Investigation of T-avoiding elements of Coxeter groups

Coxeter groups can be thought of as generalized reflections groups. In particular, a Coxeter group is generated by a set of elements of order two. Every element of a Coxeter group can be written as an expression in the generators, and if the number of generators in an expression is minimal, we say that the expression is reduced. We say that an element w of a Coxeter group is T-avoiding if w does not have a reduced expression beginning or ending with a pair of non-commuting generators. In this talk, we will state the known results concerning T-avoiding elements and discuss our current work in classifying the T-avoiding elements in Coxeter groups of type F.

This poster was presented by my undergraduate research student Selina Gilbertson on April 26, 2013 at the 2013 NAU Undergraduate Research Symposium at Northern Arizona University.

Transcript

1. Investigations of the T-avoiding elements in Coxeter groups of type 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: sjg74@nau.edu Typeset using L
   A
   TEX, TikZ, and beamerposter
   

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