Specht modules and Kazhdan--Lusztig cells in type $B_n$

TL;DR

This paper demonstrates that Specht modules for type B Hecke algebras are equivalent to Kazhdan--Lusztig cell modules under the dominance order, highlighting the importance of parameter ordering in representation theory.

Contribution

It establishes the natural equivalence between Specht modules and Kazhdan--Lusztig cell modules specifically under the dominance order in type B, extending previous asymptotic results.

Findings

Equivalence holds under dominance order

Counterexamples for other monomial orders

Highlights the role of parameter ordering in module theory

Abstract

Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig cell modules. In this paper, we show that the Specht modules are naturally equivalent to the Kazhdan--Lusztig cell modules {\em if} we choose the dominance order on the parameters, as in the ``asymptotic case'' studied by Bonnaf\'e and the second named author. We also give examples which show that such an equivalence does not hold for other choices of monomial orders.