Ordering Families using Lusztig's symbols in type B: the integer case

TL;DR

This paper extends the characterization of Lusztig family orderings in type B Weyl groups to the integer parameter case, connecting Lusztig symbols with the preorder on irreducible representations.

Contribution

It generalizes previous results by showing the order on Lusztig families coincides with Lusztig symbols in the integer parameter case for type B.

Findings

Order on Lusztig families matches Lusztig symbols in the integer case

Extension of previous type B results to a broader parameter setting

Clarifies the structure of irreducible representations in this context

Abstract

Let $\Irr(W)$ be the set of irreducible representations of a finite Weyl group $W$. Following an idea from Spaltenstein, Geck has recently introduced a preorder $\leq_L$ on $\Irr(W)$ in connection with the notion of Lusztig families. In a later paper with Iancu, they have shown that in type $B$ (in the asymptotic case and in the equal parameter case) this order coincides with the order on Lusztig symbols as defined by Geck and the second author in \cite{GJ}. In this paper, we show that this caracterisation extends to the so-called integer case, that is when the ratio of the parameters is an integer.