Bruhat Intervals and Polyhedral Cones

Lecture notes by Matthew Dyer for lectures at the workshop “Coxeter groups and convex geometry.” 1. Root labelled Bruhat graph 1.1. Bruhat order. Let (W,S) be a Coxeter group. Bruhat order is the partial order defined by the following proposition; see [3] and [13] for more details. Proposition. There is a partial order ≤ on W with the following properties. Let w ∈ W and w = s1 . . . sn, n = l0(w), si ∈ S be a reduced expression for w. Then v ≤ w if and only if there exist m ≤ n and 1 ≤ i1 < . . . < im ≤ n such that v = si1 · · · sim. This statement is also true with an extra condition m = l(v) 1.2. Root-labelled Bruhat graph. Fix a based root system (Φ,Π) with respect to (V, 〈−,−〉), with associated Coxeter system (W,S). For some purposes, more general notions of root system including those of algebraic groups, Kac-Moody Lie algebras etc would be more natural. Let l0 denote the standard length function of (W,S). We fix a length function l : W → Z by l = ±l0 (there are many other suitable l; we shall not discuss them though they are required for proofs of some results even for l0). 1.3. Bruhat graph. Bruhat order may also be defined in the following equivalent way. Definition. (a) Define a directed graph Ω = Ω(W,S,l) with vertex set W and edge set E = { (x, sαx) | x ∈ W, l(sαx) > l(x) }. Give Ω(W,S,l) an edge labelling by Φ+ such that the edge (x, y) receives label α ∈ Φ+ if y = sαx, denoted x α −→ y. (b) The order ≤(W,S,l), or ≤l, or ≤, is the partial order on W such that x ≤ y if there is a path x = x0 α1 −→ x1 α2 −→ . . . αn −→ xn = y from x to y in Ω. If l = l0, these are called the (root-labelled) Bruhat graph and Bruhat order ; for l = −l0, one has the reverse Bruhat graph and reverse Bruhat order. We write (in any poset) xl y if y covers x. In the above orders, this implies l(y) = l(x) + 1. Exercise. Describe the graph and its labelling for some (small) dihedral groups, intervals of length ≤ 3, interval [e, srts] in universal (W,S) (no braid relations). Remarks. (1) The Bruhat order has, in important special cases, geometric and representation-theoretic interpretations in terms of inclusions of Schubert varieties and embeddings of Verma modules.