------------------------------------------------ Weyl Groups, Coxeter Groups and the Bruhat Order ------------------------------------------------ Classical and affine Weyl groups -------------------------------- You can create Weyl groups and affine Weyl groups for any root system. A variety of methods are available for these. Some of these are methods are available for general Coxeter groups. By default, elements of the Weyl group are represented as matrices:: sage: WeylGroup("A3").simple_reflection(1) [0 1 0 0] [1 0 0 0] [0 0 1 0] [0 0 0 1] You may prefer a notation in which elements are written out as products of simple reflections. In order to implement this you need to specify a prefix, typically "s":: sage: W = WeylGroup("A3",prefix="s") sage: [s1,s2,s3] = W.simple_reflections() sage: (s1*s2*s1).length() 3 sage: W.long_element() s1*s2*s3*s1*s2*s1 sage: s1*s2*s3*s1*s2*s1 == s3*s2*s1*s3*s2*s3 True The Weyl group acts on the ambient space of the root lattice, which is accessed by the method domain. To illustrate this, recall that if w_0 is the long element then \alpha \mapsto -w_0(\alpha) is a permutation of the simple roots. We may compute this as follows:: sage: W = WeylGroup("E6",prefix="s") sage: w0 = W.long_element(); w0 s1*s3*s4*s5*s6*s2*s4*s5*s3*s4*s1*s3*s2*s4*s5*s6*s2*s4*s5*s3*s4*s1*s3*s2*s4*s5*s3*s4*s1*s3*s2*s4*s1*s3*s2*s1 sage: sr = W.domain().simple_roots().list(); sr [(1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (1, 1, 0, 0, 0, 0, 0, 0), (-1, 1, 0, 0, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0), (0, 0, -1, 1, 0, 0, 0, 0), (0, 0, 0, -1, 1, 0, 0, 0)] sage: [-w0.action(a) for a in sr] [(0, 0, 0, -1, 1, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0), (0, 0, -1, 1, 0, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0), (-1, 1, 0, 0, 0, 0, 0, 0), (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2)] We may ask when this permutation is trivial. If it is nontrivial it induces an automorphism of the Dynkin diagram, so it must be nontrivial when the Dynkin diagram has no automorphism. But if there is a nontrivial automorphism, the permutation might or might not be trivial:: sage: def roots_not_permuted(ct): ....: W = WeylGroup(ct) ....: w0 = W.long_element() ....: sr = W.domain().simple_roots() ....: return all(a == -w0.action(a) for a in sr) ....: sage: for ct in [CartanType(['D',r]) for r in [2..8]]: ....: print ct,roots_not_permuted(ct) ....: ['D', 2] True ['D', 3] False ['D', 4] True ['D', 5] False ['D', 6] True ['D', 7] False ['D', 8] True If \alpha is a root let r_\alpha denote the reflection in the hyperplane that is orthogonal to \alpha. We reserve the notation s_\alpha for the simple reflections, that is, the case where \alpha is a simple root. The reflections are just the conjugates of the simple reflections. The reflections are the keys in a finite family, which is a wrapper around a python dictionary. The values are the positive roots, so given a reflection, you can look up the corresponding root. If you want a list of all reflections, use the method keys for the family of reflections:: sage: W = WeylGroup("B3",prefix="s") sage: [s1,s2,s3] = W.simple_reflections() sage: ref = W.reflections(); ref Finite family {s1*s2*s1: (1, 0, -1), s2: (0, 1, -1), s3*s2*s3: (0, 1, 1), s3*s1*s2*s3*s1: (1, 0, 1), s1: (1, -1, 0), s2*s3*s1*s2*s3*s1*s2: (1, 1, 0), s1*s2*s3*s2*s1: (1, 0, 0), s2*s3*s2: (0, 1, 0), s3: (0, 0, 1)} sage: ref[s3*s2*s3] (0, 1, 1) sage: ref.keys() [s1*s2*s1, s2, s3*s2*s3, s2*s3*s1*s2*s3*s1*s2, s1, s3*s1*s2*s3*s1, s1*s2*s3*s2*s1, s2*s3*s2, s3] If instead you want a dictionary whose keys are the roots and whose values are the reflections, you may use the inverse family:: sage: W = WeylGroup("B3",prefix="s") sage: [s1,s2,s3] = W.simple_reflections() sage: altref = W.reflections().inverse_family(); altref Finite family {(1, 0, 0): s1*s2*s3*s2*s1, (1, 0, 1): s3*s1*s2*s3*s1, (0, 1, 0): s2*s3*s2, (0, 1, -1): s2, (1, 0, -1): s1*s2*s1, (0, 1, 1): s3*s2*s3, (1, 1, 0): s2*s3*s1*s2*s3*s1*s2, (0, 0, 1): s3, (1, -1, 0): s1} sage: [a1,a2,a3] = W.domain().simple_roots() sage: a1+a2+a3 (1, 0, 0) sage: altref[a1+a2+a3] s1*s2*s3*s2*s1 The Weyl group is implemented as a GAP matrix group. You therefore can display its character table. The character table is returned as a string, which you can print:: sage: print WeylGroup("D4").character_table() CT1 2 6 4 5 1 3 5 5 4 3 3 1 4 6 3 1 . . 1 . . . . . . 1 . 1 1a 2a 2b 6a 4a 2c 2d 2e 4b 4c 3a 4d 2f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 1 1 -1 -1 -1 1 1 1 X.3 2 . 2 -1 . 2 2 . . . -1 2 2 X.4 3 -1 -1 . 1 -1 3 -1 1 -1 . -1 3 X.5 3 -1 -1 . 1 3 -1 -1 -1 1 . -1 3 X.6 3 1 -1 . -1 -1 3 1 -1 1 . -1 3 X.7 3 1 -1 . -1 3 -1 1 1 -1 . -1 3 X.8 3 -1 3 . -1 -1 -1 -1 1 1 . -1 3 X.9 3 1 3 . 