Cartan invariants of symmetric groups and Iwahori-Hecke algebras

verfasst von

Christine Bessenrodt, David Hill

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 ℋ_n(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=p1^r1_ri ⋯ pk^rk, the Cartan matrix of an ℓ-block of Sn is a product of Cartan matrices associated to p_i-blocks of S_n. In particular, the invariant factors of the Cartan matrix associated to an ℓ-block of S_n can be recovered from the Cartan matrices associated to the p1^r1 _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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

University of California at Berkeley

Typ

Artikel

Journal

Journal of the London Mathematical Society

Band

81

Seiten

113-128

Anzahl der Seiten

16

ISSN

0024-6107

Publikationsdatum

23.11.2009

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Allgemeine Mathematik

Elektronische Version(en)

https://arxiv.org/abs/0809.4457 (Zugang: Offen)
https://doi.org/10.1112/jlms/jdp060 (Zugang: Geschlossen)