [X] = \sum_{n} h_{n} [X]$ (Specializations of symmetric functions and the series $σ$ ).

[F2]

$h_{n} = \sum_{λ ⊢ n} p_{λ} / z_{λ}$ , and at every honest alphabet $\sum_{n} (\sum_{λ ⊢ n} (- 1)^{ℓ (λ)} p_{λ} / z_{λ}) t^{n} = \prod_{x} (1 - x t)$ ; also $h_{k} [t] = t^{k}$ and $e_{n} [t] = 0$ for $n \geq 2$ on the one-letter alphabet $t$ (Complete homogeneous functions expand in power sums with cycle-distribution coefficients, Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

[F3]

Newton's identities and the fundamental theorem of symmetric polynomials: $k e_{k} = \sum_{i = 1}^{k} (- 1)^{i - 1} e_{k - i} p_{i}$ over $Q$ , and substitution $T_{k} \mapsto e_{k}$ is an isomorphism from a polynomial ring onto the symmetric polynomials in $n$ variables (Newton's identities: $k e_{k} = \sum_{i = 1}^{k} (- 1)^{i - 1} e_{k - i} p_{i}$ , Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_{1}, \dots, e_{n}$ ).

[F4]

$Λ_{Q} = Q [p_{1}, p_{2}, \dots]$ with ${p_{λ} : λ ⊢ d}$ a basis of $Λ_{Q}^{d}$ ; the power sums are orthogonal, $⟨ p_{ρ}, p_{σ} ⟩_{H} = δ_{ρ σ} z_{ρ}$ , and the Schur functions are orthonormal, $⟨ s_{λ}, s_{μ} ⟩_{H} = δ_{λ μ}$ (Power sums form a rational but not integral stable basis, Power sums are orthogonal for the Hall form, Schur functions form an orthonormal integral basis, Stable Schur functions from bialternants).

[F5]

In the bidegree completion the Cauchy kernel satisfies $Ω (X, Y) = \sum_{λ} s_{λ} (X) s_{λ} (Y)$ and $Ω (X, Y) = \sum_{λ} z_{λ}^{- 1} p_{λ} (X) p_{λ} (Y)$ , by diagonal bidegree; for $N \geq d$ the rank- $N$ projection sends the basis $m_{λ}$ of $Λ^{d}$ to the finite orbit sums, an isomorphism onto the degree- $d$ symmetric polynomials (Bidegree completion of two symmetric-function rings, Power-sum, complete, and Schur expansions of the Cauchy kernel, The monomial symmetric functions form the integral stable basis, Monomial symmetric polynomials form an $R$ -basis of the symmetric-polynomial ring, The stable graded ring of symmetric functions).

[F6]

$s_{λ} [U + V] = \sum_{μ, ν} c_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]$ and $s_{λ} [U V] = \sum_{μ, ν} g_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]$ for all specializations, and $⟨ s_{λ}, s_{ρ} s_{μ} ⟩_{H} = c_{ρ μ}^{λ}$ (Schur functions on sums and products of alphabets, The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients).

[F7]

The adjoints of multiplication operators and of their degreewise sums satisfy $(f g)^{⊥} = f^{⊥} g^{⊥} = g^{⊥} f^{⊥}$ (Specializations of symmetric functions and the series $σ$ ).

Proof

technique · direct

1.1F3

Transfer principle. Let $d \geq 0$ . If an identity between polynomials in the values $p_{k} [X_{1}], \dots, p_{k} [X_{r}]$ , $k \leq d$ , holds for all honest alphabets $X_{1}, \dots, X_{r}$ of at least $d$ letters, then it holds for all specializations. Indeed, the substitution $A_{k}^{(i)} \mapsto p_{k} (x_{1}^{(i)}, \dots, x_{d}^{(i)})$ from $Q [A_{k}^{(i)}]$ to $Q [x^{(1)}, \dots, x^{(r)}]$ is injective: by [F3] over $Q$ one has $Q [e_{1}, \dots, e_{d}] = Q [p_{1}, \dots, p_{d}]$ inside the symmetric polynomials in $d$ variables, and the elementary symmetric polynomials are algebraically independent by the fundamental theorem; tensor products of injective substitutions over the field $Q$ are injective.

2.1F1F2F4step 1.1

Product formula and negation (1). For an honest alphabet $X$ , $\sum_{n} h_{n} [X] t^{n} = \prod_{x \in X} (1 - x t)^{- 1}$ by [F2], so the series $σ [X]$ equals $\prod_{x \in X} (1 - x)^{- 1}$ coefficientwise at $t = 1$ . Next, $h_{n} [- U] = (- 1)^{n} e_{n} [U]$ for every specialization $U$ : both sides are polynomials in the values $p_{k} [U]$ , $k \leq n$ , by [F4], and at honest alphabets the generating function of [F2] gives $\sum_{n} h_{n} [- x] t^{n} = \exp (- \sum_{k} p_{k} (x) t^{k} / k) = \prod_{x} (1 - x t) = \sum_{n} (- 1)^{n} e_{n} (x) t^{n}$ , so the identity follows from step 1.1 with $d = n$ . Hence $σ [- U] = \sum_{n} (- 1)^{n} e_{n} [U]$ , which at an honest alphabet is $\prod_{x \in X} (1 - x)$ ; at the one-letter alphabet $t$ , $h_{n} [- t] = 0$ for $n \geq 2$ and $h_{1} [- t] = - t$ , so $σ [- t] = 1 - t$ , while $h_{n} [t] = t^{n}$ gives $σ [t] = \sum_{n} t^{n} = (1 - t)^{- 1}$ as a formal series.

2.2F1F4step 1.1

Reduction to honest alphabets. Every value $f [U]$ of $f \in Λ_{Q}^{n}$ is the value at $(p_{1} [U], \dots, p_{n} [U])$ of a universal polynomial with rational coefficients, by the power-sum basis [F4]; the same holds for $h_{n} [U + V]$ , $h_{n} [U V]$ and $s_{λ} [U + V]$ . Hence by step 1.1 such identities may be verified on honest alphabets with at least $n$ letters, and that is how the remaining steps proceed.

3.1F1F2step 2.2

(2) Multiplicativity and the one-letter scaling. For honest alphabets $X, Y$ , $\sum_{n} h_{n} [X ⊔ Y] t^{n} = \prod_{ℓ \in X ⊔ Y} (1 - ℓ t)^{- 1} = (\sum_{i} h_{i} [X] t^{i}) (\sum_{j} h_{j} [Y] t^{j})$ by [F2], so $h_{n} [A + B] = \sum_{i + j = n} h_{i} [A] h_{j} [B]$ holds on honest alphabets and hence, by step 2.2 with $d = n$ , for all specializations; summing over $n$ gives $σ [A + B] = σ [A] σ [B]$ . Since $p_{k} [t A] = t^{k} p_{k} [A]$ and $h_{n}$ is homogeneous of weight $n$ in the power sums, $h_{n} [t A] = t^{n} h_{n} [A]$ , so $σ [t A] = \sum_{n} t^{n} h_{n} [A]$ .

3.2F2F5step 2.2

(3) Cauchy identity. For honest alphabets $X, Y$ and $d \geq 0$ , apply the degree- $d$ rank- $N$ projection of [F5] (with $N \geq d$ ) to the two Cauchy expansions of $Ω$ : it produces $h_{d} [X Y] = \sum_{λ ⊢ d} s_{λ} [X] s_{λ} [Y] = \sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [X] p_{ρ} [Y]$ , the outer equality by [F2]. By step 2.2 with $d$ , the identity $h_{d} [X Y] = \sum_{λ ⊢ d} s_{λ} [X] s_{λ} [Y]$ holds for all specializations, and summing over $d$ gives $σ [X Y] = \sum_{λ} s_{λ} [X] s_{λ} [Y]$ by degree.

4.1F1F4F6F7step 3.2

(4) Adjointness. By step 3.2, $σ [A X] = \sum_{λ} s_{λ} [A] s_{λ} [X]$ , so the adjoint of multiplication by $σ [A X]$ is $\sum_{λ} s_{λ} [A] s_{λ}^{⊥}$ degree by degree. On the other hand $p_{ρ} [X^{⊥}] = p_{ρ}^{⊥} = \prod_{i} (p_{ρ_{i}})^{⊥} = (p_{ρ})^{⊥}$ by [F1] and [F7], so $h_{n} [A X^{⊥}] = \sum_{ρ ⊢ n} z_{ρ}^{- 1} p_{ρ} [A] p_{ρ}^{⊥} = h_{n} [A X]^{⊥}$ and $σ [A X^{⊥}] = σ [A X]^{⊥}$ . To evaluate this adjoint, compute on the Schur basis: by [F6], $s_{λ} [X + A] = \sum_{ρ, ν} c_{ρ ν}^{λ} s_{ρ} [X] s_{ν} [A]$ , so $⟨ s_{λ} [X + A], s_{μ} [X] ⟩_{X} = \sum_{ν} c_{μ ν}^{λ} s_{ν} [A]$ ; and $⟨ σ [A X]^{⊥} s_{λ}, s_{μ} ⟩_{X} = ⟨ s_{λ}, σ [A X] s_{μ} ⟩_{X} = \sum_{ρ} s_{ρ} [A] ⟨ s_{λ}, s_{ρ} s_{μ} ⟩_{X} = \sum_{ρ} c_{ρ μ}^{λ} s_{ρ} [A]$ by [F6] and [F4]. The two results agree for every $μ$ , so by nondegeneracy of the Hall form $σ [A X^{⊥}] s_{λ} = s_{λ} [X + A]$ for every $λ$ , and by linearity for all $f$ .

5.1F4step 3.2∎

(5) Reproducing kernel. For every partition $λ$ , $⟨ σ [A X], s_{λ} [X] ⟩_{X} = \sum_{μ} s_{μ} [A] ⟨ s_{μ} [X], s_{λ} [X] ⟩_{X} = s_{λ} [A]$ by step 3.2 and orthonormality [F4]; by linearity in $f$ this gives $⟨ σ [A X] ∣ f [X] ⟩_{X} = f [A]$ for all $f \in Λ$ .

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Vertex-operator generating series for the padded Schur functions

Statement

Let $α$ be a partition and $t$ a formal variable. With the Jacobi-Trudi convention of The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ for the integer sequences $α [n] = (n - ∣ α ∣, α_{1}, α_{2}, \dots)$ , one has an identity of formal Laurent series with coefficients in the symmetric function ring (Specializations of symmetric functions and the series $σ$ ): $\sum_{n \in Z} s_{α [n]} [X] t^{n} = σ [t X] \cdot s_{α} [t X - 1] = t^{∣ α ∣} \cdot σ [t X] \cdot s_{α} [X - t^{- 1}],$ where $s_{α} [t X - 1]$ is evaluated through $p_{k} \mapsto t^{k} p_{k} [X] - 1$ and $s_{α} [X - t^{- 1}]$ through $p_{k} \mapsto p_{k} [X] - t^{- k}$ . Equivalently, $\sum_{n \in Z} s_{α [n]} [X] t^{n} = σ [t X] \sum_{μ : α / μ a vertical strip} (- 1)^{∣ α ∣ - ∣ μ ∣} t^{∣ μ ∣} s_{μ} [X] .$

Facts & Assumptions

Given: A partition $α$ of length $k$ ; formal variables $X$ and $t$ ; a partition $ν ⊢ m$ .

