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Classiﬁcation of the T-avoiding elements in Coxeter

groups of type F

R. Cross, K. Hills-Kimball, C. Quaranta, Plymouth State University

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Abstract

In mathematics, one uses groups to study symmetry. In particular, a reﬂection group is used to study the reﬂection and rotational symmetry of an object. A Coxeter group can be thought of as a generalized reﬂection group, where the group is generated by a set

of elements of order two (i.e., reﬂections) and there are rules for how the generators interact with each other. 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. An element w of a Coxeter group is called T-avoiding if w does not have a reduced expression beginning or ending with a pair of non-commuting generators. We will present a classiﬁcation of the T-avoiding elements in the inﬁnite

Coxeter group of type F5

, as well as the ﬁnite Coxeter group of type F4

. We conjecture that our classiﬁcation holds more generally for arbitrary Fn

.

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 short braid relations

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. (Typically labels of m(s, t) = 3

are omitted.)

Note that edges correspond to non-commuting pairs of generators. Also, given X,

we can uniquely reconstruct the corresponding (W , S).

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. si

sj

si

= sj

si

sj

whenever |i − j| = 1,

3. Generators corresponding to non-connected nodes commute.

It turns out that An

is isomorphic to the symmetric group Symn+1

, where each si

corresponds to the adjacent transposition (i i + 1).

Coxeter groups of type B

The Coxeter group of type Bn

is deﬁned by the graph below.

s1

s2

s3

4

· · ·

sn−1

sn

Then Bn

is subject to:

1. s2

i

= 1 for all i,

2. s1

s2

s1

s2

= s2

s1

s2

s1

,

3. si

sj

si

= sj

si

sj

whenever |i − j| = 1 and {i, j} = {1, 2},

4. Generators corresponding to non-connected nodes commute.

In this case, Bn

is isomorphic to the group that rearranges and ﬂips n − 1 coins.

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. s2

s3

s2

s3

= s3

s2

s3

s2

,

3. si

sj

si

= sj

si

sj

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 s1

s3

s2

s1

s2

for an element w ∈ W (A3

). Note that

s1

s3

s2

s1

s2

= s1

s3

s1

s2

s1

= s3

s1

s1

s2

s1

= s3

s2

s1.

Therefore, s1

s3

s2

s1

s2

is not reduced. However, the expression on the right is reduced.

Theorem (Matsumoto/Tits)

Any two reduced expressions for w ∈ W diﬀer by a sequence of braid relations.

Heaps

One way of representing reduced expressions is via heaps. Fix a reduced expression

sx1

sx2

· · · sxm

for w. For any straight line Coxeter graph, the heap for this expression

is a set of lattice points, one for each sxi

, embedded in N × N such that:

1. The node corresponding to sxi

has vertical component equal to n + 1 − xi

(i.e.,

smaller numbers at the top),

2. If i < j and sxi

does not commute with sxj

, then sxi

occurs to the left of sxj

.

Example

Consider s1

s2

s3

s2

, s1

s3

s2

s3

, and s3

s1

s2

s3

, which are all reduced expressions of the same

element in A3

. It turns out, there are two distinct heaps:

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2 and

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2

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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 reduced expressions are bad in F5

:

Bowtie M Seagull

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We can convert Bowties ↔ M’s ↔ Seagulls via braid relations. Thus, these ex-

pressions represent the same group element and we can restrict our attention to the

bowties. By stacking bowties we can create inﬁnitely many bad elements in F5

.

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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.

Conjecture

An element is T-avoiding in Fn

for n ≥ 5 iﬀ it is a product of commuting generators

or a stack of bowties times a product of commuting generators.

Joint work with R. Cross, K. Hills-Kimball, and C. Quaranta. Research conducted under the guidance of D.C. Ernst, Plymouth State University Typeset using L

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