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Permutations Card Shuffling Representation theory Permutations, Card Shuffling, and Representation Theory Franco Saliola, Universit du Qubec Montral Based on joint work with: Victor Reiner, University of Minnesota Volkmar Welker,

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SLIDE 1

Permutations Card Shuffling Representation theory

Permutations, Card Shuffling, and Representation Theory

Franco Saliola, Université du Québec à Montréal

Based on joint work with: Victor Reiner, University of Minnesota Volkmar Welker, Philipps-Universität Marburg

13 July 2013

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SLIDE 2

Permutations Card Shuffling Representation theory

Permutations and Increasing Subsequences

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SLIDE 3

Permutations Card Shuffling Representation theory

Permutations : Definitions and Notations

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SLIDE 4

Permutations Card Shuffling Representation theory

Permutations : Definitions and Notations

Permutation : bijection σ : {1, 2, . . . , n} → {1, 2, . . . , n}

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SLIDE 5

Permutations Card Shuffling Representation theory

Permutations : Definitions and Notations

Permutation : bijection σ : {1, 2, . . . , n} → {1, 2, . . . , n} Example : σ(1) = 3 σ(2) = 1 σ(3) = 2

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SLIDE 6

Permutations Card Shuffling Representation theory

Permutations : Definitions and Notations

Permutation : bijection σ : {1, 2, . . . , n} → {1, 2, . . . , n} Example : σ(1) = 3 σ(2) = 1 σ(3) = 2 2-line notation :

  • 1

2 3 3 1 2

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SLIDE 7

Permutations Card Shuffling Representation theory

Permutations : Definitions and Notations

Permutation : bijection σ : {1, 2, . . . , n} → {1, 2, . . . , n} Example : σ(1) = 3 σ(2) = 1 σ(3) = 2 2-line notation :

  • 1

2 3 3 1 2

  • 1-line notation :

3 1 2

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SLIDE 8

Permutations Card Shuffling Representation theory

Permutations : Definitions and Notations

Permutation : bijection σ : {1, 2, . . . , n} → {1, 2, . . . , n} Example : σ(1) = 3 σ(2) = 1 σ(3) = 2 2-line notation :

  • 1

2 3 3 1 2

  • 1-line notation :

3 1 2 symmetric group Sn : group of permutations of {1, . . . , n}

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SLIDE 9

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
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SLIDE 10

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123

increasing subsequences

  • f length 2

inc2(σ)

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SLIDE 11

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123

increasing

12

subsequences

  • f length 2

inc2(σ)

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SLIDE 12

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123

increasing

12

subsequences

1 3

  • f length 2

inc2(σ)

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SLIDE 13

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123

increasing

12

subsequences

1 3

  • f length 2

23 inc2(σ)

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SLIDE 14

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123

increasing

12

subsequences

1 3

  • f length 2

23 inc2(σ) 3

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SLIDE 15

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132

increasing

12 13

subsequences

1 3

  • f length 2

23 inc2(σ) 3

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SLIDE 16

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132

increasing

12 13

subsequences

1 3 1 2

  • f length 2

23 inc2(σ) 3

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SLIDE 17

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132

increasing

12 13

subsequences

1 3 1 2

  • f length 2

23 inc2(σ) 3 2

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SLIDE 18

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132 213

increasing

12 13 2 3

subsequences

1 3 1 2 13

  • f length 2

23 inc2(σ) 3 2 2

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SLIDE 19

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132 213 231

increasing

12 13 2 3 23

subsequences

1 3 1 2 13

  • f length 2

23 inc2(σ) 3 2 2 1

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SLIDE 20

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132 213 231 312

increasing

12 13 2 3 23 12

subsequences

1 3 1 2 13

  • f length 2

23 inc2(σ) 3 2 2 1 1

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SLIDE 21

Permutations Card Shuffling Representation theory

increasing subsequences Definition inck(σ) = # increasing subsequences

  • f length k in σ
  • Example

σ 123 132 213 231 312 321

increasing

12 13 2 3 23 12

subsequences

1 3 1 2 13

  • f length 2

23 inc2(σ) 3 2 2 1 1

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SLIDE 22

Permutations Card Shuffling Representation theory

Matrix of increasing k-subsequences

Incn,k =    τ . . . σ · · · inck(τ −1σ) · · · . . .   