[F1]

$p_{k} [- 1] = - 1$ , $p_{k} [t X - 1] = t^{k} p_{k} [X] - 1$ , $p_{k} [X - t^{- 1}] = p_{k} [X] - t^{- k}$ , and $h_{n} [t X] = t^{n} h_{n} [X]$ by homogeneity of $h_{n}$ in the power sums (Specializations of symmetric functions and the series $σ$ , Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

[F2]

$σ [t X] = \sum_{n} t^{n} h_{n} [X]$ and the Cauchy identity $σ [X Y] = \sum_{λ} s_{λ} [X] s_{λ} [Y]$ hold for all specializations (The series $σ$ : product formula, Cauchy identity, adjunction and reproducing kernel).

[F3]

$s_{α} [A + B] = \sum_{μ, ν} c_{μ ν}^{α} s_{μ} [A] s_{ν} [B]$ and $s_{α} [A B] = \sum_{μ, ν} g_{μ ν}^{α} s_{μ} [A] s_{ν} [B]$ for all specializations; in particular the coproduct applies to $A = t X$ , $B = - 1$ (Schur functions on sums and products of alphabets).

[F4]

$c_{μ ν}^{λ} = ⟨ s_{λ}, s_{μ} s_{ν} ⟩_{H}$ and $c_{μ ν}^{λ} = c_{ν μ}^{λ}$ : the first by the LR product expansion and Schur orthonormality, the second because $s_{μ} s_{ν} = s_{ν} s_{μ}$ (The Littlewood–Richardson rule for products of Schur functions, Schur functions form an orthonormal integral basis); and $s_{λ / μ} = \sum_{ν} c_{μ ν}^{λ} s_{ν}$ (Skew Schur functions by Hall adjointness).

[F5]

Jacobi–Trudi: for every partition $λ$ and every $r \geq ℓ (λ)$ , $s_{λ} = \det (h_{λ_{i} - i + j})_{1 \leq i, j \leq r}$ with $h_{0} = 1$ , $h_{q} = 0$ for $q < 0$ ; and for $μ \subseteq λ$ , $s_{λ / μ} = \det (h_{λ_{i} - μ_{j} - i + j})$ , with $s_{λ / μ} = 0$ if $μ \subseteq̸ λ$ (Jacobi–Trudi and dual Jacobi–Trudi identities, Skew Jacobi–Trudi and tableau expansion).

[F7]

A Littlewood–Richardson tableau of shape $α / μ$ and content $ν$ is a semistandard skew tableau whose reading word (rows right to left, top to bottom) satisfies the lattice condition: in every prefix and for every $j$ , the number of $j$ 's is at least the number of $(j + 1)$ 's; $c_{μ ν}^{α}$ counts them (Littlewood--Richardson tableaux and coefficients).

[F8]

$α [n] = (n - ∣ α ∣, α_{1}, α_{2}, \dots)$ is a partition of $n$ exactly when $n \geq ∣ α ∣ + α_{1}$ , and $s_{α [n]}$ is the Jacobi–Trudi determinant of the integer sequence $α [n]$ , equal to $0$ for $n < 0$ (First-row padding of a partition, The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ ).

Proof

technique · direct

1.1F2F5

Negative one-letter evaluation. By [F2], $σ [- t] = 1 - t$ , so homogeneity gives $h_{0} [- 1] = 1$ , $h_{1} [- 1] = - 1$ and $h_{j} [- 1] = 0$ for $j \geq 2$ . If $ν_{1} \geq 2$ , every entry in the first row of its Jacobi–Trudi determinant [F5] has index at least two, so $s_{ν} [- 1] = 0$ . If $ν = (1^{m})$ , the matrix is lower triangular with diagonal $h_{1} [- 1] = - 1$ , hence $s_{ν} [- 1] = (- 1)^{m}$ . The empty determinant gives $s_{\emptyset} [- 1] = 1$ . Thus $s_{ν} [- 1] = (- 1)^{m} δ_{ν, (1^{m})}$ .

1.2F4

Consequences for skew functions. For all partitions $λ, μ, ν$ one has $c_{μ ν}^{λ} = c_{ν μ}^{λ}$ by [F4], so the skew expansion [F4] gives $s_{α / (1^{i})} = \sum_{μ} c_{(1^{i}), μ}^{α} s_{μ} = \sum_{μ} c_{μ, (1^{i})}^{α} s_{μ}$ for every $i \geq 0$ , where $(1^{0}) = \emptyset$ and $s_{α / \emptyset} = s_{α}$ .

2.1F1F3step 1.1step 1.2

Coproduct at $t X - 1$ . Apply [F3] with $A = t X$ , $B = - 1$ , and expand over $ν$ using step 1.1: $s_{α} [t X - 1] = \sum_{μ, ν} c_{μ ν}^{α} s_{μ} [t X] s_{ν} [- 1] = \sum_{i \geq 0} (- 1)^{i} (\sum_{μ} c_{μ, (1^{i})}^{α} s_{μ} [t X]) = \sum_{i = 0}^{k} (- 1)^{i} s_{α / (1^{i})} [t X]$ by step 1.2, using that $s_{α / (1^{i})} = 0$ for $i > ℓ (α)$ and $s_{ν} [- 1] = 0$ for $ν$ not a column. Since $s_{α / (1^{i})}$ is homogeneous of degree $∣ α ∣ - i$ and $p_{j} [t X] = t^{j} p_{j} [X]$ , homogeneity in the power sums gives $s_{α / (1^{i})} [t X] = t^{∣ α ∣ - i} s_{α / (1^{i})} [X]$ , so $s_{α} [t X - 1] = \sum_{i = 0}^{k} (- 1)^{i} t^{∣ α ∣ - i} s_{α / (1^{i})} [X]$ .

3.1F2F5F8step 2.1

Coefficient of $t^{n}$ and the Jacobi–Trudi expansion. By [F2], $σ [t X] = \sum_{m} t^{m} h_{m} [X]$ , so the coefficient of $t^{n}$ in $σ [t X] s_{α} [t X - 1]$ is $\sum_{i = 0}^{k} (- 1)^{i} h_{n - ∣ α ∣ + i} [X] s_{α / (1^{i})} [X]$ , where $h_{q} = 0$ for $q < 0$ by [F5]. Expand the Jacobi–Trudi determinant $s_{α [n]} = \det (h_{α [n]_{r} - r + j})_{1 \leq r, j \leq k + 1}$ of [F8] and [F5] along its first row: the entry in column $j = t + 1$ is $h_{α [n]_{1} - 1 + t + 1} = h_{n - ∣ α ∣ + t}$ , its cofactor sign is $(- 1)^{1 + (t + 1)} = (- 1)^{t}$ , and deleting row $1$ and column $t + 1$ leaves the matrix with entries $h_{α_{r} - r + c}$ over column offsets $c \in {0, 1, \dots, k} ∖ {t}$ , which is exactly the skew Jacobi–Trudi determinant $\det (h_{α_{r} - 1_{j \leq t} - r + j}) = s_{α / (1^{t})}$ of [F5], the first $t$ columns being shifted by one. Hence $s_{α [n]} [X] = \sum_{t = 0}^{k} (- 1)^{t} h_{n - ∣ α ∣ + t} [X] s_{α / (1^{t})} [X]$ for every $n \in Z$ , which is the coefficient of $t^{n}$ computed above; for $n < 0$ all indices $n - ∣ α ∣ + t$ are negative, both sides vanish, and for $t > ℓ (α)$ the skew functions vanish.

4.1F2F7step 1.2step 2.1step 3.1

Vertical strips and the second form. By the definition of the Littlewood–Richardson coefficients [F7], $c_{μ, (1^{i})}^{α}$ counts semistandard skew tableaux of shape $α / μ$ with content $(1^{i})$ whose reading word is lattice. A word using each of $1, \dots, i$ exactly once is lattice if and only if it is the increasing word $1, 2, \dots, i$ : the lattice condition at the position of the letter $j + 1$ forces $j$ to occur earlier, for every $j < i$ . Hence such a tableau exists precisely when each row of $α / μ$ has at most one box (a vertical strip), the filling being then forced row by row top to bottom, with the empty skew shape giving the case $i = 0$ ; so $s_{α / (1^{i})} = \sum_{μ : α / μ vertical strip, ∣ α ∣ - ∣ μ ∣ = i} s_{μ}$ . Substituting into step 2.1 and collecting by $∣ μ ∣ = ∣ α ∣ - i$ gives $s_{α} [t X - 1] = \sum_{μ : α / μ vertical strip} (- 1)^{∣ α ∣ - ∣ μ ∣} t^{∣ μ ∣} s_{μ} [X]$ , which combined with [F2] and step 3.1 is the second displayed formula.

5.1F1F5step 2.1step 3.1∎

Second evaluation of the shift. By [F1], $p_{j} [t X - 1] = t^{j} p_{j} [X] - 1 = t^{j} (p_{j} [X] - t^{- j}) = p_{j} [t (X - t^{- 1})]$ , so $s_{α} [t X - 1] = s_{α} [t (X - t^{- 1})] = t^{∣ α ∣} s_{α} [X - t^{- 1}]$ by homogeneity; this gives the middle equality of the first display. All coefficient sums are finite in each degree: for fixed $n$ only $t = 0, 1, \dots, k$ contribute, with $h_{n - ∣ α ∣ + t} = 0$ whenever $n - ∣ α ∣ + t < 0$ , and each $s_{α / (1^{t})}$ is a finite $Z$ -combination of Schur functions. No convergence of a scalar series is asserted, no value $σ [1]$ is evaluated, and no choice principle is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

The Jacobi–Trudi completion is compatible with the internal product

Statement

For partitions $α, β$ put $a = ∣ α ∣$ , $b = ∣ β ∣$ and define $R_{α, β} [Z] : = ⟨ s_{α} [A] s_{β} [B] ∣ σ [Z A B + Z A + Z B + A B] ⟩_{A, B},$ the Hall pairing in the alphabet variables $A, B$ (The Hall inner product on symmetric functions, Specializations of symmetric functions and the series $σ$ ).

