Classification of the T-avoiding permutations and generalizations to other Coxeter groups

A permutation of a set of objects is simply a rearrangement of those objects. If we have n of objects, then a permutation can be represented as a function from 1,2,,n to 1,2,,n. We say that a permutation w has property T if there exists i such that either w(i) is greater than w(i + 1), w(i + 2) or w(i + 2) is less than w(i), w(i + 1). A permutation w is T-avoiding if neither w nor its inverse have property T. We will present a classification of the T-avoiding permutations in the symmetric group, which is a Coxeter group of type A. In addition, we will discuss generalizations to other Coxeter groups and classify the T-avoiding elements in Coxeter groups of types B, which can be thought of as the group that rearranges coins and flips them over.

This poster was presented by my undergraduate research students Joseph Cormier, Zachariah Goldenberg, and Jessica Kelly (Plymouth State University) on April 27, 2012 at the 2012 PSU Research Symposium at Plymouth State University.

Transcript

1. Classification of the T-avoiding permutations
   and generalizations to other Coxeter groups
   J. Cormier, Z. Goldenberg, J. Kelly, Plymouth State University
   s4
   s4
   s3
   s1
   s2
   s5
   Symmetric Group
   The symmetric group Sn
   is the collection of bijections from {1, 2, . . . , n} to
   {1, 2, . . . , n} where the operation is function composition (left ← right). Each
   element of Sn
   is called a permutation.
   Cycle Notation
   One way of representing permutations is via cycle notation.
   Example
   If σ = (1 3 5 2), then σ(1) = 3, σ(3) = 5, σ(5) = 2, σ(2) = 1, σ(4) = 4.
   String Diagrams
   A 2nd way of representing permutations is via string diagrams. Given a permutation
   σ, there are many ways to draw associated string diagrams. Conventions:
   1. no more than 2 strings cross each other at given point,
   2. strings are drawn so as to minimize crossings.
   Example
   The following string diagram corresponds
   to the permutation σ = (1 3 5 2).
   Theorem
   Sn
   is generated by the adjacent 2-cycles: s1
   = (1 2), s2
   = (2 3), . . . , sn−1
   = (n−1 n).
   Relations of Symmetric Group
   Sn
   satisfies the following relations:
   1. s2
   i
   = 1 for all i (2-cycles have order 2)
   2. short braid relations: si
   sj
   = sj
   si
   , for |i − j| ≥ 2
   3. long braid relations: si
   sj
   si
   = sj
   si
   sj
   , for |i − j| = 1.
   Reduced Expressions
   If sx1
   sx2
   · · · sxm
   is an expression for σ ∈ Sn
   and m is minimal, then we say that the
   expression is reduced. By Matsumoto’s Theorem, any two reduced expressions for
   σ ∈ Sn
   differ by a sequence of braid relations.
   Example
   Consider σ = s2
   s1
   s2
   s3
   s1
   s2
   ∈ S4
   . We see that
   s2
   s1
   s2
   s3
   s1
   s2
   = s1
   s2
   s1
   s3
   s1
   s2
   = s1
   s2
   s1
   s1
   s3
   s2
   = s1
   s2
   s3
   s2.
   The original expression is not reduced, but it turns out that last expression is. The
   only reduced expressions for σ are: s1
   s2
   s3
   s2, s1
   s3
   s2
   s3, s3
   s1
   s2
   s3
   .
   Heaps
   A 3rd way of representing permutations is via heaps. Fix a reduced expression
   sx1
   sx2
   · · · sxm
   for σ ∈ Sn
   . Loosely speaking, the heap for this expression is a set
   of lattice points (called nodes) in N × N, one for each sxi
   such that:
   1. The node corresponding to sxi
   has vertical component equal to n + 1 − xi
   ,
   2. If i < j and sxi
   and sxj
   do not commute, then sxi
   is left of sxj
   .
   Example
   There are two distinct heaps for the reduced expressions from previous example:
   s3
   s2
   s1
   s2 and s3
   s2
   s1
   s3
   Correspondence Between String Diagrams & Heaps
   There is a 1-1 correspondence between
   string diagrams and heaps. In the absence
   of a node, the string “bounces.” s4
   s4
   s3
   s1
   s2
   Property T
   A permutation σ has Property T iff there exists i such that
   1. σ(i) > σ(i + 1), σ(i + 2),
   i
   i + 1
   i + 2
   or
   2. σ(i + 2) < σ(i), σ(i + 1).
   i
   i + 1
   i + 2
   Example
   If σ = (1 3 5 2), then σ and σ−1 have
   Property T in 1 and 2 spots, respectively.
   T-avoiding
   σ is T-avoiding iff both σ and σ−1 do not have Property T .
   Example
   The permutation σ = (1 2)(4 5) is T-avoiding.
   s4
   s1
   Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon)
   σ is T-avoiding iff σ is a product of disjoint adjacent 2-cycles (iff heap consists of a
   single column iff string diagram consists of “bars” & “X’s”).
   Sketch of Proof
   Fix a reduced expression for σ and consider its heap. The reverse implication of the
   theorem is trivial. For the forward direction, consider the contrapositive.
   The Easy Case
   The easy case occurs when a node in the 2nd column in on either side is ”blocked”
   by at most one node in the 1st column. There are 4 possibilities:
   blocked 1x
   i
   i + 1
   i + 2
   i
   i + 1
   i + 2
   blocked 1x
   i
   i + 1
   i + 2
   i
   i + 1
   i + 2
   The Hard Case
   i
   i + 1
   i + 2
   blocked 2x
   apply braids up & right
   By applying a sequence
   of long braid relations,
   you can convert a heap
   in the hard case to a
   heap in the easy case.
   Coxeter Groups
   A Coxeter group consists of a group W together with a generating set S consisting
   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 elements of order 2, 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
   Coxeter groups of Types A and B
   The symmetric group Sn+1
   with the adjacent 2-cycles as a generating set is a Coxeter
   group of type An
   . Coxeter groups of type Bn
   (n ≥ 2) having generating set S =
   {s1, s2, · · · , sn
   } and defining relations:
   1. s2
   i
   = 1 for all i,
   2. si
   sj
   = sj
   si
   if |i − j| > 1,
   3. si
   sj
   si
   = sj
   si
   sj
   if |i − j| = 1 and 1 < i, j ≤ n,
   4. s1
   s2
   s1
   s2
   = s2
   s1
   s2
   s1
   .
   Generalization of Property T to Coxeter Groups
   Let (W , S) be a Coxeter group and let w ∈ W . Then w has Property T iff w has a
   reduced expression of the form stu or uts, where m(s, t) ≥ 3 and u ∈ W . In terms
   of heaps, w is T-avoiding iff no heap for w has the property that a node in the 2nd
   column in on either side is ”blocked” by at most one node in outer column.
   Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon)
   In types A and B, w ∈ W is T-avoiding iff w is a product of commuting generators.
   Concluding Remarks
   1. Our advisor has classified the T-avoiding elements in type C, which consists of
   more than just products of commuting generators (e.g.,“sandwich stacks”).
   2. Tyson Gern (University of Colorado) has classified the T-avoiding elements in type
   D. Again, classification is more complicated than just products of commuting
   generators.
   3. Our advisor is currently working with a group of students on the classification in
   type F.
   Joint work with J. Cormier, Z. Goldenberg, J. Kelly, and C. Malbon. Research conducted under the guidance of D.C. Ernst, Plymouth State University Typeset using L
   ATEX, TikZ, PSTricks, and beamerposter
   

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