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

Abstract

Let $$\text {Irr}(W)$$ Irr ( W ) be the set of irreducible representations of a finite Weyl group $$W$$ W . Following an idea from Spaltenstein, Geck has recently introduced a preorder $$\preceq _L$$ ? L on $$\text {Irr}(W)$$ Irr ( W ) in connection with the notion of Lusztig families. In a later paper with Iancu, they have shown that in type $$B$$ B (in the asymptotic case and in the equal parameter case) this preorder coincides with the preorder on Lusztig symbols as defined by Geck and the second author in 2011. In this paper, we show that this characterisation extends to the so-called integer case, that is, when the ratio of the parameters is an integer.