Hecke Algebras with Unequal Parameters

Kazhdan–lusztig Cells and the Murphy Basis

“…There are two types of Kazhdan-Lusztig bases, denoted by {C w | w ∈ W } and {C ′ w | w ∈ W } in the original article by Kazhdan and Lusztig [10]. In [17], Lusztig writes c w instead of C ′ w , but we shall adhere to the older notation here. The precise definitions are as follows.…”

Constructible characters and canonical bases

“…In this section we review following [Lu2,Lu4] the definition of constructible characters and families for Iwahori-Hecke algebras.…”

“…Recall from 3.5 the partition of Irr W m given by the connected components of the graph Lu4] 22.2, 23.1) that χ S and χ S ′ belong to the same class if and only if the symbols S, S ′ ∈ Sy(n, k, r, m) have the same content, that is, the same elements with the same multiplicities. Using Equation (5), it is easy to deduce the following alternative description: Note that when H m (Q; q) R is semisimple, every bipartition is a Kleshchev bipartition.…”

“…Let Irr H m = {χ (λ,µ) | |λ| + |µ| = m} be the set of irreducible characters of H m and ConH m the set of constructible characters [Lu2,Lu4]. Then our main result states that the specialisation at v = 1 of B(Λ) can be identified to m ConH m , in the following sense: if we write for a vector b ∈ B(Λ) of principal degree m,…”

Equivalence classes in the Weyl groups of type B n

“…As in [10], a Coxeter group W and a weight function L on W can be used to partition W into left cells. When W is of type B n , the weight function can be identified with a positive scalar s, and the description of the corresponding left cells is known when s = 1, i.e.…”