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

Dana Ernst
January 06, 2012

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 January 6, 2012 at the 2012 Joint Mathematics Meetings in Boston, MA.

Dana Ernst

January 06, 2012
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  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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