$R_{α, β}$ is a finite symmetric-function polynomial in $Z$ of degree at most $a + b$ .
For every integer $n \geq 0$ , $(U_{n} s_{α}) * (U_{n} s_{β}) = U_{n} R_{α, β},$ where $U_{n}$ is the Jacobi–Trudi completion and $*$ the internal product (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , The internal (Kronecker) product on symmetric functions); this holds for every $n \geq 0$ , including those $n$ for which the padded sequences $α [n]$ , $β [n]$ are not partitions. Equivalently, with $t$ a one-letter formal alphabet, $\sum_{n \geq 0} (U_{n} s_{α}) * (U_{n} s_{β}) t^{n} = σ [t X] R_{α, β} [t X - 1] .$
The same polynomial is the finite Littlewood expression $R_{α, β} = \sum_{δ, ϵ, τ} ((s_{δ} s_{τ})^{⊥} s_{α}) ((s_{ϵ} s_{τ})^{⊥} s_{β}) (s_{δ} * s_{ϵ}),$ the sum over all triples of partitions, all factors evaluated at $Z$ .

Facts & Assumptions

Given: Partitions $α, β$ with $a = ∣ α ∣$ , $b = ∣ β ∣$ ; specializations $A, B, X, Z$ ; a formal variable $t$ .

[F1]

$U_{n}$ is the $Z$ -linear endomorphism with $U_{n} (s_{λ}) = s_{λ [n]}$ on the Schur basis, where $s_{λ [n]}$ is the Jacobi–Trudi determinant of the integer sequence $λ [n]$ ; $s_{λ [n]} \in Λ^{n}$ for $n \geq 0$ and $s_{λ [n]} = 0$ for $n < 0$ (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ ).

[F2]

$σ [t A] = \sum_{n} t^{n} h_{n} [A]$ , the Cauchy identity $σ [X Y] = \sum_{λ} s_{λ} [X] s_{λ} [Y] = \sum_{λ} z_{λ}^{- 1} p_{λ} [X] p_{λ} [Y]$ , the adjunction $σ [A X^{⊥}] f [X] = f [X + A]$ , and the reproducing kernel $⟨ σ [A X] ∣ f [X] ⟩_{X} = f [A]$ hold for all specializations (The series $σ$ : product formula, Cauchy identity, adjunction and reproducing kernel).

[F3]

$\sum_{n \in Z} s_{α [n]} [A] t^{n} = σ [t A] s_{α} [t A - 1]$ for every partition $α$ ; the identity is linear in the Schur index, so for every $f \in Λ$ with finite degree: the degree- $n$ component of $σ [A] f [A - 1]$ is $U_{n} f$ , and $[t^{n}] (σ [t X] f [t X - 1]) = U_{n} f$ (Vertex-operator generating series for the padded Schur functions).

[F4]

$s_{α} [U + V] = \sum_{μ, ν} c_{μ ν}^{α} s_{μ} [U] s_{ν} [V]$ and $s_{α} [U V] = \sum_{μ, ν} g_{μ ν}^{α} s_{μ} [U] s_{ν} [V]$ for all specializations; $c_{δ, σ, τ}^{α} = ⟨ s_{α}, s_{δ} s_{σ} s_{τ} ⟩_{H} = [s_{α}] s_{δ} s_{σ} s_{τ}$ and $⟨ (s_{δ} s_{τ})^{⊥} s_{α}, s_{σ} ⟩_{H} = c_{δ, σ, τ}^{α}$ (Schur functions on sums and products of alphabets, Threefold Schur coefficients $c_{a, b, c}^{δ}$ , The Littlewood–Richardson rule for products of Schur functions).

[F5]

For $f = \sum_{ρ} a_{ρ} p_{ρ}$ and $g = \sum_{ρ} b_{ρ} p_{ρ}$ of equal degree one has $⟨ f, p_{ρ} ⟩_{H} = a_{ρ} z_{ρ}$ , $p_{ρ} * p_{σ} = δ_{ρ σ} z_{ρ} p_{ρ}$ and $f * g = \sum_{ρ} a_{ρ} b_{ρ} z_{ρ} p_{ρ}$ (Power sums diagonalize the internal product, Power sums are orthogonal for the Hall form, The internal (Kronecker) product on symmetric functions).

[F6]

$s_{μ} * s_{ν} = \sum_{λ} g_{μ ν}^{λ} s_{λ}$ and $⟨ s_{λ}, s_{μ} ⟩_{H} = δ_{λ μ}$ (Kronecker coefficients and the internal tensor product, Schur functions form an orthonormal integral basis).

[F7]

All elements are $Q$ -linear combinations of the power-sum basis of $Λ_{Q}$ (Power sums form a rational but not integral stable basis).

Proof

technique · generating functions and Hall adjunction

1.1F2F5F7

Internal-product kernel. For $f, g \in Λ_{Q}^{d}$ and independent formal alphabet variables $U, V$ , with any specialization $W$ , one has $⟨ f [U] g [V] ∣ σ [U V W] ⟩_{U, V} = (f * g) [W]$ . Indeed, by the power form of the Cauchy identity [F2], $σ [U V W] = \sum_{ρ ⊢ d} z_{ρ}^{- 1} p_{ρ} [U] p_{ρ} [V] p_{ρ} [W]$ in the relevant degree and the pairing is a finite sum, since only $ρ ⊢ d$ matches the bidegree $(d, d)$ of $f [U] g [V]$ . Writing $f = \sum_{ρ} a_{ρ} p_{ρ}$ and $g = \sum_{ρ} b_{ρ} p_{ρ}$ , [F5] gives $⟨ f [U] g [V], p_{ρ} [U] p_{ρ} [V] p_{ρ} [W] ⟩_{U, V} = a_{ρ} z_{ρ} \cdot b_{ρ} z_{ρ} \cdot p_{ρ} [W]$ , so the left side is $\sum_{ρ ⊢ d} a_{ρ} b_{ρ} z_{ρ} p_{ρ} [W] = (f * g) [W]$ by [F5].

1.2F2

Finiteness of $R$ . Factor the kernel of $R$ as $σ [Z A B] σ [Z A] σ [Z B] σ [A B]$ . In the $(a, b)$ -bidegree of the pairing, the four factors contribute bidegrees $(j, j)$ , $(m, 0)$ , $(0, n)$ , $(l, l)$ with $j + m + l = a$ and $j + n + l = b$ , so $0 \leq j, m, n, l \leq a + b$ and the $Z$ -degree is $j + m + n = b + m - l \leq a + b$ . For each fixed $Z$ -degree only finitely many such $(j, m, n, l)$ occur, and each homogeneous component of $σ$ is a finite sum over partitions; hence $R_{α, β}$ is a finite polynomial in $Z$ of degree at most $a + b$ , with coefficients extracted from finitely many bidegree pieces of $σ$ . No infinite sum of symmetric functions is evaluated.

2.1F1F2F3step 1.1

The generating function $F (t) = \sum_{n \geq 0} (U_{n} s_{α}) * (U_{n} s_{β}) t^{n}$ . By [F1] and [F3], the degree- $n$ part of $σ [A] s_{α} [A - 1]$ is $U_{n} s_{α}$ , and likewise for $β$ ; the bidegree- $(n, n)$ part of the product $σ [A] s_{α} [A - 1] σ [B] s_{β} [B - 1]$ is therefore $(U_{n} s_{α}) [A] (U_{n} s_{β}) [B]$ . Pairing against $σ [t X A B] = \sum_{n} t^{n} h_{n} [X A B]$ and applying step 1.1 in each degree gives $F (t) = ⟨ σ [A] s_{α} [A - 1] σ [B] s_{β} [B - 1] ∣ σ [t X A B] ⟩_{A, B}$ , a formula in which the coefficient of $t^{n}$ is finite by [F1].

3.1F2step 2.1

Hall adjunction. For a polynomial $u$ and a formal series $W [A]$ , interpreted coefficientwise in $t$ , the adjunction of [F2] with the one-letter alphabet $1$ reads $σ [A^{⊥}] u [A] = u [A + 1]$ ; equivalently $⟨ σ [A] u [A], W [A] ⟩_{A} = ⟨ u [A], W [A + 1] ⟩_{A}$ and, in the reverse direction, $⟨ u [A - 1], W [A] ⟩_{A} = ⟨ u [A], σ [- A] W [A] ⟩_{A}$ . Applying this in the $A$ -variable and then in the $B$ -variable to the identity of step 2.1 turns $σ [A] σ [B]$ into the translation $A \mapsto A + 1$ , $B \mapsto B + 1$ of the kernel and moves the shifts $- 1$ of $s_{α}, s_{β}$ onto the kernel: $F (t) = ⟨ s_{α} [A] s_{β} [B] ∣ σ [t X (A + 1) (B + 1) - A - B] ⟩_{A, B}$ . Each coefficient of $t^{n}$ in the kernel before translation is polynomial in $A, B$ of bidegree $(n, n)$ , so both adjunctions are finite coefficientwise.

4.1F1F2F3step 3.1step 1.2

Factorization and the coefficient identity. Expanding $t X (A + 1) (B + 1) - A - B = t X + (t X - 1) A B + (t X - 1) A + (t X - 1) B + A B$ and using $σ [C + D] = σ [C] σ [D]$ from the product formula [F2], the kernel of step 3.1 is $σ [t X]$ times $σ [(t X - 1) A B + (t X - 1) A + (t X - 1) B + A B]$ , so $F (t) = σ [t X] R_{α, β} [t X - 1]$ . By [F3], the coefficient of $t^{n}$ in $σ [t X] h [t X - 1]$ is $U_{n} h$ for every finite-degree $h$ , and $R_{α, β} [t X - 1]$ is the specialization of the element $R_{α, β} \in Λ_{Z}$ introduced in the statement, so comparing the coefficient of $t^{n}$ gives $(U_{n} s_{α}) * (U_{n} s_{β}) = U_{n} R_{α, β}$ for every $n \geq 0$ ; for $n < 0$ one has $U_{n} = 0$ by [F1], so no negative index is asserted.

5.1F4F5F6step 4.1∎

The Littlewood expression. Expand the four factors of the kernel of $R$ with the Cauchy identity and the product-alphabet identity [F2] and [F4]: $σ [Z A B] = \sum_{ζ, δ, ϵ} s_{ζ} [Z] g_{δ ϵ}^{ζ} s_{δ} [A] s_{ϵ} [B]$ , $σ [Z A] = \sum_{σ} s_{σ} [Z] s_{σ} [A]$ , $σ [Z B] = \sum_{ρ} s_{ρ} [Z] s_{ρ} [B]$ and $σ [A B] = \sum_{τ} s_{τ} [A] s_{τ} [B]$ . Pairing with $s_{α} [A] s_{β} [B]$ and using the orthonormality of the Schur basis gives $R_{α, β} = \sum g_{δ ϵ}^{ζ} s_{ζ} [Z] s_{σ} [Z] s_{ρ} [Z] ⟨ s_{δ} s_{σ} s_{τ}, s_{α} ⟩_{H} ⟨ s_{ϵ} s_{ρ} s_{τ}, s_{β} ⟩_{H}$ . By [F4] the two pairings are $c_{δ, σ, τ}^{α}$ and $c_{ϵ, ρ, τ}^{β}$ , and re-summing over $σ$ with $ρ$ fixed gives $\sum_{σ} c_{δ, σ, τ}^{α} s_{σ} [Z] = (s_{δ} s_{τ})^{⊥} s_{α}$ by adjointness; similarly for $β$ . Finally $\sum_{ζ} g_{δ ϵ}^{ζ} s_{ζ} [Z] = s_{δ} * s_{ϵ} [Z]$ by [F6]. This is exactly the displayed Littlewood expression, and the sum is finite because $c$ - and $g$ -coefficients vanish outside the size constraints.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Murnaghan stability for Kronecker coefficients

Statement

Fix partitions $α, β, γ$ and work over a field $k$ of characteristic $0$ , with $V_{λ}$ denoting the irreducible $k S_{n}$ -module of shape $λ ⊢ n$ (Specht modules classify the irreducibles over every field of characteristic zero). Put $m (γ) : = \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}, ∣ γ ∣ + γ_{1}}$ and, for $n \geq m (γ)$ , let $g_{n} (α, β, γ) : = g_{α [n], β [n]}^{γ [n]}$ be the multiplicity of $V_{γ [n]}$ in $V_{α [n]} \otimes V_{β [n]}$ (Kronecker coefficients and the internal tensor product, Padded Kronecker coefficient sequences); set $g_{n} (α, β, γ) : = 0$ for $n < m (γ)$ , where the padded sequences are not partitions.