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SLIDE 23

Permutations Card Shuffling Representation theory

Inc3,1 =

  • inc1(τ −1σ)

          123 132 213 231 312 321 123 132 213 231 312 321           

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SLIDE 24

Permutations Card Shuffling Representation theory

Inc3,1 =

  • inc1(τ −1σ)

          123 132 213 231 312 321 123 3 3 3 3 3 3 132 3 3 3 3 3 3 213 3 3 3 3 3 3 231 3 3 3 3 3 3 312 3 3 3 3 3 3 321 3 3 3 3 3 3           

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SLIDE 25

Permutations Card Shuffling Representation theory

Inc3,2 =

  • inc2(τ −1σ)

          123 132 213 231 312 321 123 132 213 231 312 321           

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SLIDE 26

Permutations Card Shuffling Representation theory

Inc3,2 =

  • inc2(τ −1σ)

          123 132 213 231 312 321 123 3 132 2 213 2 231 1 312 1 321           

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SLIDE 27

Permutations Card Shuffling Representation theory

Inc3,2 =

  • inc2(τ −1σ)

          123 132 213 231 312 321 123 3 2 132 2 3 213 2 1 231 1 312 1 2 321 1           

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SLIDE 28

Permutations Card Shuffling Representation theory

Inc3,2 =

  • inc2(τ −1σ)

          123 132 213 231 312 321 123 3 2 2 1 1 132 2 3 1 2 1 213 2 1 3 2 1 231 1 2 3 1 2 312 1 2 1 3 2 321 1 1 2 2 3           

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SLIDE 29

Permutations Card Shuffling Representation theory

Inc3,3 =

  • inc3(τ −1σ)

          123 132 213 231 312 321 123 1 132 1 213 1 231 1 312 1 321 1           

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SLIDE 30

Permutations Card Shuffling Representation theory

Inc4,1 =

  • inc1(τ −1σ)

                                             4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4                                              

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SLIDE 31

Permutations Card Shuffling Representation theory

Inc4,2 =

  • inc2(τ −1σ)

                                             6 5 5 4 4 3 5 4 4 3 3 2 4 3 3 2 2 1 3 2 2 1 1 0 5 6 4 3 5 4 4 5 3 2 4 3 3 2 2 1 1 0 4 3 3 2 2 1 5 4 6 5 3 4 4 3 3 2 2 1 5 4 4 3 3 2 2 3 1 0 2 1 4 3 5 6 4 5 3 2 2 1 1 0 4 5 3 2 4 3 3 4 2 1 3 2 4 5 3 4 6 5 3 4 2 1 3 2 2 3 1 0 2 1 5 4 4 3 3 2 3 4 4 5 5 6 2 3 1 0 2 1 3 4 2 1 3 2 4 5 3 2 4 3 5 4 4 3 3 2 6 5 5 4 4 3 3 2 4 3 1 2 2 1 3 2 0 1 4 5 3 2 4 3 5 6 4 3 5 4 2 1 3 2 0 1 3 2 4 3 1 2 4 3 3 2 2 1 5 4 6 5 3 4 4 3 5 4 2 3 1 0 2 3 1 2 3 2 2 1 1 0 4 3 5 6 4 5 3 2 4 5 3 4 2 1 3 4 2 3 3 4 2 1 3 2 4 5 3 4 6 5 1 0 2 3 1 2 4 3 5 4 2 3 2 3 1 0 2 1 3 4 4 5 5 6 2 1 3 4 2 3 3 2 4 5 3 4 4 3 5 4 2 3 3 2 4 3 1 2 6 5 5 4 4 3 1 2 0 1 3 2 3 2 4 5 3 4 2 1 3 2 0 1 5 6 4 3 5 4 2 3 1 2 4 3 3 2 4 3 1 2 4 3 5 4 2 3 5 4 6 5 3 4 0 1 1 2 2 3 2 1 3 2 0 1 3 2 4 5 3 4 4 3 5 6 4 5 1 2 2 3 3 4 2 1 3 4 2 3 1 0 2 3 1 2 4 5 3 4 6 5 3 4 2 3 5 4 1 0 2 3 1 2 2 1 3 4 2 3 3 4 4 5 5 6 2 3 3 4 4 5 3 4 2 3 5 4 2 3 1 2 4 3 1 2 0 1 3 2 6 5 5 4 4 3 2 3 3 4 4 5 1 2 0 1 3 2 2 3 1 2 4 3 5 6 4 3 5 4 2 3 1 2 4 3 3 4 2 3 5 4 0 1 1 2 2 3 5 4 6 5 3 4 1 2 0 1 3 2 2 3 3 4 4 5 1 2 2 3 3 4 4 3 5 6 4 5 1 2 2 3 3 4 0 1 1 2 2 3 3 4 2 3 5 4 4 5 3 4 6 5 0 1 1 2 2 3 1 2 2 3 3 4 2 3 3 4 4 5 3 4 4 5 5 6                                              

