University of Sydney Algebra Seminar

Anne Thomas

Friday 29 August, 12-1pm, in Carslaw 175

Trivial Kazhdan-Lusztig polynomials and cubulation of the Bruhat graph

Let \( (W,S) \) be any Coxeter system and let \(y\) be any element of \(W\).  We investigate the relationship between the condition that the Kazhdan-Lusztig polynomial \(P_{x,y}\) is trivial for all \(x\) less than or equal to \(y\) in Bruhat order, and the condition that the Bruhat graph for the interval \([1,y]\) can be cubulated, meaning roughly that this graph can be spanned by a product of subintervals of the integers. In one direction, we combine results of Carrell-Peterson and Elias-Williamson to prove that if the Bruhat graph for \([1,y]\) can be cubulated, then \(P_{x,y} = 1\) for all \(x \leq y\). We then consider the converse, using word combinatorics in Coxeter groups, methods from geometric group theory, and computer experiments. We establish that the converse is not true in some exceptional types, but it is true in many other cases.