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Positive-characteristic decomposition polynomials factor through adjustment

Statement

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, for every $λ \in R P_{α}$ and every partition $μ$ of content $α$ , the graded decomposition polynomial factors as $[S (μ) : D (λ)]_{q} = \sum_{ν \in R P_{α}} a_{λ, ν} (q) d_{ν, μ} (q),$ where $a_{λ, ν}$ are the adjustment polynomials of The graded adjustment polynomials in positive characteristic and $d_{ν, μ}$ the characteristic-independent global-basis transition polynomial of Level-one global-basis transition polynomials in the symmetric-group convention; equivalently $[J (ν)] = [D (ν)] + \sum_{λ ⊲ ν} a_{λ, ν} (q) [D (λ)]$ . In particular the characteristic- $p$ decomposition matrix is the product of the dominance-unitriangular adjustment matrix and the (characteristic-independent) matrix of global-basis coefficients, and no claim is made that the adjustment matrix is the identity. The finite coefficient substitution uses no additional choice.

Facts & Assumptions

Given: Assume the Axiom of Choice. Branch (CP) of the regime with a content $α$ , the graded Specht modules $S (μ)$ , the reduced modules $J (ν)$ and the normalized simples $D (λ)$ .

[F1]

The Specht classes realize the standard Fock vectors: $[S (μ)]$ maps to $S_{μ}$ under $Ψ_{c a n}^{H e c k e}$ (Graded Specht classes realize the standard Fock vectors), and $S_{μ} = \sum_{ν \in R P_{α}} d_{ν, μ} (q) D_{ν}$ in the upper family (Standard Fock vectors expand in the upper global basis), with the transition polynomials of Level-one global-basis transition polynomials in the symmetric-group convention.

[F2]

The reduced modules satisfy $Ψ_{c a n}^{H e c k e} ([J (ν)]) = D_{ν}$ over the positive-characteristic field, and the labels agree with the cellular heads in this branch (Reduction of characteristic-zero simple modules modulo p, Positive-characteristic Specht head and cyclotomic crystal labels agree).

[F3]

The adjustment polynomials are defined by $[J (ν)] = \sum_{λ} a_{λ, ν} (q) [D (λ)]$ , with $a_{ν, ν} = 1$ and $a_{λ, ν} = 0$ unless $λ ⊴ ν$ (The graded adjustment polynomials in positive characteristic); the graded Grothendieck group has the shift-orbit basis of the normalized simples, so all expansions are unique (Graded module and projective Grothendieck groups of the KLR and Hecke towers).

[F4]

The ungraded cellular triangularity of the decomposition matrix is The type-A Hecke decomposition matrix is dominance unitriangular; the regime is branch (CP) of The nondegenerate root-of-unity level-one Hecke regime.

Proof

technique · direct

1.1F1F2

Expand the Specht class in the reduced basis. By [F1] the transported isomorphism sends $[S (μ)]$ to $S_{μ} = \sum_{ν \in R P_{α}} d_{ν, μ} (q) D_{ν}$ ; applying $(Ψ_{c a n}^{H e c k e})^{- 1}$ and using that the reduced modules map to the dual-canonical vectors by [F2] gives $[S (μ)] = \sum_{ν \in R P_{α}} d_{ν, μ} (q) [J (ν)]$ in $G_{0}^{g r} (H_{α}^{Λ} (F))$ [F1, F2].

2.1F3step 1.1

Substitute the adjustment expansion. Substituting the defining expansion $[J (ν)] = \sum_{λ \in R P_{α}} a_{λ, ν} (q) [D (λ)]$ of [F3] into step 1.1 and collecting the coefficients of the basis elements $[D (λ)]$ yields $[S (μ)] = \sum_{λ} (\sum_{ν} a_{λ, ν} (q) d_{ν, μ} (q)) [D (λ)]$ [F3].

3.1F3F4step 2.1

Read off the multiplicities. By [F3] the simple classes form the shift-orbit basis, so the coefficient of $[D (λ)]$ in the expansion of $[S (μ)]$ is the graded composition multiplicity $[S (μ) : D (λ)]_{q}$ ; hence the displayed factorization, and the second displayed form is the same expansion written at the diagonal $a_{ν, ν} = 1$ and the dominance support $a_{λ, ν} = 0$ for $λ ⋬ ν$ [F3]. This factorization alone neither computes off-diagonal adjustment entries nor forces the adjustment matrix to be the identity [F3, F4].

4.1givenF1F2F3∎

Choice. AC is inherited through the transported global-basis identifications of [F1, F2] and the CP crystal-label comparison of [F2, F3]. Substitution and coefficient comparison in the finite simple-class basis use no additional choice.

Depends on

Used by

Canonical bases do not solve arbitrary positive-characteristic decomposition numbers Remark

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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)
Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, Bull. Amer. Math. Soc. 47 (2010) 419-481 (complete 66-page survey) (standard reference, not scraped)