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SLIDE 32

Permutations Card Shuffling Representation theory

Inc4,3 =

  • inc3(τ −1σ)

                                             4 2 2 1 1 0 2 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 2 4 1 0 2 1 0 2 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 2 1 4 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 0 1 0 2 4 1 2 1 0 0 0 0 0 0 2 0 0 1 1 0 1 0 0 0 0 1 2 0 1 4 2 0 1 0 0 0 0 0 1 0 0 0 0 2 0 1 1 0 0 0 1 1 2 2 4 0 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 1 2 0 1 1 0 0 4 2 2 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 2 0 0 1 1 2 4 1 0 2 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 2 1 4 2 0 1 1 1 2 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 2 4 1 2 0 0 0 2 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 2 0 1 4 2 0 0 0 1 0 0 1 1 2 0 0 0 0 1 0 0 0 0 0 1 1 2 2 4 0 0 0 1 0 0 0 0 0 2 1 1 1 1 2 0 0 0 0 0 1 0 0 0 4 2 2 1 1 0 0 0 0 0 1 0 0 0 0 2 1 1 0 0 1 0 0 0 2 4 1 0 2 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 2 0 0 0 2 1 4 2 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 1 0 2 4 1 2 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 2 0 1 4 2 1 1 0 0 2 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 2 2 4 0 0 1 1 0 2 1 1 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 4 2 2 1 1 0 0 0 1 1 0 2 0 0 0 0 1 0 0 0 0 0 1 0 2 4 1 0 2 1 0 0 0 0 1 0 1 1 0 0 2 0 0 0 0 0 0 1 2 1 4 2 0 1 0 0 0 0 1 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 2 4 1 2 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 2 0 1 2 0 1 4 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 2 0 1 1 2 2 4                                              

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SLIDE 33

Permutations Card Shuffling Representation theory

Inc4,4 =

  • inc4(τ −1σ)

                                             1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                                              

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SLIDE 34

Permutations Card Shuffling Representation theory

Motivation : Mysteries

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SLIDE 35

Permutations Card Shuffling Representation theory

Motivation : Mysteries

  • for each n, we have a family of n matrices

Incn,1, Incn,2, . . . , Incn,n each of which is n! × n!.

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SLIDE 36

Permutations Card Shuffling Representation theory

Motivation : Mysteries

  • for each n, we have a family of n matrices

Incn,1, Incn,2, . . . , Incn,n each of which is n! × n!.

  • these arose from a problem in computer science
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SLIDE 37

Permutations Card Shuffling Representation theory

Motivation : Mysteries

  • for each n, we have a family of n matrices

Incn,1, Incn,2, . . . , Incn,n each of which is n! × n!.

  • these arose from a problem in computer science
  • experimentation suggested some intriguing properties :
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SLIDE 38

Permutations Card Shuffling Representation theory

Motivation : Mysteries

  • for each n, we have a family of n matrices

Incn,1, Incn,2, . . . , Incn,n each of which is n! × n!.

  • these arose from a problem in computer science
  • experimentation suggested some intriguing properties :
  • 1. Incn,i Incn,j = Incn,j Incn,i
slide-39
SLIDE 39

Permutations Card Shuffling Representation theory

Motivation : Mysteries

  • for each n, we have a family of n matrices

Incn,1, Incn,2, . . . , Incn,n each of which is n! × n!.

  • these arose from a problem in computer science
  • experimentation suggested some intriguing properties :
  • 1. Incn,i Incn,j = Incn,j Incn,i
  • 2. the eigenvalues are non-negative integers
slide-40
SLIDE 40

Permutations Card Shuffling Representation theory

Motivation : Mysteries

  • for each n, we have a family of n matrices

Incn,1, Incn,2, . . . , Incn,n each of which is n! × n!.

