Specht Modules and the Irreducibles of the Symmetric Group

Given: $n\ge 0$, $\lambda ⊢n$, a $\lambda$-tableau $t$, $\sigma \in S_n$, and $\gamma \in C_t$.

The column antisymmetrizer is the finite group-algebra sum ${\kappa }_t={\sum }_{c\in C_t}\mathrm{sgn}(c)c$ (Column antisymmetrizers, polytabloids, and Specht modules).

The polytabloid is $e_t={\kappa }_t\cdot \{t\}$ (Column antisymmetrizers, polytabloids, and Specht modules).

$S^{\lambda }$ is the complex span of all $e_s$ for $\lambda$-tableaux $s$ (Column antisymmetrizers, polytabloids, and Specht modules).

$C_t$ is the direct product of the symmetric groups on its pairwise disjoint column sets (Row and column stabilizers).

$C_{\sigma \cdot t}=\sigma C_t{\sigma }^{-1}$ (Tableau stabilizers transform by conjugation).

Sign is a group homomorphism (The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$).

A $\lambda$-tableau is a bijection from $[\lambda ]$ to $\{1,\dots ,n\}$ (Tableaux and standard tableaux).

The tabloid module has the tabloids as basis and carries the linear extension of their left $S_n$-action (Young subgroups, tabloids, and permutation modules).

Given: An integer $n\ge 0$, partitions $\lambda ,\mu ⊢n$, a $\lambda$-tableau $t$, a $\mu$-tabloid $\{s\}$, and distinct labels $a,b$ lying in one row of $s$ and one column of $t$.

The row stabilizer preserves each row set, and the column stabilizer preserves each column set (Row and column stabilizers).

A tabloid records row sets, so the order of entries within each row is forgotten (Young subgroups, tabloids, and permutation modules).

The column antisymmetrizer is ${\kappa }_t={\sum }_{\gamma \in C_t}\mathrm{sgn}(\gamma )\gamma$ (Column antisymmetrizers, polytabloids, and Specht modules).

The sign function is a group homomorphism (The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$).

Sign is defined by $\mathrm{sgn}(\sigma )=(-1{)}^{\mathrm{inv}(\sigma )}$ (Inversions, inversion number, the sign $\mathrm{sgn}(\sigma )=(-1{)}^{\mathrm{inv}(\sigma )}$, and even and odd permutations).

The library's $1,\dots ,n$ sign convention is transported by the canonical relabelling from the finite-ordinal convention (Column antisymmetrizers, polytabloids, and Specht modules).

Given: $n\ge 0$, $\lambda ,\mu ⊢n$, a $\lambda$-tableau $t$, and the hypothesis ${\kappa }_t{M}^{\mu }\mathrm{\ne }0$.

The $\mu$-tabloids form a basis of $M^{\mu }$, with the linear extension of the left action of $S_n$ (Young subgroups, tabloids, and permutation modules).

The column antisymmetrizer is the group-algebra element ${\kappa }_t={\sum }_{\gamma \in C_t}\mathrm{sgn}(\gamma )\gamma$ (Column antisymmetrizers, polytabloids, and Specht modules).

If two entries in one row of $s$ lie in one column of $t$, then ${\kappa }_t\cdot \{s\}=0$ (Column collision cancels antisymmetrization).

If every row of $s$ meets each column of $t$ in at most one entry, then for tableaux of shapes $\lambda$ and $\mu$ one has $\lambda ⊵\mu$ (Basic row-column incidence lemma).

The notation $\lambda ⊵\mu$ means that every prefix sum of $\lambda$ is at least the corresponding prefix sum of $\mu$, with both partitions padded by zeros (Dominance order on partitions).

Given: $n\ge 0$, $\lambda ⊢n$, and a $\lambda$-tableau $t$.

The column antisymmetrizer is ${\kappa }_t={\sum }_{\gamma \in C_t}\mathrm{sgn}(\gamma )\gamma$ (Column antisymmetrizers, polytabloids, and Specht modules).

The polytabloid is $e_t={\kappa }_t\cdot \{t\}$ (Column antisymmetrizers, polytabloids, and Specht modules).

The coefficient of $\{t\}$ in $e_t$ is $1$ (Column antisymmetrizers, polytabloids, and Specht modules).

