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

A

TEX, TikZ, and beamerposter