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The graded adjustment polynomials in positive characteristic

Definition

Assume the Axiom of Choice (The Axiom of Choice), inherited through the crystal-label comparison and transported global-basis suppliers below.

In branch (CP) of the regime of The nondegenerate root-of-unity level-one Hecke regime, with the reduced modules $J (ν)$ of Reduction of characteristic-zero simple modules modulo p and the normalized simples $D (λ)$ with crystal label $λ \in R P_{α}$ of the general-field normalization/crystal clauses of The QG-8 categorification transports across the blockwise Hecke-KLR bridge and Positive-characteristic Specht head and cyclotomic crystal labels agree (in branch (C0) these are the heads of the graded Specht modules by Graded Specht heads classify the graded simple modules (3)), define the graded adjustment polynomials $a_{λ, ν} (q) \in Z [q, q^{- 1}]$ for $λ, ν \in R P_{α}$ by $[J (ν)] = \sum_{λ \in R P_{α}} a_{λ, ν} (q) [D (λ)] ⟺ a_{λ, ν} (q) = [J (ν) : D (λ)]_{q} .$ Then the matrix $(a_{λ, ν})$ is the graded adjustment matrix; its entries are bar-invariant, have nonnegative integer coefficients, satisfy $a_{ν, ν} (q) = 1$ and vanish unless $λ ⊴ ν$ (dominance); equivalently $a_{λ, ν}$ is the coefficient of $[D (λ)]$ in $(Ψ_{c a n}^{H e c k e})^{- 1} (D_{ν})$ .

Source comparison (recorded, not used). For level one and $ξ = 1$ the source states that this adjustment matrix specializes at $q = 1$ to the adjustment matrix of James; that degenerate comparison lies outside the nondegenerate branch of this page and is deferred (see the batch coverage record). The finite multiplicity expansion uses no additional choice.

Facts & Assumptions

Given: Assume the Axiom of Choice. Branch (CP) of the regime, a content $α$ , the reduced modules $J (ν)$ and the normalized positive-characteristic simples $D (λ)$ for $λ, ν \in R P_{α}$ .

[F1]

The reduced modules $J (ν) = F \otimes_{Z} L (ν)$ have field-independent $q$ -characters and well-defined classes in $G_{0}^{g r} (H_{α}^{Λ} (F))$ (Reduction of characteristic-zero simple modules modulo p).

[F2]

The normalized simples $D (λ)$ have crystal labels in $R P_{α}$ and their labels agree with the cellular head labels in positive characteristic. The reduction $J (ν)$ contains $D (ν)$ once, has no constituents with labels not dominated by $ν$ , and its class is fixed by finite-module diagram duality (Positive-characteristic Specht head and cyclotomic crystal labels agree). The normalized simples are fixed by this duality in both field branches (The QG-8 categorification transports across the blockwise Hecke-KLR bridge); in branch (C0) they are the graded Specht heads (Graded Specht heads classify the graded simple modules (3)).

[F3]

The graded Grothendieck group has the shift-orbit basis of the normalized simples, so expansions in that basis are unique (Graded module and projective Grothendieck groups of the KLR and Hecke towers).

[F4]

The dominance-unitriangularity of the transition data is that of Level-one global-basis transition polynomials in the symmetric-group convention and The type-A Hecke decomposition matrix is dominance unitriangular; the Specht classes realize the standard Fock vectors (Graded Specht classes realize the standard Fock vectors), and the graded Specht modules are those of The graded Specht module by homogeneous standard tableaux.

Proof

technique · direct

1.1F1F3

The defining expansion is unique with nonnegative integral coefficients. By [F1] each $J (ν)$ is a finite-dimensional graded module in a finite block, and by [F3] the normalized simples form a shift-orbit basis of $G_{0}^{g r}$ ; therefore the coefficients $a_{λ, ν} (q)$ are uniquely determined Laurent polynomials, and being multiplicities of graded composition factors they have nonnegative integer coefficients [F1, F3].

1.2F2F4

Diagonal one and dominance support. The local positive-characteristic comparison identifies $J (ν)$ with $D (ν)$ plus terms whose labels are strictly smaller in the dominance order (the labels agreeing with the cellular heads by [F2]); hence $a_{ν, ν} = 1$ and $a_{λ, ν} = 0$ unless $λ ⊴ ν$ [F2, F4].

2.1F2F3step 1.1

Bar invariance. By [F2], $[D_{ψ} J (ν)] = [J (ν)]$ and $D_{ψ} D (λ) ≅ D (λ)$ . Applying finite graded duality to the defining expansion inverts grading shifts, giving $[J (ν)] = \sum_{λ} a_{λ, ν} (q^{- 1}) [D (λ)]$ . Uniqueness in step 1.1 therefore gives $a_{λ, ν} (q) = a_{λ, ν} (q^{- 1})$ .

3.1givenF1F2F3F4∎

The equivalent description and the recorded comparison. By [F1], $Ψ_{c a n}^{H e c k e} ([J (ν)]) = D_{ν}$ ; therefore $a_{λ, ν}$ is the coefficient of $[D (λ)]$ in $(Ψ_{c a n}^{H e c k e})^{- 1} (D_{ν})$ . The recorded comparison at $ξ = 1$ lies outside this branch. AC is inherited through the crystal-label comparison in [F2] and the transported global-basis identification in [F1, F4]; the finite expansion and duality calculation use no additional choice.

Depends on

Used by

Dependency tree · two levels

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Sources

Jonathan Brundan and Alexander Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, arXiv:0901.4450v3, Adv. Math. 222 (2009) 1883-1942 (complete 57-page article) (standard reference, not scraped)
Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete 79-page prepublication manuscript (standard reference, not scraped)