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Graded decomposition numbers equal canonical-basis coefficients in characteristic zero

Statement

In branch (C0) of the regime of The nondegenerate root-of-unity level-one Hecke regime, for every partition $μ$ with content $α$ and every $λ \in R P_{α}$ , the graded composition multiplicity satisfies $[S (μ) : D (λ)]_{q} = d_{λ, μ} (q),$ where $d_{λ, μ}$ is the level-one global-basis transition polynomial of Level-one global-basis transition polynomials in the symmetric-group convention; equivalently $[S (μ)] = \sum_{λ \in R P_{α}} d_{λ, μ} (q) [D (λ)]$ in $G_{0}^{g r} (H_{α}^{Λ})$ , and $d_{λ, μ} = 0$ unless $λ ⊴ μ$ . Specializing $q = 1$ gives Ariki's ungraded decomposition-number theorem $[S (μ) : D (λ)] = d_{λ, μ} (1)$ . Choice is inherited only through the QG-8 global-basis suppliers.

Facts & Assumptions

Given: Branch (C0) of the regime, a partition $μ$ with content $α$ , the graded Specht module $S (μ)$ , and the transported identification $Ψ_{c a n}^{H e c k e}$ .

[F1]

The transported identification sends $[S (μ)]$ to the standard Fock vector $S_{μ} = π (M_{μ})$ (Graded Specht classes realize the standard Fock vectors); and $S_{μ} = \sum_{λ} d_{(λ, μ)} (q) G^{*} (λ)$ over the restricted labels, with the same coefficients as the lower expansion, by Standard Fock vectors expand in the upper global basis.

[F2]

In branch (C0) the transported normalized self-dual simples $D (λ)$ map to the upper global basis vectors $G^{*} (λ)$ under $Ψ_{c a n}^{H e c k e}$ (Ariki's canonical basis theorem in characteristic zero (2)), and the simple classes form a basis of the free $A$ -module $G_{0}^{g r} (H_{α}^{Λ})$ (The QG-8 categorification transports across the blockwise Hecke-KLR bridge (2)).

[F3]

The transition polynomials have off-diagonal entries in $q Z [q]$ and are supported on $λ ⊴ μ$ with diagonal $1$ (Level-one global-basis transition polynomials in the symmetric-group convention); the graded module group is the restricted direct sum of the blockwise groups with forgetful maps $ζ$ satisfying $ζ (q m) = ζ (m)$ (Graded module and projective Grothendieck groups of the KLR and Hecke towers).

[F4]

The graded Specht module and its basis are those of The graded Specht module by homogeneous standard tableaux; the regime is branch (C0) of The nondegenerate root-of-unity level-one Hecke regime.

[C1]

Choice is inherited only through the QG-8 global-basis and Fock suppliers (The Axiom of Choice, Ariki's canonical basis theorem in characteristic zero).

Proof

technique · direct

1.1F1F3

Transport the Specht class. Applying $Ψ_{c a n}^{H e c k e}$ to $[S (μ)]$ gives $S_{μ}$ by [F1], and the expansion of the standard vector in the upper family is $S_{μ} = \sum_{λ \in R P_{α}} d_{(λ, μ)} (q) G^{*} (λ)$ by [F1]; the coefficients are the transition polynomials of [F3].

2.1F1F2step 1.1

Replace the upper basis by the simple classes. By [F2] the vectors $G^{*} (λ)$ are the images of the simple classes $[D (λ)]$ under the same isomorphism; applying $(Ψ_{c a n}^{H e c k e})^{- 1}$ to step 1.1 therefore gives the expansion $[S (μ)] = \sum_{λ} d_{(λ, μ)} (q) [D (λ)]$ in $G_{0}^{g r} (H_{α}^{Λ})$ [F1, F2].

3.1F2F3step 2.1

Read off multiplicities and support. Since the simple classes form a basis of the free $A$ -module $G_{0}^{g r} (H_{α}^{Λ})$ by [F2], the coefficient of $[D (λ)]$ in the expansion of step 2.1 is the graded composition multiplicity $[S (μ) : D (λ)]_{q}$ , and it equals $d_{(λ, μ)} (q)$ ; the support statement $d_{(λ, μ)} = 0$ unless $λ ⊴ μ$ is the triangularity of [F3].

4.1C1F3F4given∎

The ungraded specialization. The forgetful map $ζ$ specializes $q$ to $1$ and sends a graded composition series to an ungraded one with the same multiplicities [F3]; applying $ζ$ to the expansion of step 2.1 gives $[S (μ) : D (λ)] = d_{(λ, μ)} (1)$ , Ariki's ungraded decomposition-number theorem. No new choice is made: [C1] is inherited.

Depends on

Used by

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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)
Susumu Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), J. Math. Kyoto Univ. 36 (1996) 789-808 (standard reference, not scraped)