Dana Ernst
April 26, 2013
140

# 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.

April 26, 2013

## 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: [email protected] Typeset using L
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