Then the sequence $n \mapsto g_{n} (α, β, γ)$ is eventually constant. More precisely, there is a finite set $S (α, β)$ of partitions and nonnegative integers ${\overset{ˉ}{g}}_{α, β}^{γ}$ for $γ \in S (α, β)$ , with ${\overset{ˉ}{g}}_{α, β}^{γ} : = 0$ for $γ \notin S (α, β)$ , such that $g_{n} (α, β, γ) = {\overset{ˉ}{g}}_{α, β}^{γ}$ for all sufficiently large $n$ and every partition $γ$ . No explicit bound on the stable rank is asserted here; the sharp global bound is a separate claim.

Facts & Assumptions

Given: Partitions $α, β, γ$ , a field $k$ of characteristic $0$ , and the FI-module machinery over $k$ .

[F1]

There is a finitely generated FI-module $V (α)$ over $k$ with $V (α)_{n} ≅ V_{α [n]}$ for $n \geq ∣ α ∣ + α_{1}$ and $V (α)_{n} = 0$ below; it has weight $∣ α ∣$ and stability degree $α_{1}$ (Each padded irreducible family is realised by a finitely generated FI-module).

[F2]

The pointwise tensor product of finitely generated FI-modules is finitely generated, with $(V \otimes W)_{S} = V_{S} \otimes_{k} W_{S}$ ; if $V$ is generated in degree $\leq m_{1}$ and $W$ in degree $\leq m_{2}$ , then $V \otimes W$ is generated in degree $\leq m_{1} + m_{2}$ (Tensor products of finitely generated FI-modules are finitely generated).

[F3]

A finitely generated FI-module over a characteristic-zero field with finite-dimensional values is uniformly representation stable, so the multiplicity of each admissible padded irreducible family is constant in a common stable range (Finite generation is equivalent to uniform representation stability, Uniform representation stability for consistent sequences of $S_{n}$ -modules).

[F4]

For partitions $λ \neq λ^{'}$ of possibly different sizes and $n \geq \max (∣ λ ∣ + λ_{1}, ∣ λ^{'} ∣ + λ_{1}^{'})$ , the irreducible modules $V_{λ [n]}$ and $V_{λ^{'} [n]}$ of $k S_{n}$ are nonisomorphic; and $V_{μ} \otimes V_{ν} ≅ ⨁_{γ} g_{μ, ν}^{γ} V_{γ}$ with $g_{μ, ν}^{γ}$ the Kronecker coefficient (Kronecker coefficients and the internal tensor product, First-row padding of a partition, Specht modules classify the irreducibles over every field of characteristic zero).

[F5]

Generation in a finite degree $d$ bounds the weight by $d$ : only tails $λ$ with $∣ λ ∣ \leq d$ can occur (Bounded generation degree bounds surjectivity degree and weight, Weight of an FI-module over a field of characteristic zero).

Proof

technique · direct

1.1F1

By [F1] the assignment $n \mapsto V_{α [n]}$ (and $0$ below the threshold) is a finitely generated FI-module $V (α)$ over $k$ with finite-dimensional values, and likewise $V (β)$ .

2.1F2F4step 1.1

By [F2] the pointwise tensor product $T : = V (α) \otimes V (β)$ is finitely generated and has finite-dimensional values; at every rank where both paddings are admissible its value is $V_{α [n]} \otimes_{k} V_{β [n]}$ . Its tensor multiplicities are the usual complex Kronecker coefficients: by the rational-model construction in Specht modules classify the irreducibles over every field of characteristic zero, $V_{μ} = k \otimes_{Q} S_{Q}^{μ}$ . The map $(c \otimes u) \otimes (e \otimes v) \mapsto c e \otimes (u \otimes v)$ identifies its tensor product with the scalar extension of the rational tensor product. Extend a rational Specht decomposition to $k$ and to $C$ ; tensoring carries a finite direct sum to the same sum of extended Specht modules (by the displayed map on bases), so the two fields have identical multiplicities.

3.1F3F5step 2.1

Since $T$ is finitely generated, it is generated in some finite degree $d$ . By [F5] only the finitely many partitions $λ$ with $∣ λ ∣ \leq d$ can label constituents. By [F3] their multiplicities are constant from some rank $N$ onward. Increase $N$ to make all these finitely many labels admissible. Then $T_{n} ≅ ⨁_{λ \in S} c_{λ} V_{λ [n]}$ for $n \geq N$ , where $S$ is the finite set of labels with positive multiplicity; put ${\overset{ˉ}{g}}_{α, β}^{γ} : = c_{γ}$ on $S$ and zero outside $S$ .

4.1F4step 3.1

Put $N_{0} : = \max {N, ∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}, \max ({0} \cup {∣ λ ∣ + λ_{1} : λ \in S})}$ , finite because $S$ is finite. For $n \geq N_{0}$ the equality of step 3.1 holds with every $V_{γ [n]}$ , $γ \in S$ , an honest irreducible of $k S_{n}$ ; and the rank- $n$ value also decomposes by [F4] as $⨁_{γ} g_{α [n], β [n]}^{γ [n]} V_{γ [n]}$ , the sum over admissible tails $γ$ with $n \geq ∣ γ ∣ + γ_{1}$ , which label all partitions of $n$ uniquely by deleting their first row. By the nonisomorphism clause of [F4], the two decompositions have the same labels and multiplicities: $g_{n} (α, β, γ) = {\overset{ˉ}{g}}_{α, β}^{γ}$ for every partition $γ$ and every $n \geq N_{0}$ , where for $γ \notin S$ this says $g_{n} (α, β, γ) = 0$ .

5.1givenstep 4.1

The case $γ \notin S$ and $n < m (γ)$ also gives $g_{n} (α, β, γ) = 0$ by the convention in the statement, so making $N_{0}$ larger if necessary the equality $g_{n} (α, β, γ) = {\overset{ˉ}{g}}_{α, β}^{γ}$ holds for all $n \geq N_{0}$ and all partitions $γ$ ; this is eventual constancy with the finite support $S (α, β) : = S$ . The empty partition is included: $V_{\emptyset [n]}$ is the trivial module, the thresholds $∣ \emptyset ∣ + \emptyset_{1} = 0$ cause no special case, and the argument did not use any bound on $N_{0}$ sharper than existence.

6.1F1F2F3step 4.1∎

No choice principle is used: the FI-module suppliers are $Z$ -linear constructions on finite-dimensional representation categories, the finite family $S$ and its multiplicities come from [F3], and the comparison of irreducibles in step 4.1 is a finite-dimensional decomposition statement over $k$ ; the statement is asserted for every field of characteristic $0$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$

Definition

For partitions $α, β, γ$ define the reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ to be the eventual value of the padded sequence $n \mapsto g_{n} (α, β, γ)$ furnished by Murnaghan stability for Kronecker coefficients; equivalently ${\overset{ˉ}{g}}_{α, β}^{γ} : = \lim_{n \to \infty} g_{α [n], β [n]}^{γ [n]},$ where the sequence is defined for $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}, ∣ γ ∣ + γ_{1}}$ and set to $0$ below by the convention in Murnaghan stability for Kronecker coefficients (the size-indexed coefficients of Padded Kronecker coefficient sequences themselves are defined only on that common domain; see also Kronecker coefficients and the internal tensor product).

Well-definedness. The sequence is eventually constant by the Murnaghan stability theorem, and the eventual value of an eventually constant sequence of nonnegative integers is a unique nonnegative integer; no choice is involved. Two immediate conventions: (i) ${\overset{ˉ}{g}}_{α, β}^{γ} = {\overset{ˉ}{g}}_{β, α}^{γ}$ for all $α, β, γ$ , because $V_{μ} \otimes V_{ν} ≅ V_{ν} \otimes V_{μ}$ and the Kronecker coefficients are symmetric in their two lower indices; (ii) for fixed $α, β$ only finitely many $γ$ have ${\overset{ˉ}{g}}_{α, β}^{γ} \neq 0$ , because the finite set $S (α, β)$ of the stability theorem carries the whole support. The values used later and verified in the source's introduction for $α = β = (2)$ are ${\overset{ˉ}{g}}_{(2), (2)}^{(2)} = 2$ and ${\overset{ˉ}{g}}_{(2), (2)}^{(4)} = 1$ ; they are read off from the displayed stable row and are not used as logical inputs anywhere on this page.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Brion's formula for the reduced Kronecker coefficients

Statement

For all partitions $α, β, γ$ , ${\overset{ˉ}{g}}_{α, β}^{γ} = ⟨ s_{α} [X] s_{β} [Y] ∣ σ [X Y] s_{γ} [X Y + X + Y] ⟩_{X, Y},$ the inner product being the Hall inner product extended to symmetric functions in the two alphabet variables $X, Y$ , and ${\overset{ˉ}{g}}_{α, β}^{γ}$ the reduced Kronecker coefficient (The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ ).

Equivalently, introducing a third alphabet $Z$ , $\sum_{α, β, γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{α} [X] s_{β} [Y] s_{γ} [Z] = σ [X Y Z + X Y + X Z + Y Z],$ the symmetric Cauchy-type identity for the reduced coefficients, summed over all triples of partitions and by total degree. Equivalently again, in the notation of The Jacobi–Trudi completion is compatible with the internal product, $R_{α, β} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ},$ the finite Schur expansion of the polynomial $R_{α, β}$ .

Facts & Assumptions

Given: Partitions $α, β, γ$ , alphabet variables $X, Y, Z, A, B$ and integers $a = ∣ α ∣$ , $b = ∣ β ∣$ .

