An open problem of the symmetric group

Many people are often surprised to hear that mathematicians do research. What is mathematical research? Research in mathematics takes many forms, but one common theme is that the research seeks to answer an open question concerning some collection of mathematical objects. The goal of this talk will be to introduce you to one of the many open questions in mathematics: how many commutation classes does the longest element in the symmetric group have? This problem has been nicknamed "Heroin Hero" by my advisor (Richard M. Green) in honor of a game from the TV show "South Park" in which the character Stan obsesses over chasing a dragon. We will review the basics of the symmetric group and introduce all of the necessary terminology, so that we can understand this problem.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar on Friday, September 14, 2012.

Transcript

1. An open problem of the symmetric group
   Dana C. Ernst
   Northern Arizona University
   Mathematics & Statistics Department
   http://danaernst.com
   Friday Afternoon Mathematics Undergraduate Seminar
   September 14, 2012
   D.C. Ernst An open problem of the symmetric group 1 / 11
   

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2. The symmetric group
   Intuitive definition
   Start with a static collection of objects (a set), throw in a method for combining two
   objects together (a binary operation) so that it satisfies some reasonable
   requirements (associative, identity, and inverses), and you’ve got yourself a group.
   Here’s a classic example of a group.
   Definition
   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 rearranges a string of n objects; called a permutation.
   Things to think about:
   • What is the identity permutation?
   • Given a permutation, what is its inverse?
   • How do you compose two permutations?
   D.C. Ernst An open problem of the symmetric group 2 / 11
   

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3. Permutation diagrams & cycle notation
   Two ways of representing elements from Sn
   are via permutation diagrams and via
   cycle notation.
   Example
   Here are some examples from S5
   .
   α =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   = (1 2 3 4 5)
   β =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   = (2 4 3)
   σ =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   = (1 3)(2 5 4)
   γ =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   = (1 5)
   D.C. Ernst An open problem of the symmetric group 3 / 11
   

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4. Permutation diagrams & cycle notation (continued)
   Let’s try multiplying.
   Example
   Using diagrams:
   αβ =
   β
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   α
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   βα =
   α
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   β
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   Using cycle notation:
   αβ = (1 2 3 4 5)(2 4 3) = (1 2 5) βα = (2 4 3)(1 2 3 4 5) = (1 4 5)
   We see that products of permutations do not necessarily commute (order matters).
   D.C. Ernst An open problem of the symmetric group 4 / 11
   

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5. Permutation diagrams & cycle notation (continued)
   However, sometimes permutations do commute.
   Example
   βγ = (2 4 3)(1 5) =
   γ
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   β
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   γβ = (1 5)(2 4 3) =
   β
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   γ
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   So, β and γ commute.
   We’ve stumbled upon the following general fact.
   Theorem
   Products of disjoint cycles commute.
   D.C. Ernst An open problem of the symmetric group 5 / 11
   

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6. The adjacent 2-cycles
   Definition
   In Sn
   , the adjacent 2-cycles are as follows.
   (1 2), (2 3), (3 4), . . . , (n − 2 n − 1), (n − 1 n)
   Example
   The adjacent 2-cycles in S5
   are (1 2), (2 3), (3 4), and (4 5).
   (1 2) =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   (2 3) =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   (3 4) =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   (4 5) =
   q
   q
   q
   q
   q
   q
   q
   q
   q
   q
   Theorem
   Every element in Sn
   can be written as a product of the adjacent 2-cycles. That is,
   the adjacent 2-cycles generate Sn
   .
   D.C. Ernst An open problem of the symmetric group 6 / 11
   

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7. The adjacent 2-cycles (continued)
   It is important to note that there are potentially many different ways to express a
   given permutation as a product of adjacent 2-cycles, but we only need a few tools to
   get from one expression for a permutation to another.
   Theorem
   The symmetric group Sn
   is generated by the adjacent 2-cycles subject only to the
   following relations.
   1. (i i + 1)2 = (1) (2-cycles have order two)
   2. (i i + 1)(j j + 1) = (j j + 1)(i i + 1), where |i − j| > 1 (disjoint cycles commute)
   3. (i i + 1)(i + 1 i + 2)(i i + 1) = (i + 1 i + 2)(i i + 1)(i + 1 i + 2) (braid relations)
   If we express a permutation as a product of adjacent 2-cycles in the most efficient
   way possible, then we call the expression a reduced expression.
   There may be many different reduced expressions for a given permutation, but all of
   them can be written in terms of the same number of adjacent 2-cycles occurring in
   the product (called the length).
   D.C. Ernst An open problem of the symmetric group 7 / 11
   

