BLOCKS OF CYCLOTOMIC HECKE ALGEBRAS

By S. Lyle, Andrew Mathas. 2006

122 citations recorded by the source corpus.

Abstract This paper classifies the blocks of the cyclotomic Hecke algebras of type G ( r , 1 , n ) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of ‘Jantzen equivalence.’ We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999. Unfortunately, Cox has discovered that this previous proof is incomplete.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3ad4bee1d8d0ecac37f0779f043db08b521660cd

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In Discrete Mathematics and Combinatorics

1. Jonathan Brundan
2. Alexander S. Kleshchev
3. Arun Ram
4. Raphaël Rouquier
5. Andrew Mathas
6. E. Vasserot
7. P. Shan
8. Stephen Griffeth

In Geometry and Topology

1. Alexander S. Kleshchev
2. Jonathan Brundan
3. Ben Webster
4. Arun Ram
5. Raphaël Rouquier
6. Andrew Mathas
7. E. Vasserot
8. Aaron D. Lauda

Citations in the corpus

• Seminormal forms and Gram determinants for cellular algebras (2006, methodology): “[14] for Hk;qðSnÞ and the Ariki-Koike algebras, and [3] for the degenerate Ariki-Koike algebras.”
• Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A (2009, unknown): “Then by the main result of [25], H Λ β is a block of H Λ n .”
• Universal graded Specht modules for cyclotomic Hecke algebras (2011, background): “By [20] and [2, Theorem 1], eα is either zero or it is a primitive central idempotent in H Λ d .”
• Generalized Jack polynomials and the representation theory of rational Cherednik algebras (2010, background): “In [ 12 ] Lyle and Mathas finished the classification of the blocks of cyclotomic Hecke algebras (and showed that they are the same as those for the affine Hecke algebra).”
• Centers of degenerate cyclotomic Hecke algebras and parabolic category (2006, methodology): “Nevertheless, Lyle and Mathas [ LM ] have recently managed to solve the problem of classifying the blocks of the cyclotomic Hecke algebras too, using the analogue of Jantzen’s sum formula for cyclotomic q-Schur algebras from [JM].”
• Weights of multipartitions and representations of Ariki–Koike algebras (2006, background): “This classification has now been proved by Lyle and Mathas [ 15, Theorem 2.11 ], but their proof uses the results of the present paper.”
• Categorifications and cyclotomic rational double affine Hecke algebras (2013, background): “, the algebra H s K,α “ 1αHsK,d is either zero or a single block of HK,d [6, 38].”
• Representation theory of symmetric groups and related Hecke algebras (2009, unknown): “As a consequence of [143] or [28, Theorem 1], eα is either zero or it is a primitive central idempotent in HΛ d .”
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras (2008, background): “sition 4.1]. So, given also 20 JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV α ∈Q+ of height d, the idempotent eα from (1.3) is either zero or it is a central idempotent in HΛ d . In fact, it follows from [LM] that the nonzero eα’s are precisely the primitive central idempotents of HΛ d (although we don’”