[F1]

The element $R_{α, β} [Z] = ⟨ s_{α} [A] s_{β} [B] ∣ σ [Z A B + Z A + Z B + A B] ⟩_{A, B}$ is a finite polynomial in $Z$ of degree at most $a + b$ , and the identity of graded series $\sum_{n \geq 0} (U_{n} s_{α}) * (U_{n} s_{β}) t^{n} = σ [t X] R_{α, β} [t X - 1]$ holds (The Jacobi–Trudi completion is compatible with the internal product).

[F2]

For every $γ$ in the finite support and all sufficiently large $n$ , $g_{α [n], β [n]}^{γ [n]} = {\overset{ˉ}{g}}_{α, β}^{γ}$ ; for fixed $α, β$ only finitely many ${\overset{ˉ}{g}}_{α, β}^{γ}$ are nonzero (Murnaghan stability for Kronecker coefficients, The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ , Padded Kronecker coefficient sequences).

[F3]

The Cauchy identity $σ [U V] = \sum_{λ} s_{λ} [U] s_{λ} [V]$ and the reproducing kernel $⟨ σ [A X] ∣ f [X] ⟩_{X} = f [A]$ hold for all specializations, summed by degree (The series $σ$ : product formula, Cauchy identity, adjunction and reproducing kernel).

[F4]

$U_{n} s_{λ} = s_{λ [n]}$ ; for $n \geq ∣ λ ∣ + λ_{1}$ these are honest Schur functions, and increasing $n$ by $k \geq 0$ increases their first part by $k$ , so $s_{λ [n + k]} = V^{k} (s_{λ [n]})$ . Distinct admissible tails give distinct padded partitions, since deleting the first row recovers the tail (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , Padded Kronecker coefficient sequences).

[F5]

$s_{μ} * s_{ν} = \sum_{δ} g_{μ, ν}^{δ} s_{δ}$ for partitions of a common size, and $g_{μ, ν}^{δ} = 0$ unless $δ$ has that size (Kronecker coefficients and the internal tensor product).

Proof

technique · comparison of generating functions at a large rank

1.1F1

The polynomial $R$ and its Schur coefficients. By [F1], $R_{α, β}$ is a finite element of $Λ_{Z}$ of degree at most $a + b$ , so it has a unique finite Schur expansion $R_{α, β} = \sum_{γ} r_{γ} s_{γ}$ with $r_{γ} = ⟨ R_{α, β}, s_{γ} ⟩_{Z}$ .

2.1F1F2F4F5step 1.1

Identification of the coefficients. Choose $n$ large enough that (i) $n \geq ∣ α ∣ + α_{1}$ and $n \geq ∣ β ∣ + β_{1}$ , (ii) $n \geq ∣ λ ∣ + λ_{1}$ for every $λ$ occurring in the finite support of $R_{α, β}$ and for every $γ$ in the finite support $S (α, β)$ of the reduced coefficients, and (iii) $n$ is at least the Murnaghan stability threshold of [F2]; this is possible by [F1] and [F2]. By [F1], [F4] and the definition of $U_{n}$ , $(U_{n} s_{α}) * (U_{n} s_{β}) = σ [t X] R [t X - 1]$ -coefficient at $t^{n}$ , which is $\sum_{γ} r_{γ} s_{γ [n]}$ , a sum over honest Schur functions by (ii) with pairwise distinct labels; by [F2] and [F5], the same element is $s_{α [n]} * s_{β [n]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]}$ , again over honest Schur functions, with the same set of labels by (ii). Comparing the two expansions in the Schur basis of $Λ^{n}$ gives $r_{γ} = {\overset{ˉ}{g}}_{α, β}^{γ}$ for every $γ$ , treating $γ$ outside both finite supports as zero on each side.

3.1F1F3step 2.1

First form. Factor the kernel of $R_{α, β}$ as $σ [A B] σ [Z (A B + A + B)]$ and expand the second factor with the Cauchy identity [F3]: $σ [Z (A B + A + B)] = \sum_{γ} s_{γ} [Z] s_{γ} [A B + A + B]$ . Reading off the coefficient of $s_{γ} [Z]$ in $R_{α, β}$ gives $r_{γ} = ⟨ s_{α} [A] s_{β} [B] ∣ σ [A B] s_{γ} [A B + A + B] ⟩_{A, B}$ ; combining with step 2.1 yields the displayed Brion formula after renaming the pairing variables $A, B$ to $X, Y$ .

4.1F3step 2.1step 3.1

Three-alphabet form. Sum the first form against $s_{α} [X] s_{β} [Y] s_{γ} [Z]$ and use the reproducing kernel [F3] twice: for fixed $γ$ , $\sum_{α, β} s_{α} [X] s_{β} [Y] ⟨ s_{α} [A] s_{β} [B], σ [A B] s_{γ} [A B + A + B] ⟩_{A, B} = ⟨ σ [X A] σ [Y B], σ [A B] s_{γ} [A B + A + B] ⟩_{A, B} = σ [X Y] s_{γ} [X Y + X + Y]$ , where the last equality applies $⟨ σ [A X] ∣ h [X] ⟩_{X} = h [A]$ in each variable after truncating to each fixed output bidegree; the pairings then involve only finitely many homogeneous polynomial components. Hence $\sum_{α, β, γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{α} [X] s_{β} [Y] s_{γ} [Z] = σ [X Y] \sum_{γ} s_{γ} [Z] s_{γ} [X Y + X + Y] = σ [X Y] σ [Z (X Y + X + Y)]$ by [F3], and $X Y + Z (X Y + X + Y) = X Y Z + X Y + X Z + Y Z$ .

5.1F1F2step 2.1step 4.1∎

Finiteness and choice. In step 2.1 the set of labels is finite by [F1] and [F2]; in step 4.1 each total degree receives finitely many triples with nonzero coefficient, because $\overset{ˉ}{g}$ vanishes outside the finite support and the Schur expansions are homogeneous. The argument uses no choice principle: the comparably large rank $n$ is a single existential instantiation, and the expansions are finite.

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The nonnegative expansion of a reduced Kronecker coefficient

Statement

For all partitions $α, β, γ$ one has the finite nonnegative expansion ${\overset{ˉ}{g}}_{α, β}^{γ} = \sum_{μ, λ, η, ρ, θ, π} c_{μ, θ, π}^{γ} c_{ρ, λ, θ}^{α} c_{ρ, η, π}^{β} g_{λ, η}^{μ},$ the sum over all six-tuples of partitions (only finitely many summands are nonzero), where the $c$ 's are the threefold Schur coefficients (Threefold Schur coefficients $c_{a, b, c}^{δ}$ ) and the $g$ 's are the ordinary Kronecker coefficients (Kronecker coefficients and the internal tensor product).

Consequences.

${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ if and only if there exist partitions $μ, λ, η, ρ, θ, π$ with all four coefficients $c_{μ, θ, π}^{γ}$ , $c_{ρ, λ, θ}^{α}$ , $c_{ρ, η, π}^{β}$ and $g_{λ, η}^{μ}$ positive.
For every nonzero summand the degree equations $∣ α ∣ = ∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣$ , $∣ β ∣ = ∣ ρ ∣ + ∣ η ∣ + ∣ π ∣$ , $∣ γ ∣ = ∣ μ ∣ + ∣ θ ∣ + ∣ π ∣$ and $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ hold.
For fixed $α, β$ the index set is finite: for a nonzero summand $ρ, λ, θ$ are contained in $α$ , $ρ, η, π$ are contained in $β$ , and $∣ μ ∣ = ∣ λ ∣$ .
All summands are nonnegative integers, so no cancellation occurs.

Facts & Assumptions

Given: Partitions $α, β, γ$ and six-tuples of partitions $(μ, λ, η, ρ, θ, π)$ .

[F1]

${\overset{ˉ}{g}}_{α, β}^{γ} = ⟨ s_{α} [X] s_{β} [Y] ∣ σ [X Y] s_{γ} [X Y + X + Y] ⟩_{X, Y}$ (Brion's formula for the reduced Kronecker coefficients).

[F2]

$s_{λ} [U + V] = \sum_{μ, ν} c_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]$ and $s_{λ} [U V] = \sum_{μ, ν} g_{μ ν}^{λ} s_{μ} [U] s_{ν} [V]$ for all specializations (Schur functions on sums and products of alphabets), and $σ [X Y] = \sum_{ρ} s_{ρ} [X] s_{ρ} [Y]$ degreewise (The series $σ$ : product formula, Cauchy identity, adjunction and reproducing kernel).

[F3]

$c_{μ, θ, π}^{γ} = ⟨ s_{γ}, s_{μ} s_{θ} s_{π} ⟩_{H} = [s_{γ}] s_{μ} s_{θ} s_{π}$ , and $c_{μ, θ, π}^{γ} = \sum_{ν} c_{μ, ν}^{γ} c_{θ, π}^{ν}$ (Threefold Schur coefficients $c_{a, b, c}^{δ}$ ).

[F4]

The Schur basis is orthonormal (Schur functions form an orthonormal integral basis), the coefficients $c$ and $g$ are nonnegative integers, and $c_{μ ν}^{λ} = 0$ unless $μ \subseteq λ$ and $∣ λ ∣ = ∣ μ ∣ + ∣ ν ∣$ , while $g_{μ ν}^{λ} = 0$ unless $∣ λ ∣ = ∣ μ ∣ = ∣ ν ∣$ (Littlewood--Richardson tableaux and coefficients, The Littlewood–Richardson rule for products of Schur functions, Kronecker coefficients and the internal tensor product).

Proof

technique · direct

1.1F1F2F3F4

Expanding Brion's pairing. Insert the four expansions of [F2] into the pairing of [F1]: $σ [X Y] = \sum_{ρ} s_{ρ} [X] s_{ρ} [Y]$ , and by the coproduct identity at the sum $X Y + (X + Y)$ , $s_{γ} [X Y + X + Y] = \sum_{μ, ν} c_{μ ν}^{γ} s_{μ} [X Y] s_{ν} [X + Y]$ , with $s_{μ} [X Y] = \sum_{λ, η} g_{λ η}^{μ} s_{λ} [X] s_{η} [Y]$ and $s_{ν} [X + Y] = \sum_{θ, π} c_{θ π}^{ν} s_{θ} [X] s_{π} [Y]$ . Multiplying out and pairing with $s_{α} [X] s_{β} [Y]$ using orthonormality [F4] extracts $⟨ s_{α}, s_{ρ} s_{λ} s_{θ} ⟩_{H} = c_{ρ, λ, θ}^{α}$ and $⟨ s_{β}, s_{ρ} s_{η} s_{π} ⟩_{H} = c_{ρ, η, π}^{β}$ by [F3], so ${\overset{ˉ}{g}}_{α, β}^{γ} = \sum_{μ, ν, λ, η, ρ, θ, π} c_{μ ν}^{γ} c_{θ π}^{ν} g_{λ η}^{μ} c_{ρ λ θ}^{α} c_{ρ η π}^{β}$ .

