Type: Seminar

Distribution: World

CalTitle1: Burrull - The Bruhat graph of a Coxeter group

Auth: romanova@203.54.36.254 (arom8272) in SMS-SAML

Let (W,S) be a Coxeter system with length function l. Here arises a very natural question in combinatorics. For each positive integer n, how many types of intervals [x, y] of length n (i.e. l(y)-l(x)=n) are possible? In 1991, Dyer provided a partial answer for finite Coxeter groups by introducing the "Bruhat graph" associated to (W, S) [Dye91]. He proved that there are only finitely many types of intervals of fixed length. On the other hand, the seminal paper [KL79] introduced for elements x,y of the Coxeter group the so-called Kazhdan-Lusztig polynomials Px,y(q). In this paper, they conjectured that the polynomials Px,y only depend on the poset type of [x, y], this conjecture is known as the "combinatorial invariance conjecture". Dyer’s answer gives support to this conjecture, which is nowadays - even in finite type - a major unsolved problem in combinatorics. In this talk I will define the Bruhat graph (which carries more information than the Bruhat order) and reflection subgroups. I will state the main theorem of [Dye90] and other useful propositions. Finally, I will explain the proof of the main result of [Dye91] thus providing an answer to our original question. I will provide many illustrative examples to keep your feet on the ground during the talk.