  • these arose from a problem in computer science
  • experimentation suggested some intriguing properties :
  • 1. Incn,i Incn,j = Incn,j Incn,i
  • 2. the eigenvalues are non-negative integers
  • Questions : Is this true ? Why ? What are these integers ?
slide-41
SLIDE 41

Permutations Card Shuffling Representation theory

Some Surprises

slide-42
SLIDE 42

Permutations Card Shuffling Representation theory

Some Surprises

they do commute !

  • 2011 : we gave an enumerative, inductive proof
slide-43
SLIDE 43

Permutations Card Shuffling Representation theory

Some Surprises

they do commute !

  • 2011 : we gave an enumerative, inductive proof

eigenvalues ?

  • 2013 : recent work with Ton Dieker : explicit formulas
slide-44
SLIDE 44

Permutations Card Shuffling Representation theory

Some Surprises

they do commute !

  • 2011 : we gave an enumerative, inductive proof

eigenvalues ?

  • 2013 : recent work with Ton Dieker : explicit formulas

additional families of intriguing matrices :

  • second family of matrices with similar properties

(obtained by replacing inck with another permutation statistic)

slide-45
SLIDE 45

Permutations Card Shuffling Representation theory

Some Surprises

they do commute !

  • 2011 : we gave an enumerative, inductive proof

eigenvalues ?

  • 2013 : recent work with Ton Dieker : explicit formulas

additional families of intriguing matrices :

  • second family of matrices with similar properties

(obtained by replacing inck with another permutation statistic)

connections with probability and representation theory :

  • card shuffling and related random walks
  • representation theory of the symmetric group
slide-46
SLIDE 46

Permutations Card Shuffling Representation theory

Card Shuffling

slide-47
SLIDE 47

Permutations Card Shuffling Representation theory

random-to-random shuffle

deck of cards :

σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ9

slide-48
SLIDE 48

Permutations Card Shuffling Representation theory

random-to-random shuffle

deck of cards :

σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ9

remove a card at random :

σ3 σ1 σ2 ↑ σ4 σ5 σ6 σ7 σ8 σ9

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SLIDE 49

Permutations Card Shuffling Representation theory

random-to-random shuffle

deck of cards :

σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ9

remove a card at random :

σ3 σ1 σ2 ↑ σ4 σ5 σ6 σ7 σ8 σ9

insert the card at random :

↓ σ1 σ2 σ4 σ5 σ6 σ7 σ3 σ8 σ9

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SLIDE 50

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123 132 213 231 312 321           

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SLIDE 51

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123

3 9

132 213 231 312 321           

3 ways to obtain 123 from 123 : 1 2 3 1 2 3 1 2 3

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SLIDE 52

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123

3 9 2 9

132 213 231 312 321           

2 ways to obtain 132 from 123 : 1 2 3 1 2 3

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SLIDE 53

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123

3 9 2 9 2 9

132 213 231 312 321           

2 ways to obtain 213 from 123 : 1 2 3 1 2 3

slide-54
SLIDE 54

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123

3 9 2 9 2 9 1 9

132 213 231 312 321           

1 way to obtain 231 from 123 : 1 2 3

slide-55
SLIDE 55

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123

3 9 2 9 2 9 1 9 1 9

132 213 231 312 321           

1 way to obtain 312 from 123 : 1 2 3

slide-56
SLIDE 56

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123

3 9 2 9 2 9 1 9 1 9 9

132 213 231 312 321           

slide-57
SLIDE 57

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123 3 2 2 1 1 132 2 3 1 2 1 213 2 1 3 2 1 231 1 2 3 1 2 312 1 2 1 3 2 321 1 1 2 2 3            × 1 9

slide-58
SLIDE 58

Permutations Card Shuffling Representation theory

Transition matrix of the random-to-random shuffle

entries : probability of going from σ to τ using one shuffle

           123 132 213 231 312 321 123 3 2 2 1 1 132 2 3 1 2 1 213 2 1 3 2 1 231 1 2 3 1 2 312 1 2 1 3 2 321 1 1 2 2 3            × 1 9 Incn,n−1

(renorm.)