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8. Commutation classes
   Definition
   Two reduced expressions are commutation equivalent if we can obtain one from the
   other by only commuting disjoint adjacent 2-cycles (no need to apply any braid
   relations). A commutation class of a permutation is the subset of all reduced
   expressions that can be obtained from one another by commuting disjoint cycles.
   Example
   There are 11 reduced expressions for (1 3 5 4) that split into 2 commutation classes:
   (1 2)(2 3)(1 2)(4 5)(3 4) (1 2)(2 3)(4 5)(1 2)(3 4) (1 2)(4 5)(2 3)(1 2)(3 4)
   (1 2)(2 3)(4 5)(3 4)(1 2) (1 2)(4 5)(2 3)(3 4)(1 2) (4 5)(1 2)(2 3)(3 4)(1 2)
   (4 5)(1 2)(2 3)(1 2)(3 4)
   (2 3)(1 2)(2 3)(4 5)(3 4) (2 3)(1 2)(4 5)(2 3)(3 4) (2 3)(4 5)(1 2)(2 3)(3 4)
   (4 5)(2 3)(1 2)(2 3)(3 4)
   D.C. Ernst An open problem of the symmetric group 8 / 11
   

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9. The longest element
   Definition
   The longest element in Sn
   is the (unique) element having maximal length. The
   longest element is usually denoted by w0
   .
   The permutation diagram for w0
   is of the form (n = 5 here):
   r
   r
   r
   r
   r
   r
   r
   r
   r
   r
   The number of reduced expressions for w0
   is known. But what we don’t know is:
   Open problem
   How many commutation classes does the longest element in the symmetric group
   have?
   For a given n, we could work really hard to figure out the answer (the bigger n is, the
   harder we’d have to work). But what we want is a general solution. A (good)
   solution would either be a function of n or a recurrence relation.
   D.C. Ernst An open problem of the symmetric group 9 / 11
   

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10. Example of commutation classes for the longest word
    Example
    In S4
    , the longest element is (1 4)(2 3). In this case, there are 8 commutation classes.
    (2 3)(1 2)(3 4)(2 3)(3 4)(1 2) (2 3)(3 4)(1 2)(2 3)(3 4)(1 2)
    (2 3)(3 4)(1 2)(2 3)(1 2)(3 4) (2 3)(1 2)(3 4)(2 3)(1 2)(3 4)
    (1 2)(3 4)(2 3)(3 4)(1 2)(2 3) (3 4)(1 2)(2 3)(3 4)(1 2)(2 3)
    (1 2)(3 4)(2 3)(1 2)(3 4)(2 3) (3 4)(1 2)(2 3)(1 2)(3 4)(2 3)
    (1 2)(2 3)(3 4)(1 2)(2 3)(1 2) (1 2)(2 3)(1 2)(3 4)(2 3)(1 2)
    (3 4)(2 3)(3 4)(1 2)(2 3)(3 4) (3 4)(2 3)(1 2)(3 4)(2 3)(3 4)
    (2 3)(1 2)(2 3)(3 4)(2 3)(1 2) (2 3)(3 4)(2 3)(1 2)(2 3)(3 4)
    (1 2)(2 3)(3 4)(2 3)(1 2)(2 3) (3 4)(2 3)(1 2)(2 3)(3 4)(2 3)
    D.C. Ernst An open problem of the symmetric group 10 / 11
    

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11. Closing remarks
    • According to the On-Line Encyclopedia of Integer Sequences, the number of
    commutation classes of the longest element in S1, S2, . . . , S11
    is 1, 1, 2, 8, 62,
    908, 24698, 1232944, 112018190, 18410581880, 5449192389984, respectively.
    • Many people have worked on this problem: R. Stanley (& some of his students),
    B. Tenner, A. Bj¨
    orner, D. Knuth, S. Elnitsky, R. Bedard, Bailly, Mosseri,
    Destainville, Widom, Kassel, Lascoux, Reutenauer, H. Denoncourt, me (but only
    a little), etc.
    • My academic brother, Hugh Denoncourt, has spent quite a bit of time working
    on this problem. In fact, he was so obsessed with it that my advisor, Richard M.
    Green, nicknamed the problem “Heroin Hero” after the game by the same name
    that occurred in an episode of South Park in which the character Stan obsesses
    over chasing a dragon that cannot be caught.
    • This problem is related to primitive sorting networks (computer science),
    oriented matroids (math), pseudoline arrangements (math), rhombic tilings
    (math/physics), Schubert cells (math), and stability of quasicrystals (physics).
    • Lastly, please come talk to me if you come up with a solution for arbitrary n.
    D.C. Ernst An open problem of the symmetric group 11 / 11
    

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