ar X iv : m at h / 06 07 45 1 v 4 [ m at h . R T ] 1 7 Ju n 20 07 BLOCKS OF CYCLOTOMIC HECKE ALGEBRAS

ar X iv : m at h / 06 07 45 1 v 1 [ m at h . R T ] 1 9 Ju l 2 00 6 BLOCKS OF AFFINE AND CYCLOTOMIC HECKE ALGEBRAS

This paper classifies the blocks of the affine Hecke algebras of type A and the blocks of the cyclotomic Hecke algebras of type G(r, 1, n) over an arbitrary algebraically closed 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 char...

X iv : m at h . Q A / 0 50 40 89 v 4 1 2 Ju n 20 06 GENERALIZED DOUBLE AFFINE HECKE ALGEBRAS OF HIGHER RANK

ar X iv : m at h / 04 05 17 6 v 4 [ m at h . R T ] 1 6 Ju n 20 06 QUANTIZED SYMPLECTIC OSCILLATOR ALGEBRAS OF RANK ONE

A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with Uq(sl2). We study its representation theory, and in particular, its category O.

ar X iv : 0 90 1 . 02 18 v 1 [ m at h . R T ] 2 J an 2 00 9 GRADED SPECHT MODULES

Recently, the first two authors have defined a Z-grading on group algebras of symmetric groups and more generally on the cyclotomic Hecke algebras of type G(l, 1, d). In this paper we explain how to grade Specht modules over these algebras.

ar X iv : m at h . R T / 0 11 13 06 v 1 2 9 N ov 2 00 1 CARTAN DETERMINANTS AND SHAPOVALOV FORMS

We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of p-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity.