F Selina Gilbertson, Directed by Dana C. Ernst Department of Mathematics & Statistics, Northern Arizona University Deﬁnition A Coxeter system consists of a group W (called a Coxeter group) generated by a set S of elements of order 2 with presentation W = S : s2 = 1, (st)m(s,t) = 1 , where m(s, t) ≥ 2 for s = t. Since s and t are their own inverses, the relation (st)m(s,t) = 1 can be rewritten as m(s, t) = 2 =⇒ st = ts commutations m(s, t) = 3 =⇒ sts = tst m(s, t) = 4 =⇒ stst = tsts . . . long braid relations Deﬁnition We can encode (W , S) with a unique Coxeter graph X having: 1. vertex set S; 2. edges {s, t} labeled m(s, t) whenever m(s, t) ≥ 3. (Omit labels of m(s, t) = 3) Note that edges correspond to non-commuting pairs of generators. Coxeter groups of type A The Coxeter group of type An is deﬁned by the graph below. s1 s2 s3 · · · sn−1 sn Then An is subject to: 1. s2 i = 1 for all i, 2. sisjsi = sjsisj whenever |i − j| = 1, 3. Generators corresponding to non-connected nodes commute. Coxeter groups of type F The Coxeter group of type Fn (n ≥ 4) is deﬁned by the graph below. s1 s2 s3 s4 4 · · · sn−1 sn Then Fn is subject to: 1. s2 i = 1 for all i, 2. s2s3s2s3 = s3s2s3s2 , 3. sisjsi = sjsisj whenever |i − j| = 1 and {i, j} = {2, 3}, 4. Generators corresponding to non-connected nodes commute. It turns out that F4 is a ﬁnite group while Fn for n ≥ 5 is an inﬁnite group. Deﬁnition A word sx1 sx2 · · · sxm ∈ S∗ is called an expression for w ∈ W if it is equal to w when considered as a group element. If m is minimal, it is a reduced expression. Example Consider the expression s1s3s2s1s2s3 for an element w ∈ W (A3 ). Note that s1s3s2s1s2s3 = s1s3s1s2s1s3 = s3s1s1s2s1s3 = s3s2s1s3 = s3s2s3s1 = s2s3s2s1 reduced . Therefore, s1s3s2s1s2 is not reduced. However, the expressions on the right are reduced. Theorem (Matsumoto/Tits) Any two reduced expressions for w ∈ W diﬀer by a sequence of braid relations and commutations. Heaps One way of representing reduced expressions is via heaps. Fix a reduced expression sx1 sx2 · · · sxm for w ∈ W (for any straight line Coxeter graph). Loosely speaking, the heap for this expression is a set of lattice points, one for each sxi , embedded in N×N, subject to contraints illustrated by example. Example Consider s1s2s3s2 , s1s3s2s3 , and s3s1s2s3 , which are all reduced expressions of the same element in A3 . It turns out, there are two distinct heaps: 1 2 3 2 and 1 3 2 3 Deﬁnition We say that w ∈ W has Property T iﬀ some reduced expression begins or ends with a product of non-commuting generators. That is, w = s t (other generators) or w = (other generators) t s Deﬁnition We say that w is T-avoiding iﬀ w does not have Property T. Proposition Products of commuting generators are T-avoiding. Question Are there other T-avoiding elements besides products of commuting generators? Deﬁnition An element is bad iﬀ it is T-avoiding, but not a product of commuting generators. Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon) In types A and B, there are no bad elements. That is, in types A and B, w is T-avoiding iﬀ w is a product of commuting generators. Comment The answer is not so simple in other Coxeter groups. In particular, there are bad elements in types C (Ernst) and D (Tyson Gern). Proposition (Cross, Ernst, Hills-Kimball, Quaranta) The following heap (called a bowtie) corresponds to a bad element in F5 : 1 3 5 2 4 3 2 4 1 3 5 Stacking in type F5 We can also stack bowties to create inﬁnitely many bad elements in F5 . 1 3 5 2 4 3 2 4 1 3 5 2 4 3 2 4 1 3 5 · · · 1 3 5 2 4 3 2 4 1 3 5 Theorem (Cross, Ernst, Hills-Kimball, Quaranta) An element is T-avoiding in F5 iﬀ it is a product of commuting generators or a stack of bowties. Moreover, there are no bad elements in F4 . That is, the only T-avoiding elements in F4 are products of commuting generators. Proposition (Ernst, Gilbertson) The following heap corresponds to a bad element in F6 : 2 4 6 3 5 2 4 1 3 2 1 3 2 4 3 5 2 4 6 As in F5 , we can stack these elements to create inﬁnitely many bad elements in F6 . 2 4 6 3 5 2 4 1 3 2 1 3 2 4 3 5 2 4 6 3 5 2 4 1 3 2 1 3 2 4 3 5 2 4 6 · · · 2 4 6 3 5 2 4 1 3 If n is even, we can create bad elements in Fn using a similar construction. (However, when n is large, the outer walls of each heap block do not need to be the same size.) Stacking in type F6 Stacking in Fn with n > 5 gets more complicated. 2 4 6 3 5 2 4 1 3 2 1 3 2 4 3 5 2 4 6 3 5 7 2 4 6 8 3 5 7 2 4 6 3 5 2 4 1 3 2 1 3 2 4 3 5 2 4 6 · · · The above heap shows how we can connect heaps with diﬀerently sized outer walls. An open question to this problem is whether we can add ones in the open spaces in the middle and ensure the element still does not have Property-T. Future work If n is even, are there other bad elements in Fn that we have not thought of? Proof? We have noticed that when n is large and even, we can insert some extra 1’s. How awful can this get? What happens with Fn when n is odd and larger than 5? What happens in other types? Email:

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