The $\lambda$-tabloids form a basis of $M^{\lambda }$, and its $S_n$ action extends linearly from the left action on tabloids (Young subgroups, tabloids, and permutation modules).

The stabilizer of the tabloid $\{s\}$ is $R_s$ (Young subgroups, tabloids, and permutation modules).

If a row of a tabloid contains two entries from one column of $t$, then ${\kappa }_t$ sends that tabloid to zero (Column collision cancels antisymmetrization).

If every row of a tableau $s$ meets every column of $t$ in at most one entry and $s,t$ have shape $\lambda$, then there are $\rho \in R_s$ and $\gamma \in C_t$ with $\rho \cdot s=\gamma \cdot t$ (Basic row-column incidence lemma).

The sign function is a group homomorphism to $\{+1,-1\}$ (The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$).

Given: $n\ge 0$, $\lambda ⊢n$, and an $S_n$-submodule $U\subseteq M^{\lambda }$.

The tabloids form a basis of $M^{\lambda }$, and the $S_n$-action extends linearly from the tabloid action (Young subgroups, tabloids, and permutation modules).

An $S_n$-submodule is a linear subspace stable under each element of $S_n$ (Subrepresentations, direct sums of representations, and irreducibility).

The column antisymmetrizer, polytabloid, and Specht space are

for each $\lambda$-tableau $t$ (Column antisymmetrizers, polytabloids, and Specht modules).

For every $t$, ${\kappa }_t{M}^{\lambda }=\mathbb{C}{e}_t$ and $e_t\mathrm{\ne }0$ (The antisymmetrizer image in its own tabloid module is one-dimensional).

The Specht space is generated as an $S_n$-module by any one polytabloid $e_t$ (Polytabloid covariance and the column sign rule).

The tabloid product is conjugate-linear in its first argument and linear in its second (Invariant Hermitian product on a tabloid module).

Each ${\kappa }_t$ is self-adjoint for the tabloid product (Invariant Hermitian product on a tabloid module).

This Hermitian product is positive definite: if $x\mathrm{\ne }0$, then $⟨x,x⟩>0$ (Invariant Hermitian product on a tabloid module).

The orthogonal complement is $(S^{\lambda }{)}^{\perp }=\{v:⟨v,s⟩=0\text{ for every }s\in S^{\lambda }\}$ (Orthogonality and the orthogonal complement).

Every shape has a canonical standard row-filled tableau $t_0$; for $n=0$ it is the empty tableau (Young subgroups, tabloids, and permutation modules).

No form of the Axiom of Choice is used. The first branch uses only witnesses to one existential statement, and all group-algebra sums are finite.

Given: $n\ge 0$ and $\lambda ⊢n$.

The tabloid-basis Hermitian product is positive definite: $⟨x,x⟩={\sum }_T\mathrm{\mid }{a}_T{\mathrm{\mid }}^2>0$ whenever $x={\sum }_T{a}_{T}T\mathrm{\ne }0$ (Invariant Hermitian product on a tabloid module).

$S^{\lambda }$ is the complex span of the polytabloids $e_s$ (Column antisymmetrizers, polytabloids, and Specht modules).

For every tableau $t$, the coefficient of $\{t\}$ in $e_t$ is $1$ (Column antisymmetrizers, polytabloids, and Specht modules).

For every partition, the canonical row-filled $\lambda$-tableau exists (Young subgroups, tabloids, and permutation modules).

The orthogonal complement is defined by vanishing of the inner product against every vector of the subspace (Orthogonality and the orthogonal complement).

Given: $n\ge 0$, $\lambda ⊢n$, and a nonzero $S_n$-subrepresentation $U\subseteq S^{\lambda }$.

$S^{\lambda }$ is the complex span of its polytabloids, which lie in the tabloid module $M^{\lambda }$ (Column antisymmetrizers, polytabloids, and Specht modules).

$M^{\lambda }$ is a finite-dimensional complex representation of $S_n$ (Young subgroups, tabloids, and permutation modules).

$S^{\lambda }$ is an $S_n$-subrepresentation of $M^{\lambda }$ (Polytabloid covariance and the column sign rule).

