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.

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
   . . .
   
   
   
   
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   
   
   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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   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
   

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