Dana Ernst
April 27, 2012
150

# Classification of the T-avoiding elements in Coxeter groups of type F

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 classify the T-avoiding elements of type F_5. We conjecture that our classification holds more generally for F_n with n ≥ 5.

This poster was presented by my undergraduate research students Ryan Cross, Katie Hills-Kimball, and Christie Quaranta (Plymouth State University) on April 27, 2012 at the 2012 PSU Research Symposium at Plymouth State University.

April 27, 2012

## Transcript

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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:
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 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
ATEX, TikZ, and beamerposter