Ordering Lusztig’s families in type B n

Abstract

Let W be a finite Coxeter group and L be a weight function on W in the sense of Lusztig. We have recently introduced a pre-order relation L on the set of irreducible characters of W which extends Lusztig's definition of "families" and which, conjecturally, corresponds to the ordering given by Kazhdan-Lusztig cells. Here, we give an explicit description of L for W of type B n and any L. (All other cases are known from previous work.) This crucially relies on some new combinatorial constructions around Lusztig's "symbols". Combined with previous work, we deduce general compatibility results between L and Lusztig's a-function, valid for any W, L.

Ordering families using Lusztig’s symbols in type $$B$$ B : the integer case

“…We start by giving an explicit characterisation of the preorder L as in [5], which is in spirit very similar to the one given in Example 2.5 in type A. Then, using our characterisation of the adjacency of bipartitions we will prove the main result of this paper, that is, the order L on bipartition is the same as the order in the integer case.…”

“…has already been proved in [5,Theorem 7.11]. So this article is devoted to the proof of the reverse implication.…”

“…The fact that this order gives rise to families (in other words that the converse of 1. holds) is not straightforward by any means: it is one of the main result of [3] and [5].…”

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

“…The description of the Lusztig families in the degenerate case is given in [21,Example 7.13] and follows from the general theory in [22, §2.4.3]. Lemma 6.12.…”

Orderings on Calogero–Moser partition of imprimitive groups

“…One of the most important areas of representation theory and combinatorics for the last 30 years has been Kazhdan-Lusztig theory, which produces remarkable bases for Iwahori-Hecke algebras of Coxeter groups. A function plays an important role in the representation theory of these algebras: the a-function which gives a partition of Irr W in families and whose order is related to the partial ordering that then exists on this partition when W is a Coxeter group (see [19] and [12]). Now, for the last 15 years it has been understood that it would be extremely interesting to generalize as much of the above theory as possible to Hecke algebras with unequal parameters and to Hecke algebras of complex reflection groups.…”