On the decomposition numbers of the Hecke algebra of $G(m, 1, n)$

Monodromy of Partial KZ Functors for Rational Cherednik Algebras

“…If we calculate by lifting paths to h r , then a representative of ψ(Σ) is given by the path γ n It follows from the argument in [2, Theorem 6.1] that this crystal equals the crystal defined from Uglov's canonical basis of the Fock space provided that we know that the decomposition matrix of the Hecke algebra is given by the evaluation of Uglov's canonical basis at 1. This is the well-known result of Ariki, [1]. Thus we have an explicit identification of the so-called KZ-component of the crystal with the combinatorial crystal on Uglov multipartitions.…”

A diagrammatic approach to categorification of quantum groups I

“…The work of Ariki [1] can be viewed as a categorification of the restricted dual of U − (g) for g = sl N and g = sl N and a categorification of all irreducible integrable representations of these Lie algebras (see also [27], [2], [3], [37]). An integral version of the restricted dual of U − (g) becomes the sum of Grothendieck groups of suitable blocks of affine Hecke algebra representations.…”

Generalized Blob Algebras and Alcove Geometry

“…In general, we may not assume that these representations will factor through any particular Hecke quotient, but if one does, then it could provide a generalisation of the extension of 1 2 F to B n -braid. (40) verifies equation (1). We also require that R(U 1 e − U 1 − k − U 1 ) = 0 for some k − (the relation (5) is not sufficient to ensure this).…”

“…Affine Hecke algebras are currently the subject of widespread interest in representation theory [1,31]. It is customary to study them through certain families of quotient algebras, among which the usual choice is the cyclotomic Hecke algebras [2,10].…”