2.1F2F3step 1.1

Collapsing the middle index. By the iterated identity of [F3], $\sum_{ν} c_{μ ν}^{γ} c_{θ π}^{ν} = c_{μ, θ, π}^{γ}$ ; substituting into step 1.1 and relabelling the summation indices gives the displayed expansion. The threefold coefficients are symmetric in their three subscripts by their definition as coefficients of the commutative product $s_{μ} s_{θ} s_{π}$ , so the displayed index pattern is the same as the source's equation (8) with the four factors assigned to the four faces of the index tetrahedron in a different order. All coefficients are nonnegative integers, so no cancellation occurs in the sum.

3.1F4step 2.1

Consequences (2) and (4). A nonzero summand has all four coefficients positive (they are nonnegative integers), and then the degree equations of (2) follow from the size constraints in [F4]: $∣ γ ∣ = ∣ μ ∣ + ∣ θ ∣ + ∣ π ∣$ , $∣ α ∣ = ∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣$ , $∣ β ∣ = ∣ ρ ∣ + ∣ η ∣ + ∣ π ∣$ and $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ ; conversely, any summand with all four coefficients positive contributes positively, giving the criterion (1) since the total is a sum of nonnegative integers.

4.1F3F4step 3.1∎

Consequence (3), finiteness. If $c_{ρ, λ, θ}^{α} > 0$ then, by the iterated form of [F3], there is a partition $ϕ$ with $c_{ρ, λ}^{ϕ} > 0$ and $c_{ϕ, θ}^{α} > 0$ ; by [F4] this forces $ρ \subseteq ϕ \subseteq α$ , $λ \subseteq ϕ \subseteq α$ and $θ \subseteq α$ , so $ρ, λ, θ$ range over the finitely many subpartitions of $α$ . Similarly $c_{ρ, η, π}^{β} > 0$ forces $ρ, η, π \subseteq β$ , and $g_{λ, η}^{μ} > 0$ forces $∣ μ ∣ = ∣ λ ∣$ , so $μ$ ranges over the finitely many partitions of the bounded number $∣ λ ∣$ . Hence only finitely many six-tuples contribute, and the expansion is a finite sum; the proof uses finite expansions and no choice principle.

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The Murnaghan expansion at every admissible rank

Statement

For all partitions $α, β$ and every integer $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}}$ : $s_{α [n]} * s_{β [n]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]},$ an identity in the symmetric function ring, where $s_{γ [n]}$ is the Jacobi–Trudi determinant attached to the integer sequence $γ [n] = (n - ∣ γ ∣, γ_{1}, γ_{2}, \dots)$ , equal to $0$ or $\pm 1$ times a Schur function (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , First-row padding of a partition), and $*$ is the internal product (Kronecker coefficients and the internal tensor product).

For an ordinary partition $δ ⊢ n$ the coefficient of $s_{δ}$ is $g_{α [n], β [n]}^{δ} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} ⟨ s_{γ [n]}, s_{δ} ⟩_{H},$ where each Hall factor is $0$ or a sign coming from Jacobi–Trudi straightening (Schur functions form an orthonormal integral basis). At small admissible ranks the signed terms can cancel or coalesce; individual coefficients need not yet equal their reduced values. The range displayed is precisely the set of $n$ for which $α [n]$ and $β [n]$ are partitions.

Facts & Assumptions

Given: Partitions $α, β, γ, δ$ with $δ ⊢ n$ , and integers $k \geq 0$ .

[F1]

$R_{α, β} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ}$ , a finite Schur expansion (Brion's formula for the reduced Kronecker coefficients, The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ ).

[F2]

$(U_{n} s_{α}) * (U_{n} s_{β}) = U_{n} R_{α, β}$ for every $n \geq 0$ , including nonadmissible ranks, and $\sum_{n \geq 0} (U_{n} s_{α}) * (U_{n} s_{β}) t^{n} = σ [t X] R_{α, β} [t X - 1]$ (The Jacobi–Trudi completion is compatible with the internal product).

[F3]

$U_{n} (s_{λ}) = s_{λ [n]}$ on the Schur basis and $U_{n}$ is $Z$ -linear; for $n \geq ∣ α ∣ + α_{1}$ the sequence $α [n]$ is a partition and $U_{n} s_{α} = s_{α [n]}$ is the honest Schur function (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , First-row padding of a partition).

[F4]

$s_{μ} * s_{ν} = \sum_{δ} g_{μ, ν}^{δ} s_{δ}$ for partitions of equal size, the Schur basis is orthonormal, and $s_{γ [n]}$ is an integer combination of Schur functions of degree $n$ that is $0$ or $\pm 1$ times a Schur function (Kronecker coefficients and the internal tensor product, Schur functions form an orthonormal integral basis, The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ ).

[F5]

The vertex-operator series identifies the coefficients of $σ [t X] R_{α, β} [t X - 1]$ , so $U_{n} R_{α, β}$ is the coefficient of $t^{n}$ there (Vertex-operator generating series for the padded Schur functions).

[F6]

The finite Littlewood expression for $R_{α, β}$ is given in The Jacobi–Trudi completion is compatible with the internal product; LR coefficients count lattice semistandard tableaux (The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients), and $s_{(m)}$ is the internal-product unit in degree $m$ (The internal (Kronecker) product on symmetric functions).

Proof

technique · direct

1.1F1

By [F1], $R_{α, β} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ}$ with finitely many nonzero terms.

1.2F1F6

Local reduced coefficients for $(2), (2)$ . In the Littlewood expression of [F6], $(s_{δ} s_{τ})^{⊥} s_{(2)}$ is nonzero only when $δ = (m)$ and $τ = (r)$ are rows with $m + r \leq 2$ (a row LR target forces every factor to be a row); it then equals $s_{(2 - m - r)}$ . The analogous condition holds for $ϵ$ , and $s_{δ} * s_{ϵ} = 0$ unless their degrees agree, in which case it is $s_{(m)}$ by the unit rule. Here $(0)$ means $\emptyset$ . Hence $R_{(2), (2)} = \sum_{m + r \leq 2} s_{(2 - m - r)}^{2} s_{(m)} = s_{(2)}^{2} + s_{(1)}^{2} + 1 + s_{(1)}^{3} + s_{(1)} + s_{(2)}$ . Counting row LR tableaux gives $s_{(2)}^{2} = s_{(4)} + s_{(3, 1)} + s_{(2, 2)}$ and $s_{(1)}^{2} = s_{(2)} + s_{(1, 1)}$ ; adding one box to these two shapes gives $s_{(1)}^{3} = s_{(3)} + 2 s_{(2, 1)} + s_{(1, 1, 1)}$ . Thus $R_{(2), (2)} = 1 + s_{(1)} + 2 s_{(2)} + s_{(1, 1)} + s_{(1, 1, 1)} + 2 s_{(2, 1)} + s_{(2, 2)} + s_{(3)} + s_{(3, 1)} + s_{(4)}$ , whose coefficients are exactly the reduced coefficients by [F1].

2.1F2F3F5step 1.1

By [F2], $(U_{n} s_{α}) * (U_{n} s_{β}) = U_{n} R_{α, β}$ for every $n \geq 0$ ; by [F5], $U_{n} R_{α, β}$ is the coefficient of $t^{n}$ in $σ [t X] R_{α, β} [t X - 1]$ , and since $U_{n}$ is $Z$ -linear with $U_{n} (s_{γ}) = s_{γ [n]}$ by [F3], it equals $\sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]}$ , a finite sum. Hence the displayed identity holds for every $n \geq 0$ at the level of Jacobi–Trudi completions.

3.1F3step 2.1

In the admissible range $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}}$ the padded sequences $α [n], β [n]$ are partitions, so by [F3] the left side of step 2.1 is the internal product $s_{α [n]} * s_{β [n]}$ of two honest Schur functions; this proves the first display in exactly the stated range and shows that the range is the full set of $n$ for which both inputs are partitions.

3.2F3F4step 2.1step 1.2

Rank-four cancellation. Apply $U_{4}$ to the ten terms of step 1.2. The tails $\emptyset, (1), (2), (1, 1), (1, 1, 1)$ give $s_{(4)}, s_{(3, 1)}, s_{(2, 2)}, s_{(2, 1, 1)}, s_{(1, 1, 1, 1)}$ respectively. The tails $(2, 1), (2, 2)$ give zero by repeated straightening indices. The tails $(3), (3, 1), (4)$ give $- s_{(2, 2)}, - s_{(2, 1, 1)}, - s_{(3, 1)}$ respectively by [F3]–[F4]. With their multiplicities from step 1.2, the sum is $s_{(4)} + s_{(1, 1, 1, 1)} + s_{(2, 2)}$ ; in particular its $(2, 2)$ coefficient is $2 - 1 = 1$ , whereas the reduced $(2)$ coefficient is $2$ . This proves the announced cancellation locally.

4.1F4step 3.1

Coefficient form. For $δ ⊢ n$ , pair the identity of step 2.1 against $s_{δ}$ using orthonormality [F4]: the left side gives $⟨ s_{α [n]} * s_{β [n]}, s_{δ} ⟩_{H} = g_{α [n], β [n]}^{δ}$ by [F4], and the right side gives $\sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} ⟨ s_{γ [n]}, s_{δ} ⟩_{H}$ , where each factor is a Hall pairing of the Jacobi–Trudi determinant $s_{γ [n]}$ with a Schur function, hence a signed multiplicity of a Schur constituent, $0$ or $\pm 1$ by [F4]. This is the second display, and it shows that individual coefficients can receive both positive and negative contributions from different $γ$ and need not equal ${\overset{ˉ}{g}}_{α, β}^{γ}$ .

5.1F1F2step 4.1∎

Choice and finiteness. Every step uses finite sums: the Schur expansion of $R_{α, β}$ is finite by [F1], the sum over $γ$ in step 2.1 is finite, and the coefficient extraction of step 4.1 uses the finite degree- $n$ part of the Schur basis. No form of the axiom of choice is used; the identity is asserted for all admissible ranks, not only for the eventually stable ones.

Remark

For $α = β = (2)$ , every rank $n \geq 4$ product is $U_{n} R_{(2), (2)}$ , where $R_{(2), (2)} = 1 + s_{(1)} + 2 s_{(2)} + s_{(1, 1)} + s_{(1, 1, 1)} + 2 s_{(2, 1)} + s_{(2, 2)} + s_{(3)} + s_{(3, 1)} + s_{(4)} .$ Its coefficients are the reduced coefficients, and all other reduced coefficients vanish. This finite polynomial and its rank-four cancellation are proved in steps 1.2 and 3.2 above.

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Murnaghan–Littlewood: top-degree reduced coefficients are Littlewood–Richardson coefficients

Statement

Let $α, β, γ$ be partitions with $∣ γ ∣ = ∣ α ∣ + ∣ β ∣$ . Then ${\overset{ˉ}{g}}_{α, β}^{γ} = c_{α, β}^{γ},$ the Littlewood–Richardson coefficient (Littlewood--Richardson tableaux and coefficients). In particular ${\overset{ˉ}{g}}_{α, β}^{α + β} = c_{α, β}^{α + β} = 1,$ where $α + β = (α_{1} + β_{1}, α_{2} + β_{2}, \dots)$ .