= random-to-random shuffle

slide-59
SLIDE 59

Permutations Card Shuffling Representation theory

Properties of the transition matrix

The transition matrix T governs properties of the random walk. typical questions ← → algebraic properties probability after m steps ← → entries of Tm long-term behaviour (limiting distribution) ← → eigenvectors π s.t. π T = π rate of convergence

  • v Tn −

→ π ← → governed by eigenvalues of T

slide-60
SLIDE 60

Permutations Card Shuffling Representation theory

random walks on the chambers

  • f a hyperplane arrangement
slide-61
SLIDE 61

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b
slide-62
SLIDE 62

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

the origin

slide-63
SLIDE 63

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

rays emanating from the origin

slide-64
SLIDE 64

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

rays emanating from the origin

slide-65
SLIDE 65

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

rays emanating from the origin

slide-66
SLIDE 66

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

rays emanating from the origin

slide-67
SLIDE 67

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

rays emanating from the origin

slide-68
SLIDE 68

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

rays emanating from the origin

slide-69
SLIDE 69

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

chambers cut out by the hyperplanes

slide-70
SLIDE 70

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

chambers cut out by the hyperplanes

slide-71
SLIDE 71

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

chambers cut out by the hyperplanes

slide-72
SLIDE 72

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

chambers cut out by the hyperplanes

slide-73
SLIDE 73

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

chambers cut out by the hyperplanes

slide-74
SLIDE 74

Permutations Card Shuffling Representation theory

faces of a hyperplane arrangement a set of hyperplanes partitions Rn into faces :

b

chambers cut out by the hyperplanes

slide-75
SLIDE 75

Permutations Card Shuffling Representation theory

product of two faces xy :=

  • the face first encountered after a small

movement along a line from x toward y x y

slide-76
SLIDE 76

Permutations Card Shuffling Representation theory

product of two faces xy :=

  • the face first encountered after a small

movement along a line from x toward y x y

b b

slide-77
SLIDE 77

Permutations Card Shuffling Representation theory

product of two faces xy :=

  • the face first encountered after a small

movement along a line from x toward y x y

b b

slide-78
SLIDE 78

Permutations Card Shuffling Representation theory

product of two faces xy :=

  • the face first encountered after a small

movement along a line from x toward y xy x y

b b

slide-79
SLIDE 79

Permutations Card Shuffling Representation theory

Special Case : The “Braid” Arrangement

hyperplanes :

Hi,j = { v ∈ Rn : vi = vj}

slide-80
SLIDE 80

Permutations Card Shuffling Representation theory

Special Case : The “Braid” Arrangement

hyperplanes :

Hi,j = { v ∈ Rn : vi = vj}

  • consider a vector

v that belongs to a chamber

slide-81
SLIDE 81

Permutations Card Shuffling Representation theory

Special Case : The “Braid” Arrangement

hyperplanes :

Hi,j = { v ∈ Rn : vi = vj}

  • consider a vector

v that belongs to a chamber

  • we can order the entries of

v in increasing order ; e.g. :

v5 < v1 < v3 < v2 < v4

slide-82
SLIDE 82

Permutations Card Shuffling Representation theory

Special Case : The “Braid” Arrangement

hyperplanes :

Hi,j = { v ∈ Rn : vi = vj}

  • consider a vector

v that belongs to a chamber

  • we can order the entries of

v in increasing order ; e.g. :

v5 < v1 < v3 < v2 < v4

  • so chambers correspond to permutations :

v5 < v1 < v3 < v2 < v4 ← →

  • 5, 1, 3, 2, 4
slide-83
SLIDE 83

Permutations Card Shuffling Representation theory

Special Case : The “Braid” Arrangement

hyperplanes :

Hi,j = { v ∈ Rn : vi = vj}

  • consider a vector

v that belongs to a chamber

  • we can order the entries of

v in increasing order ; e.g. :

v5 < v1 < v3 < v2 < v4

  • so chambers correspond to permutations :

v5 < v1 < v3 < v2 < v4 ← →

  • 5, 1, 3, 2, 4
  • if

v lies on Hi,j, then vi < vj becomes vi = vj :

v1 = v5 < v2 = v3 < v4 ← →

  • {1, 5}, {2, 3}, {4}
slide-84
SLIDE 84

Permutations Card Shuffling Representation theory

Special Case : The “Braid” Arrangement combinatorial description : faces ↔ ordered set partitions of {1, . . . , n} :

