Type $A$ admissible cells are Kazhdan–Lusztig

Van Minh Nguyen

Abstract

Admissible $W$-graphs were defined and combinatorially characterised by Stembridge in A finiteness theorem for $W$-Graphs (Adv. Math. 229 (2012)). The theory of admissible $W$-graphs was motivated by the need to construct $W$-graphs for Kazhdan–Lusztig cells, which play an important role in the representation theory of Hecke algebras, without computing Kazhdan–Lusztig polynomials. In this paper we show that type $A$ admissible $W$-cells are Kazhdan–Lusztig, as conjectured by Stembridge in his original paper.

Keywords: Coxeter group, Hecke algebra, W-graph, cell.

AMS Subject Classification: Primary 20C08; secondary 20C30.