Facts & Assumptions

Given: Partitions $α, β, γ$ with $∣ γ ∣ = ∣ α ∣ + ∣ β ∣$ , and six-tuples of partitions $(μ, λ, η, ρ, θ, π)$ .

[F1]

${\overset{ˉ}{g}}_{α, β}^{γ} = \sum_{μ, λ, η, ρ, θ, π} c_{μ, θ, π}^{γ} c_{ρ, λ, θ}^{α} c_{ρ, η, π}^{β} g_{λ, η}^{μ}$ , and for every nonzero summand $∣ α ∣ = ∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣$ , $∣ β ∣ = ∣ ρ ∣ + ∣ η ∣ + ∣ π ∣$ , $∣ γ ∣ = ∣ μ ∣ + ∣ θ ∣ + ∣ π ∣$ , $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ , with all summands nonnegative integers (The nonnegative expansion of a reduced Kronecker coefficient).

[F2]

$c_{a, b, c}^{δ} = [s_{δ}] s_{a} s_{b} s_{c}$ , symmetric in $a, b, c$ ; moreover $c_{a, b, \emptyset}^{δ} = [s_{δ}] s_{a} s_{b}$ , equal to the Littlewood–Richardson coefficient $c_{a, b}^{δ}$ (Threefold Schur coefficients $c_{a, b, c}^{δ}$ , The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients).

[F3]

LR tableaux have weakly increasing rows, strictly increasing columns, the prescribed content and a lattice reading word (Littlewood--Richardson tableaux and coefficients).

Proof

technique · direct

1.1F1

Degree collapse. Take a nonzero summand of the expansion in [F1]; by the degree equations of [F1], $∣ α ∣ + ∣ β ∣ - ∣ γ ∣ = (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) + (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) - (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) = 2 ∣ ρ ∣ + ∣ λ ∣ + ∣ η ∣ - ∣ μ ∣ = 2 ∣ ρ ∣ + ∣ μ ∣$ , using $∣ λ ∣ = ∣ η ∣ = ∣ μ ∣$ . Since $∣ γ ∣ = ∣ α ∣ + ∣ β ∣$ , this gives $2 ∣ ρ ∣ + ∣ μ ∣ = 0$ , so $ρ = μ = \emptyset$ and then $∣ λ ∣ = ∣ η ∣ = 0$ , so $λ = η = \emptyset$ .

2.1F1F2step 1.1

Collapse of the sum. With $ρ = λ = η = μ = \emptyset$ , the only possibly nonzero summands have $c_{\emptyset, \emptyset, θ}^{α} = [s_{α}] s_{θ} = δ_{α, θ}$ and $c_{\emptyset, \emptyset, π}^{β} = [s_{β}] s_{π} = δ_{β, π}$ , and $g_{\emptyset, \emptyset}^{\emptyset} = 1$ ; hence ${\overset{ˉ}{g}}_{α, β}^{γ} = c_{\emptyset, α, β}^{γ} = c_{α, β}^{γ}$ by the symmetry of the threefold coefficient in [F2], which is the first display.

3.1F2F3step 2.1

The top coefficient equals one. For $γ = α + β$ we have $c_{α, β}^{α + β} = 1$ . If $β = \emptyset$ , the skew shape is empty and its unique empty tableau proves this; this is also the induction base. Suppose $β \neq \emptyset$ . Existence: fill row $i$ of the skew shape $(α + β) / α$ , which has $β_{i}$ cells, with the letter $i$ ; the rows are constant and hence weakly increasing, the columns are strictly increasing because the letters increase down the rows, and the reading word is $1^{β_{1}} 2^{β_{2}} \dots$ , which is lattice because $β$ is weakly decreasing. Uniqueness: let $T$ be a Littlewood–Richardson tableau of shape $(α + β) / α$ and content $β$ . The first letter of its reading word is the rightmost cell of the first skew row; the lattice condition on the prefix of length one forces that letter to be $1$ , and since the row is weakly increasing every cell of the first row is $1$ . The first row is read first and consists of exactly $β_{1}$ cells, so it exhausts the $β_{1}$ letters $1$ of the content. Deleting the first row leaves the shape $(α^{'} + β^{'}) / α^{'}$ with $α^{'} = (α_{2}, α_{3}, \dots)$ , $β^{'} = (β_{2}, β_{3}, \dots)$ , carrying no letter $1$ ; its reading word, with every letter decreased by one, satisfies the lattice condition: the comparisons among letters $\geq 2$ hold because they hold for the corresponding prefixes of the original lattice word, and the comparison with letter $1$ is not required after the shift. Induction on the number of rows now shows that row $i$ consists of $β_{i}$ copies of $i$ , so $T$ is the displayed filling and $c_{α, β}^{α + β} = 1$ . With [F2] and step 2.1 this gives ${\overset{ˉ}{g}}_{α, β}^{α + β} = c_{α, β}^{α + β} = 1$ .

4.1F1F2step 2.1∎

Empty cases. If $α = \emptyset$ , step 2.1 gives ${\overset{ˉ}{g}}_{\emptyset, β}^{γ} = c_{\emptyset, β}^{γ} = [s_{γ}] s_{β} = δ_{γ, β}$ under the top-degree hypothesis. Thus equal size alone does not force $γ = β$ : the value is zero for every other partition of that size. Similarly, if $β = \emptyset$ the value is $δ_{γ, α}$ . If both inputs are empty then the top-degree hypothesis forces $γ = \emptyset$ and the coefficient is one. No choice principle is used.

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The support of the reduced Kronecker coefficients satisfies the size triangle inequalities

Statement

If ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ then $∣ α ∣ \leq ∣ β ∣ + ∣ γ ∣, ∣ β ∣ \leq ∣ α ∣ + ∣ γ ∣, ∣ γ ∣ \leq ∣ α ∣ + ∣ β ∣$ (Murnaghan's inequalities). Hence the support $Supp (α, β)$ of the nonzero reduced coefficients with fixed $α, β$ lies in the triangle region of the weight lattice.

Facts & Assumptions

Given: Partitions $α, β, γ$ with ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ and a nonzero summand $(μ, λ, η, ρ, θ, π)$ of the expansion.

[F1]

${\overset{ˉ}{g}}_{α, β}^{γ} = \sum c_{μ, θ, π}^{γ} c_{ρ, λ, θ}^{α} c_{ρ, η, π}^{β} g_{λ, η}^{μ}$ , a sum of nonnegative integers, and for every nonzero summand $∣ α ∣ = ∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣$ , $∣ β ∣ = ∣ ρ ∣ + ∣ η ∣ + ∣ π ∣$ , $∣ γ ∣ = ∣ μ ∣ + ∣ θ ∣ + ∣ π ∣$ and $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ (The nonnegative expansion of a reduced Kronecker coefficient).

Proof

technique · direct

1.1F1

Since ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ is a sum of nonnegative integers, some summand is positive; fix such a summand, whose four coefficients are positive and whose sizes satisfy the degree equations of [F1].

2.1F1step 1.1

The three differences. Using the degree equations and $∣ λ ∣ = ∣ η ∣ = ∣ μ ∣$ , compute $∣ α ∣ + ∣ β ∣ - ∣ γ ∣ = (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) + (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) - (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) = 2 ∣ ρ ∣ + ∣ λ ∣ + ∣ η ∣ - ∣ μ ∣ = 2 ∣ ρ ∣ + ∣ μ ∣ \geq 0$ ; next $∣ α ∣ + ∣ γ ∣ - ∣ β ∣ = (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) + (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) - (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) = ∣ λ ∣ + ∣ μ ∣ + 2 ∣ θ ∣ - ∣ η ∣ = ∣ μ ∣ + 2 ∣ θ ∣ \geq 0$ ; and symmetrically $∣ β ∣ + ∣ γ ∣ - ∣ α ∣ = (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) + (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) - (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) = ∣ μ ∣ + 2 ∣ π ∣ \geq 0$ . Each difference is a sum of nonnegative integers, giving the three inequalities for this summand.

3.1F1step 2.1∎

Since these inequalities depend only on the sizes of $α, β, γ$ and not on the chosen positive summand, they hold whenever ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ , that is, for every $γ \in Supp (α, β)$ . This is precisely the assertion that the support lies in the triangle region cut out by the three size inequalities. The argument is a finite computation with nonnegative integers and uses no choice principle; the empty partition is allowed, and then the corresponding inequality is the trivial nonnegativity bound.

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The exact global stabilization threshold for a fixed Kronecker product

Statement

For all partitions $α, β$ with missing parts read as zero, $stab (α, β) = N : = ∣ α ∣ + ∣ β ∣ + α_{1} + β_{1},$ with $stab$ as in The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ .

More precisely. At every $n \geq N$ all support labels $γ [n]$ of the finite support of ${\overset{ˉ}{g}}_{α, β}^{∙}$ are ordinary partitions, the full product has the fixed nonnegative Schur expansion $s_{α [n]} * s_{β [n]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]}$ , and the family is $V$ -stable from $N$ onward. If both $α$ and $β$ are nonempty, $N - 1$ is admissible and the product at $N$ is not $V$ of the product at $N - 1$ : the coefficient of $s_{(α + β) [N]}$ is $1$ , its first two parts are equal, and no such Schur basis vector lies in the image of $V$ . If exactly one input is empty, $N$ is already the least ordinary input rank, and the same image obstruction at $N$ excludes any earlier stabilization index in the Jacobi–Trudi convention of the Definition. If both inputs are empty, $N = 0$ , $s_{(n)} * s_{(n)} = s_{(n)}$ (including $s_{(0)} = 1$ ) and $V^{n} (1) = s_{(n)}$ , so stabilization begins at zero and no negative-rank product is invoked. No sharpness for an individual coefficient is claimed.

Facts & Assumptions

Given: Partitions $α, β, γ$ with ${\overset{ˉ}{g}}_{α, β}^{γ} \neq 0$ ; a nonzero summand $(μ, λ, η, ρ, θ, π)$ of the nonnegative expansion.

[F1]

${\overset{ˉ}{g}}_{α, β}^{γ} = \sum c_{μ, θ, π}^{γ} c_{ρ, λ, θ}^{α} c_{ρ, η, π}^{β} g_{λ, η}^{μ}$ with the size equations $∣ α ∣ = ∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣$ , $∣ β ∣ = ∣ ρ ∣ + ∣ η ∣ + ∣ π ∣$ , $∣ γ ∣ = ∣ μ ∣ + ∣ θ ∣ + ∣ π ∣$ , $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ for every nonzero summand (The nonnegative expansion of a reduced Kronecker coefficient).

[F2]

Two-factor coefficients count LR tableaux and vanish outside containment and size compatibility; commutativity makes them symmetric in the two inputs. A positive threefold coefficient has a positive iterated two-factor summand (The Littlewood–Richardson rule for products of Schur functions, Threefold Schur coefficients $c_{a, b, c}^{δ}$ , Littlewood--Richardson tableaux and coefficients).

