Cartan invariants of symmetric groups and Iwahori-Hecke algebras

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Authors

  • Christine Bessenrodt
  • David Hill

External Research Organisations

  • University of California at Berkeley
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Details

Original languageEnglish
Pages (from-to)113-128
Number of pages16
JournalJournal of the London Mathematical Society
Volume81
Issue number1
Early online date23 Nov 2009
Publication statusPublished - 23 Nov 2009

Abstract

Külshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the ℓ-Cartan matrix for S n (equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke algebra ℋn(q), where q is a primitive ℓth root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when ℓ=p r, p is prime and r≤p, and went on to conjecture that the formulae should hold for all r. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition ℓ=p1r1ri ⋯ pkrk, the Cartan matrix of an ℓ-block of Sn is a product of Cartan matrices associated to pi-blocks of Sn. In particular, the invariant factors of the Cartan matrix associated to an ℓ-block of Sn can be recovered from the Cartan matrices associated to the p1r1 ri-blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of Sn, not only for the full Cartan matrix, but also for an individual block. We collect evidence for this conjecture by showing that the formulae predict the correct determinant of the ℓ-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of Külshammer, Olsson and Robinson.

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Cartan invariants of symmetric groups and Iwahori-Hecke algebras. / Bessenrodt, Christine; Hill, David.
In: Journal of the London Mathematical Society, Vol. 81, No. 1, 23.11.2009, p. 113-128.

Research output: Contribution to journalArticleResearchpeer review

Bessenrodt C, Hill D. Cartan invariants of symmetric groups and Iwahori-Hecke algebras. Journal of the London Mathematical Society. 2009 Nov 23;81(1):113-128. Epub 2009 Nov 23. doi: 10.1112/jlms/jdp060
Bessenrodt, Christine ; Hill, David. / Cartan invariants of symmetric groups and Iwahori-Hecke algebras. In: Journal of the London Mathematical Society. 2009 ; Vol. 81, No. 1. pp. 113-128.
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