  • {2, 3}, {4}, {1, 5}
  • =
  • {4}, {1, 5}, {2, 3}
  • chambers ↔ partitions into singletons
  • {2}, {3}, {4}, {1}, {5}
  • product ↔ intersection of sets in the partition
slide-85
SLIDE 85

Permutations Card Shuffling Representation theory

Product of set compositions

slide-86
SLIDE 86

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
slide-87
SLIDE 87

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {2, 5} ∩ {4}
slide-88
SLIDE 88

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • ∅
slide-89
SLIDE 89

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
slide-90
SLIDE 90

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}
  • {1}{5}{6}{3}{2}
  • =
  • {2, 5} ∩ {1}
slide-91
SLIDE 91

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}
  • {1}{5}{6}{3}{2}
  • =
slide-92
SLIDE 92

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}
  • {5}{6}{3}{2}
  • =
  • {2, 5} ∩ {5}
slide-93
SLIDE 93

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}
  • {5}{6}{3}{2}
  • =
  • {5}
slide-94
SLIDE 94

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}
  • {5}{6}{3}{2}
  • =
  • {5}
slide-95
SLIDE 95

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}
  • {6}{3}{2}
  • =
  • {5}{2, 5} ∩ {6}
slide-96
SLIDE 96

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}
  • {6}{3}{2}
  • =
  • {5}
slide-97
SLIDE 97

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}
  • {3}{2}
  • =
  • {5}{2, 5} ∩ {3}
slide-98
SLIDE 98

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}
  • {2}
  • =
  • {5}{2, 5} ∩ {2}
slide-99
SLIDE 99

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}
  • {2}
  • =
  • {5}{2}
slide-100
SLIDE 100

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {5}{2}{1, 3, 4, 6} ∩ {4}
slide-101
SLIDE 101

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}
slide-102
SLIDE 102

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}
  • {1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}{1, 3, 4, 6} ∩ {1}
slide-103
SLIDE 103

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}
  • {1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}{1}
slide-104
SLIDE 104

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}
  • {5}{6}{3}{2}
  • =
  • {5}{2}{4}{1}{1, 3, 4, 6} ∩ {5}
slide-105
SLIDE 105

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}
  • {6}{3}{2}
  • =
  • {5}{2}{4}{1}{1, 3, 4, 6} ∩ {6}
slide-106
SLIDE 106

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}
  • {6}{3}{2}
  • =
  • {5}{2}{4}{1}{6}
slide-107
SLIDE 107

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}
  • {3}{2}
  • =
  • {5}{2}{4}{1}{6}{1, 3, 4, 6} ∩ {3}
slide-108
SLIDE 108

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}
  • {3}{2}
  • =
  • {5}{2}{4}{1}{6}{3}
slide-109
SLIDE 109

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}
  • {1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}
  • {2}
  • =
  • {5}{2}{4}{1}{6}{3}{1, 3, 4, 6} ∩ {2}
slide-110
SLIDE 110

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}{1}{6}{3}
slide-111
SLIDE 111

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}{1}{6}{3}
slide-112
SLIDE 112

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}{1}{6}{3}
slide-113
SLIDE 113

Permutations Card Shuffling Representation theory

Product of set compositions

  • {2, 5}{1, 3, 4, 6}
  • ·
  • {4}{1}{5}{6}{3}{2}
  • =
  • {5}{2}{4}{1}{6}{3}
slide-114
SLIDE 114

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c.

slide-115
SLIDE 115

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [1, 2, 3, 4, 5, 6]

slide-116
SLIDE 116

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6]

slide-117
SLIDE 117

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6]

slide-118
SLIDE 118

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [2, 5, 1, 3, 4, 6]

slide-119
SLIDE 119

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6]

slide-120
SLIDE 120

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4]

slide-121
SLIDE 121

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4] [2, 1, 6, 5, 3, 4]

slide-122
SLIDE 122

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4] [{3}{1, 2, 4, 5, 6}] × [2, 1, 6, 5, 3, 4]

slide-123
SLIDE 123

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4] [{3}{1, 2, 4, 5, 6}] × [2, 1, 6, 5, 3, 4] = [3, 2, 1, 6, 5, 4]

slide-124
SLIDE 124

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4] [{3}{1, 2, 4, 5, 6}] × [2, 1, 6, 5, 3, 4] = [3, 2, 1, 6, 5, 4]