For every $S_n$-submodule $V\subseteq M^{\lambda }$, either $S^{\lambda }\subseteq V$ or $V\subseteq (S^{\lambda }{)}^{\perp }$ (James's submodule theorem over the complex numbers).

$S^{\lambda }\mathrm{\ne }0$ and $S^{\lambda }\cap (S^{\lambda }{)}^{\perp }=\{0\}$ (Complex Specht modules have nondegenerate Hermitian self-pairing).

A subrepresentation is a linear subspace stable under every group element (Subrepresentations, direct sums of representations, and irreducibility).

A representation is irreducible when it is nonzero and its only subrepresentations are $0$ and the whole representation (Subrepresentations, direct sums of representations, and irreducibility).

No form of the Axiom of Choice is used.

Given: $n\ge 0$, partitions $\lambda ,\mu ⊢n$, and a complex $\mathbb{C}[S_n]$-module homomorphism $\varphi :S^{\lambda }\to M^{\mu }$.

$S_n$ is the group of bijections of $\{1,\dots ,n\}$, with $S_0$ trivial (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition, $\mathrm{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements, Young subgroups, tabloids, and permutation modules).

Each $M^{\nu }$ is a finite-dimensional complex representation of $S_n$ (Young subgroups, tabloids, and permutation modules).

The $\nu$-tabloids form a basis of $M^{\nu }$, with the linear extension of the left $S_n$-action (Young subgroups, tabloids, and permutation modules).

For a $\lambda$-tableau $t$, ${\kappa }_t={\sum }_{\gamma \in C_t}\mathrm{sgn}(\gamma )\gamma$, $e_t={\kappa }_t\cdot \{t\}$, and $S^{\lambda }$ is the complex span of the $e_s$ (Column antisymmetrizers, polytabloids, and Specht modules).

Every $e_t$ is nonzero, since its coefficient at $\{t\}$ is $1$ (Column antisymmetrizers, polytabloids, and Specht modules).

$S^{\lambda }$ is an $S_n$-subrepresentation generated by any one $e_t$ (Polytabloid covariance and the column sign rule).

A subrepresentation is a linear subspace stable under each group element (Subrepresentations, direct sums of representations, and irreducibility).

For a finite group over a field whose characteristic does not divide the group order, every subrepresentation has a complementary subrepresentation; this applies over $\mathbb{C}$ (Maschke's theorem for finite groups over fields whose characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$).

The space ${\mathrm{Hom}}_{S_n}(V,W)$ consists of complex-linear $S_n$-equivariant maps and is a subspace of the complex vector space of linear maps (Intertwiners, the spaces ${\mathrm{Hom}}_G(V,W)$ and ${\mathrm{End}}_G(V)$, equivalent representations, and faithful representations).

A complex-linear map is $S_n$-equivariant exactly when it is a $\mathbb{C}[S_n]$-module homomorphism (For a commutative ring $R$, $R$-linear $G$-actions are exactly the compatible left $R[G]$-module structures).

If ${\kappa }_t{M}^{\mu }\mathrm{\ne }0$, then $\lambda ⊵\mu$ (Nonzero antisymmetrizer image detects dominance).

For every $\lambda$-tableau $t$, ${\kappa }_t{M}^{\lambda }=\mathbb{C}{e}_t$ (The antisymmetrizer image in its own tabloid module is one-dimensional).

$\lambda ⊵\mu$ means every prefix sum of $\lambda$ is at least the corresponding prefix sum of $\mu$ (Dominance order on partitions).

Every shape has a canonical standard row-filled tableau $t_0$ (Young subgroups, tabloids, and permutation modules).

The label set $\{1,\dots ,n\}$ is finite (empty when $n=0$), and the set of bijections of a finite set is finite; hence $S_n$ is finite (The natural numbers $\mathbb{N}$ (von Neumann), Young subgroups, tabloids, and permutation modules, A finite set $A$ with $\mid A\mid =n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality).

No form of the Axiom of Choice is used. The proof uses one complement supplied by Maschke's theorem and existential witnesses from spanning families; it does not choose from an arbitrary indexed family.

Given: $n\ge 0$, partitions $\lambda ,\mu ⊢n$, and an isomorphism $f:S^{\lambda }\to S^{\mu }$ of complex $S_n$-representations.