1 -1 -1 1 -1 -1 . -1 3 X.10 4 -2 . -1 . . . 2 . . 1 . -4 X.11 4 2 . -1 . . . -2 . . 1 . -4 X.12 6 . -2 . . -2 -2 . . . . 2 6 X.13 8 . . 1 . . . . . . -1 . -8 Affine Weyl groups ------------------ Affine Weyl groups may be created the same way. You simply begin with an affine Cartan type:: sage: W = WeylGroup(['A',2,1],prefix="s") sage: W.cardinality() +Infinity sage: [s0,s1,s2] = W.simple_reflections() sage: s0*s1*s2*s1*s0 s0*s1*s2*s1*s0 The affine Weyl group differs from a classical Weyl group since it is infinite. The associated classical Weyl group is a subgroup that may be extracted as follows:: sage: W = WeylGroup(['A',2,1],prefix="s") sage: W1 = W.classical(); W1 Parabolic Subgroup of the Weyl Group of type ['A', 2, 1] (as a matrix group acting on the root space) sage: W1.simple_reflections() Finite family {1: s1, 2: s2} Although W1 in this example is isomorphic to WeylGroup("A2") it has a different matrix realization:: sage: for s in WeylGroup(['A',2,1]).classical().simple_reflections(): ....: print s ....: print ... [ 1 0 0] [ 1 -1 1] [ 0 0 1] [ 1 0 0] [ 0 1 0] [ 1 1 -1] sage: for s in WeylGroup(['A',2]).simple_reflections(): ....: print s ....: print ... [0 1 0] [1 0 0] [0 0 1] [1 0 0] [0 0 1] [0 1 0] Bruhat order ------------ The Bruhat partial order on the Weyl group may be defined as follows. If u,v \in W, find a reduced expression of v into a product of simple reflections: v = s_1 \cdots s_n. (It is not assumed that the s_i are distinct.) If omitting some of the s_i gives a product that represents u, then u \le v. The Bruhat order is implemented in Sage as a method of Coxeter groups, and so it is available for Weyl groups, classical or affine. If u, v \in W then u.bruhat_le(v) returns true of u \le v in the Bruhat order. If u \le v then The *Bruhat interval* [u,v] is defined to be the set of all t such that u \le t \le v. One might try to implement this as follows:: sage: W = WeylGroup("A2",prefix="s") sage: [s1,s2] = W.simple_reflections() sage: def bi(u,v) : return [t for t in W if u.bruhat_le(t) and t.bruhat_le(v)] ... sage: bi(s1,s1*s2*s1) [s1*s2*s1, s1*s2, s2*s1, s1] This would not be a good definition since it would fail if W is affine and be inefficient of W is large. Sage has a Bruhat interval method:: sage: W = WeylGroup("A2",prefix="s") sage: [s1,s2] = W.simple_reflections() sage: W.bruhat_interval(s1,s1*s2*s1) [s1*s2*s1, s2*s1, s1*s2, s1] This works even for affine Weyl groups. The Bruhat graph ---------------- References: - [Carrell1994]_ - [Deodhar1977]_ - [Dyer1993]_ - [BumpNakasuji2010]_ The *Bruhat Graph* is a structure on the Bruhat interval. Suppose that u \le v. The vertices of the graph are x with u \le x \le v. There is a vertex connecting x,y \in [x,y] if x = y.r where r is a reflection. If this is true then either x < y or y < x. If W is a classical Weyl group the Bruhat graph is implemented in Sage:: sage: W = WeylGroup("A3",prefix="s") sage: [s1,s2,s3] = W.simple_reflections() sage: bg = W.bruhat_graph(s2,s2*s1*s3*s2); bg Digraph on 10 vertices sage: bg.show3d() The Bruhat graph has interesting regularity properties that were investigated by Carrell and Peterson. It is a regular graph if both the Kazhdan Lusztig polynomials P_{u,v} and P_{w_0v,w_0u} are 1, where w_0 is the long Weyl group element. It is closely related to the *Deodhar conjecture* which was proved by Deodhar, Carrell and Peterson, Dyer and Polo. Deodhar proved that if u < v then the Bruhat interval [u,v] contains as many elements of odd length as it does of even length. We observe that often this can be strengthened: if there exists a reflection r such that left (or right) multiplication by r takes the Bruhat interval [u,v] to itself, then this gives an explicit bijection between the elements of odd and even length in [u,v]. Let us search for such reflections. Put the following commands in a file and load the file:: W = WeylGroup("A3",prefix="s") [s1,s2,s3] = W.simple_reflections() ref = W.reflections().keys() def find_reflection(u,v): bi = W.bruhat_interval(u,v) ret = [] for r in ref: if all( r*x in bi for x in bi): ret.append(r) return ret for v in W: for u in W.bruhat_interval(1,v): if u != v: print u,v,find_reflection(u,v) This shows that the Bruhat interval is stabilized by a reflection for all pairs (u,v) with u < v except the following two: s_3s_1,s_1s_2s_3s_2s_1 and s_2,s_2s_3s_1s_2. Now these are precisely the pairs such that u\prec v in the notation of Kazhdan and Lusztig, and l(v)-l(u) > 1. One should not rashly suppose that this is a general characterization of the pairs (u,v) such that no reflection stabilizes the Bruhat interval, for this is not true, but it does suggest that the question is worthy of further investigation.