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# 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.

## Dana Ernst

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
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
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
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
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
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
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
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
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
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
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