[F3]

$s_{α [n]} * s_{β [n]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]}$ for every admissible $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}}$ (The Murnaghan expansion at every admissible rank).

[F4]

For $n \geq ∣ γ ∣ + γ_{1}$ , increasing $n$ by $k \geq 0$ adds $k$ to the first part, so $s_{γ [n + k]} = V^{k} (s_{γ [n]})$ ; the support of ${\overset{ˉ}{g}}_{α, β}^{∙}$ is finite; and for $∣ γ ∣ = ∣ α ∣ + ∣ β ∣$ , ${\overset{ˉ}{g}}_{α, β}^{γ} = c_{α, β}^{γ}$ with ${\overset{ˉ}{g}}_{α, β}^{α + β} = 1$ (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , Murnaghan–Littlewood: top-degree reduced coefficients are Littlewood–Richardson coefficients, The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ ).

[F5]

If ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ then $∣ α ∣ \leq ∣ β ∣ + ∣ γ ∣$ , $∣ β ∣ \leq ∣ α ∣ + ∣ γ ∣$ , $∣ γ ∣ \leq ∣ α ∣ + ∣ β ∣$ (The support of the reduced Kronecker coefficients satisfies the size triangle inequalities).

[F6]

The image of $V$ is contained in the span of the Schur functions $s_{ν}$ with $ν_{1} > ν_{2}$ , and these are linearly independent; the Schur functions of each degree form a $Z$ -basis (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , Schur functions form an orthonormal integral basis).

[F7]

On $Λ^{n}$ the internal product has unit $h_{n} = s_{(n)}$ , and $s_{(n)} * s_{(n)} = s_{(n)}$ (The internal (Kronecker) product on symmetric functions, Kronecker coefficients and the internal tensor product).

Proof

technique · direct

1.1F1F2

Upper bound on the support. Let $γ$ have ${\overset{ˉ}{g}}_{α, β}^{γ} \neq 0$ and choose a nonzero summand of [F1]. First derive the needed row bounds from [F2]: if $c_{a, b}^{ζ} > 0$ and its first skew row is nonempty, the rightmost entry is the first reading-word letter and must be $1$ ; weak row increase forces all added first-row entries to be $1$ . Thus $ζ_{1} - a_{1} \leq b_{1}$ ; an empty first skew row gives the same inequality. Iterating gives $γ_{1} \leq μ_{1} + θ_{1} + π_{1}$ . Containment, applied with $θ$ (respectively $π$ ) first by symmetry, gives $θ_{1} \leq α_{1}$ , $π_{1} \leq β_{1}$ ; the degree equations give $∣ λ ∣ + ∣ θ ∣ \leq ∣ α ∣$ and $∣ η ∣ + ∣ π ∣ \leq ∣ β ∣$ . Therefore $γ_{1} \leq μ_{1} + θ_{1} + π_{1} \leq ∣ μ ∣ + θ_{1} + π_{1}$ , so $∣ γ ∣ + γ_{1} \leq (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) + (∣ μ ∣ + θ_{1} + π_{1}) = 2 ∣ μ ∣ + ∣ θ ∣ + θ_{1} + ∣ π ∣ + π_{1}$ ; using $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ this is $(∣ λ ∣ + ∣ θ ∣ + θ_{1}) + (∣ η ∣ + ∣ π ∣ + π_{1}) \leq (∣ α ∣ + α_{1}) + (∣ β ∣ + β_{1})$ by [F2]. Hence every nonzero reduced coefficient has $∣ γ ∣ + γ_{1} \leq N$ .

1.2F6F7

Both inputs empty. Here $N = 0$ , $α [n] = β [n] = (n)$ for every $n \geq 0$ , and $s_{(n)} * s_{(n)} = s_{(n)}$ by [F7]; also $V^{n} (1) = V^{n} (s_{\emptyset}) = s_{(n)}$ by [F6], since $V$ adds a box to the first row and $s_{\emptyset} = 1$ . Hence $s_{α [n + k]} * s_{β [n + k]} = s_{(n + k)} = V^{k} (s_{α [n]} * s_{β [n]})$ for all $n \geq 0, k > 0$ , with no negative-rank product invoked, so $stab (\emptyset, \emptyset) = 0$ , the least possible value of the defining index.

2.1F4step 1.1

Attainment and identification of the maximum. The partition $γ^{*} : = α + β$ satisfies $∣ γ^{*} ∣ + γ_{1}^{*} = ∣ α ∣ + ∣ β ∣ + (α_{1} + β_{1}) = N$ and ${\overset{ˉ}{g}}_{α, β}^{γ^{*}} = c_{α, β}^{γ^{*}} = 1$ by [F4]; hence $\max {∣ γ ∣ + γ_{1} : {\overset{ˉ}{g}}_{α, β}^{γ} \neq 0} = N$ . In particular $∣ γ ∣ + γ_{1} \leq n$ for every support label and every $n \geq N$ , so all support labels $γ [n]$ are ordinary partitions at such $n$ ; this proves the stated support maximum.

3.1F3F4step 2.1

$V$ -stability from $N$ . For $n \geq N$ the identity of [F3] is available (the thresholds $∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}$ are $\leq N$ ) and every $γ [n]$ in the finite support is a partition by step 2.1; applying [F3] at $n + k$ and using [F4] term by term gives $s_{α [n + k]} * s_{β [n + k]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n + k]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} V^{k} (s_{γ [n]}) = V^{k} (s_{α [n]} * s_{β [n]})$ for all $k > 0$ , together with the fixed nonnegative Schur expansion at every rank $n \geq N$ . Hence $stab (α, β) \leq N$ in the sense of the Definition.

4.1F4F6step 3.1

$N$ is not in the image of $V$ , when $N > 0$ . If $N > 0$ then $α$ or $β$ is nonempty and the partition $(α + β) [N] = (N - ∣ α ∣ - ∣ β ∣, (α + β)_{1}, \dots) = (α_{1} + β_{1}, α_{1} + β_{1}, \dots)$ has equal first two parts, which are positive because at least one of $α, β$ is nonempty. At rank $N$ the coefficient of $s_{(α + β) [N]}$ in the product is ${\overset{ˉ}{g}}_{α, β}^{α + β} = 1$ by [F4] and step 3.1. If $N > 0$ and the product at $N$ lay in the image of $V$ , then by [F6] its coefficient of $s_{(α + β) [N]}$ would be $0$ , a contradiction.

5.1F6step 3.1step 4.1

Least-ness, both inputs nonempty. Suppose $α \neq \emptyset \neq β$ . Then $N - 1 = ∣ α ∣ + ∣ β ∣ + α_{1} + β_{1} - 1 \geq ∣ α ∣ + α_{1}$ and $\geq ∣ β ∣ + β_{1}$ , since $∣ β ∣ + β_{1} \geq 1$ and $∣ α ∣ + α_{1} \geq 1$ ; so $N - 1$ is an admissible rank and the product at $N - 1$ is defined. If some $n_{0} < N$ satisfied the stabilization condition of the Definition, then the product at $N$ would equal $V^{N - n_{0}}$ of the product at $n_{0}$ , which lies in the image of $V$ , contradicting step 4.1; taking $n_{0} = N - 1$ shows in particular that the product at $N$ is not $V$ of the product at $N - 1$ . Hence $stab (α, β) = N$ in this case.

5.2F4F5step 3.1step 4.1

Least-ness, exactly one input empty. Say $β = \emptyset \neq α$ , so $N = ∣ α ∣ + α_{1}$ is the least rank at which $α [n]$ (and $β [n] = (n)$ ) is an ordinary partition. By [F5], every $γ$ with ${\overset{ˉ}{g}}_{α, \emptyset}^{γ} \neq 0$ satisfies $∣ γ ∣ \leq ∣ α ∣ \leq ∣ \emptyset ∣ + ∣ γ ∣ = ∣ γ ∣$ , hence $∣ γ ∣ = ∣ α ∣$ , and then [F4] gives ${\overset{ˉ}{g}}_{α, \emptyset}^{γ} = c_{α, \emptyset}^{γ} = δ_{γ, α}$ ; so the product at rank $N$ is exactly $s_{α [N]}$ , whose first two parts are $α_{1}, α_{1}$ with $α_{1} \geq 1$ . The obstruction of step 4.1 therefore applies at $N$ : the product at $N$ does not lie in the image of $V$ , so no $n_{0} < N$ can satisfy the stabilization condition, while $n_{0} = N$ does by step 3.1. The case $α = \emptyset \neq β$ is symmetric.

6.1F3F4step 3.1step 5.1step 5.2step 1.2∎

Conclusion. Combining steps 3.1, 5.1, 5.2 and 1.2 gives $stab (α, β) = N = ∣ α ∣ + ∣ β ∣ + α_{1} + β_{1}$ in all cases, with the fixed nonnegative expansion and $V$ -stability from $N$ onward; the statement claims nothing about sharpness for an individual coefficient, and the proof used only finite sums, the proved all-rank expansion and the unit of the internal product. No choice principle is used.

5 · Examples, counterexamples and false statements

None yet.

Murnaghan Stability via FI-Modules

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Specializations of symmetric functions and the series σσ

Definition

Threefold Schur coefficients ca,b,cδca,b,cδ​

Definition

The Jacobi–Trudi completion sμsμ​, the operators UnUn​ and VV, and stab⁡(α,β)stab(α,β)

Definition

Schur functions on sums and products of alphabets

Statement

Facts & Assumptions

Proof

The series σσ: product formula, Cauchy identity, adjunction and reproducing kernel

Statement

Facts & Assumptions

Proof

Vertex-operator generating series for the padded Schur functions

Statement

Facts & Assumptions

Proof

The Jacobi–Trudi completion is compatible with the internal product

Statement

Facts & Assumptions

Proof

Murnaghan stability for Kronecker coefficients

Statement

Facts & Assumptions

Proof

The reduced (stable) Kronecker coefficient gˉα,βγgˉ​α,βγ​

Definition

Brion's formula for the reduced Kronecker coefficients

Statement

Facts & Assumptions

Proof

The nonnegative expansion of a reduced Kronecker coefficient

Statement

Facts & Assumptions

Proof

The Murnaghan expansion at every admissible rank

Statement

Facts & Assumptions

Proof

Remark

Murnaghan–Littlewood: top-degree reduced coefficients are Littlewood–Richardson coefficients

Statement

Facts & Assumptions

Proof

The support of the reduced Kronecker coefficients satisfies the size triangle inequalities

Statement

Facts & Assumptions

Proof

The exact global stabilization threshold for a fixed Kronecker product

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources

Specializations of symmetric functions and the series $σ$

Threefold Schur coefficients $c_{a, b, c}^{δ}$

The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$

The series $σ$ : product formula, Cauchy identity, adjunction and reproducing kernel

The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$