Each shape $\nu ⊢n$ has a canonical standard row-filled tableau $t_{\nu }$ (Young subgroups, tabloids, and permutation modules).

The Specht space $S^{\nu }$ is the complex span of its polytabloids, each lying in the tabloid module $M^{\nu }$ (Column antisymmetrizers, polytabloids, and Specht modules).

For every tableau $t$, the coefficient of $\{t\}$ in $e_t$ is $1$; in particular $e_t\mathrm{\ne }0$ (Column antisymmetrizers, polytabloids, and Specht modules).

For each $\nu ⊢n$, $S^{\nu }$ is an $S_n$-submodule of $M^{\nu }$ (Polytabloid covariance and the column sign rule).

An isomorphism of representations is an invertible intertwiner (Intertwiners, the spaces ${\mathrm{Hom}}_G(V,W)$ and ${\mathrm{End}}_G(V)$, equivalent representations, and faithful representations).

A complex-linear map of $S_n$-representations is equivariant exactly when it is a $\mathbb{C}[S_n]$-module homomorphism (For a commutative ring $R$, $R$-linear $G$-actions are exactly the compatible left $R[G]$-module structures).

A nonzero $\mathbb{C}[S_n]$-module map $S^{\alpha }\to M^{\beta }$ implies $\alpha ⊵\beta$ (Homomorphisms from Specht to Young permutation modules obey dominance).

The dominance relation on partitions of $n$ is antisymmetric: $\lambda ⊵\mu$ and $\mu ⊵\lambda$ imply $\lambda =\mu$ (Dominance order on partitions).

No form of the Axiom of Choice is used. The proof uses only the canonical tableaux in [F1] and the given isomorphism.

Given: $n\ge 0$. Let ${\mathcal{P}}_n$ be the set of partitions of $n$, ${\mathcal{C}}_n$ the set of conjugacy classes of $S_n$, and ${\mathcal{I}}_n$ the set of isomorphism classes of finite-dimensional irreducible complex $S_n$-representations.

A partition is a finite weakly decreasing list of positive integers with sum $n$ (Partitions, English diagrams, and conjugation).

$S_n=\mathrm{Sym}(\{1,\dots ,n\})$ and $S_0=\mathrm{Sym}(\mathrm{\varnothing })=\{1\}$ (Partitions, English diagrams, and conjugation, The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition); $\mathrm{Sym}(X)$ is a group ($\mathrm{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements).

$S_n$ is finite and $\mathrm{\mid }{S}_n\mathrm{\mid }=n!$ (The Lehmer code gives $\mathrm{\mid }{S}_n\mathrm{\mid }=n!$ again).

$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field containing the embedded copy $\iota :\mathbb{R}\hookrightarrow \mathbb{C}$ (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$); this field embedding preserves $1$ and addition (Field homomorphism and embedding).

$\mathbb{R}$ is an ordered field and, for every integer $m\ge 1$, its canonical natural $m\cdot 1_{\mathbb{R}}$ is positive (The reals form a totally ordered field, Canonical naturals are positive and strictly increasing).

A ring has characteristic zero exactly when no positive integer multiple of its identity is zero (The characteristic of a ring is the additive order of $1_R$, with $0$ recording infinite order; $n\cdot 1_R=0$ holds exactly when $\mathrm{char}(R)\mid n$; and in an integral domain every nonzero element has the same additive order as $1_R$).

The natural-to-integer embedding preserves order, and zero divides no positive integer (The naturals embed in the integers, Invertibility of a positive natural scalar in a field).

$\mathbb{C}$ is algebraically closed (The complex numbers are algebraically closed).

For a finite group $G$ and algebraically closed field $k$ with $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, the finite set of irreducible representation classes has cardinality equal to the finite set of conjugacy classes (If $k$ is algebraically closed and $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, the number of irreducible representations of $G$ equals the number of conjugacy classes).

Conjugacy classes of $S_n$ are in bijection with the tuples of nonnegative integers $(c_1,\dots ,c_n)$ satisfying ${\sum }_{k=1}^{n}k{c}_k=n$; when $n=0$ the unique empty tuple indexes the identity class (The conjugacy classes of $S_n$ are indexed by the tuples $(c_1,\dots ,c_n)$ with $\sum k{c}_k=n$).