Cyclotomic Carter-Payne homomorphisms

Lyle, Sinéad; Mathas, Andrew

doi:10.1090/s1088-4165-2014-00450-3

Cited by 2 publications

(3 citation statements)

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“…, L n (Brundan [19] in the degenerate case when v 2 = 1 and announced when v 2 = 1 by Graham and Francis building on [42]). • The blocks of H n are indexed by the same combinatorial data (Lyle and Mathas [96] when v 2 = 1 and Brundan [19] when v 2 = 1). • The irreducible H n -modules are indexed by the crystal graph of the integral highest weight module L(Λ) for U q ( sl e ) (Ariki [3] when v 2 = 1 and Brundan and Kleshchev [23] when v 2 = 1).…”

Section: Cyclotomic Hecke Algebras Of Type Amentioning

confidence: 99%

“…. , L n (Ariki [3]); if ℓ > 1 then they have only one block (Lyle and Mathas [96]); their irreducible modules are indexed by a different set (Mathas [103]); they do not categorify L(Λ) and no non-trivial grading on these algebras is known. In this sense, the definition of the Ariki-Koike algebras from [10] gives the wrong algebras when v 2 = 1.…”

Section: Cyclotomic Hecke Algebras Of Type Amentioning

confidence: 99%

“…Parts (a) and (b) are equivalent by the general theory of cellular algebras [48] whereas the equivalence of parts (b) and (c) is a general property of Jantzen filtrations from [96]. (In fact, part (c) is general property of the standard modules of a quasi-hereditary algebra.)…”

Section: Multipartitions and Tableauxmentioning

confidence: 99%

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CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

Mathas

2015

Modular Representation Theory of Finite and p-Adic Groups

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Contents n 1.6. Semisimple cyclotomic Hecke algebras of type A 1.7. Gram determinants and the Jantzen sum formula 1.8. The blocks of H F n 2. Cyclotomic quiver Hecke algebras of type A 2.1. Graded algebras 2.2. Cyclotomic quiver Hecke algebras 2.3. Nilpotence and small representations 2.4. Semisimple KLR algebras 2.5. The nil-Hecke algebra 3. Isomorphisms, Specht modules and categorification 3.1. The Graded Isomorphism Theorem 3.2. Graded Specht modules 3.3. Blocks and dual Specht modules 3.4. Induction and restriction 3.5. Grading Ariki's Categorification Theorem 3.6. Homogeneous Garnir relations 3.7. Graded adjustment matrices 4. Seminormal bases and the KLR grading 4.1. Gram determinants and graded dimensions 4.2. A deformation of the KLR grading 4.3. A distinguished homogeneous basis 4.4. A simple conjecture References

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Section: Cyclotomic Hecke Algebras Of Type Amentioning

confidence: 99%

Section: Cyclotomic Hecke Algebras Of Type Amentioning

confidence: 99%

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CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

Mathas

2015

Modular Representation Theory of Finite and p-Adic Groups

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Modular Decomposition Numbers of Cyclotomic Hecke and Diagrammatic Cherednik Algebras: A Path Theoretic Approach

Bowman

Cox

2018

Forum of Mathematics, Sigma

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We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a "super-strong linkage principle" which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalise the notion of homomorphisms between Weyl/Specht modules which are "generically" placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic. arXiv:1706.07128v2 [math.RT] 3 May 2018Inspired by these results, we provide the first conjectural framework for calculating the (graded) decomposition numbers of cyclotomic Hecke algebras over fields of sufficiently large characteristic in Subsection 10.2. We verify these conjectures in the cases of (i) maximal finite parabolic orbits with k arbitrary -generalising all the results of [Kle97, CMT08, TT13a] (ii) = 2 or 3 with e > n and k arbitrary -generalising [HM15, Theorems B3 & B5] and [BS11, Section 9] (iii) = 2 and h = 1 and k arbitrary [CGM03, Section 8] (iv) k = C. For = 1 the conjecture was formulated by Andersen [And98, Section 5] and proved by Riche and Williamson in [RW16, Theorem 1.9]. Conjecture A. Let e > h , κ ∈ I be an h-admissible multicharge, and k be field of characteristic p h . The decomposition numbers of A h (n, θ, κ) are given by d λµ (t) = n λµ (t)for λ, µ ∈ P n (h) in the first ep-alcove and n λµ (t) the associated affine parabolic Kazhdan-Lusztig polynomial of type DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS 3Our treatment covers the quotient algebras Q ,h,n (κ) for e ∈ Z >0 uniformly alongside the algebras H n (κ) for e = ∞ (of more generally e > n). The alcove geometries controlling the latter family of algebras are seen as simple sub-cases of those controlling the former family of algebras. The following conjecture (a refinement of [KR11, Conjecture 7.3] which was debunked in [Wil14, Section 4.2]) is the corresponding simplification of Conjecture A.Conjecture B. Let e = ∞ or more generally e > n and suppose κ ∈ I has no repeated entries. Let k be field of characteristic p . The decomposition numbers of A(n, θ, κ) are given by d λµ (t) = n λµ (t)for λ, µ ∈ P n and n λµ (t) the associated parabolic Kazhdan-Lusztig polynomials of type A

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Cyclotomic Carter-Payne homomorphisms

Cited by 2 publications

References 20 publications

CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

Modular Decomposition Numbers of Cyclotomic Hecke and Diagrammatic Cherednik Algebras: A Path Theoretic Approach

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