slide-125
SLIDE 125

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4] [{3}{1, 2, 4, 5, 6}] × [2, 1, 6, 5, 3, 4] = [3, 2, 1, 6, 5, 4] (Inverse) Riffle Shuffle

slide-126
SLIDE 126

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements A step in the random walk : starting from an element c, pick an element y at random, and move to y × c. Example : [{2, 5}{1, 3, 4, 6}] × [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6] [{1, 2, 6}{3, 4, 5}] × [2, 5, 1, 3, 4, 6] = [2, 1, 6, 5, 3, 4] [{3}{1, 2, 4, 5, 6}] × [2, 1, 6, 5, 3, 4] = [3, 2, 1, 6, 5, 4] (Inverse) Riffle Shuffle and Random-to-Top

slide-127
SLIDE 127

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements

Introduced by Bidigare–Hanlon–Rockmore (1999) :

  • computed eigenvalues of the transition matrices
  • presents a unified approach to several random walks
slide-128
SLIDE 128

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements

Introduced by Bidigare–Hanlon–Rockmore (1999) :

  • computed eigenvalues of the transition matrices
  • presents a unified approach to several random walks

Further developed by Brown–Diaconis (1998) :

  • described stationary distribution
  • proved diagonalizability of transition matrices
slide-129
SLIDE 129

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements

Introduced by Bidigare–Hanlon–Rockmore (1999) :

  • computed eigenvalues of the transition matrices
  • presents a unified approach to several random walks

Further developed by Brown–Diaconis (1998) :

  • described stationary distribution
  • proved diagonalizability of transition matrices

Extended by Brown (2000) :

  • extended results to larger class of examples
  • used algebraic techniques and representation theory
slide-130
SLIDE 130

Permutations Card Shuffling Representation theory

Random walks on hyperplane arrangements

Introduced by Bidigare–Hanlon–Rockmore (1999) :

  • computed eigenvalues of the transition matrices
  • presents a unified approach to several random walks

Further developed by Brown–Diaconis (1998) :

  • described stationary distribution
  • proved diagonalizability of transition matrices

Extended by Brown (2000) :

  • extended results to larger class of examples
  • used algebraic techniques and representation theory

Others : Björner, Athanasiadis-Diaconis, Chung-Graham, . . .

slide-131
SLIDE 131

Permutations Card Shuffling Representation theory

Incn,k and hyperplane chamber walks Theorem (Reiner–S–Welker 2011)

Incn,k = Tk Tt

k

where Tk is the transition matrix of a random walk on the braid arrangement.

slide-132
SLIDE 132

Permutations Card Shuffling Representation theory

Incn,k and hyperplane chamber walks Theorem (Reiner–S–Welker 2011)

Incn,k = Tk Tt

k

where Tk is the transition matrix of a random walk on the braid arrangement. Consequently, ker(Incn,k) = ker(Tk)

slide-133
SLIDE 133

Permutations Card Shuffling Representation theory

Representation theory

slide-134
SLIDE 134

Permutations Card Shuffling Representation theory

Representation Theory

A representation of a group G is a group homomorphism ρ : G − → Matd

slide-135
SLIDE 135

Permutations Card Shuffling Representation theory

Representation Theory

A representation of a group G is a group homomorphism ρ : G − → Matd That is,

  • ρ(g) is a (d × d)–matrix
slide-136
SLIDE 136

Permutations Card Shuffling Representation theory

Representation Theory

A representation of a group G is a group homomorphism ρ : G − → Matd That is,

  • ρ(g) is a (d × d)–matrix
  • ρ(gh) = ρ(g)ρ(h)
slide-137
SLIDE 137

Permutations Card Shuffling Representation theory

Representation Theory

A representation of a group G is a group homomorphism ρ : G − → Matd That is,

  • ρ(g) is a (d × d)–matrix
  • ρ(gh) = ρ(g)ρ(h)

Examples :

  • trivial representation :

ρ(g) = [1]

slide-138
SLIDE 138

Permutations Card Shuffling Representation theory

Regular representation of S3

ρ(123) =  

1 · · · · · · 1 · · · · · · 1 · · · · · · 1 · · · · · · 1 · · · · · · 1

  ρ(132) =  

· 1 · · · · 1 · · · · · · · · 1 · · · · 1 · · · · · · · · 1 · · · · 1 ·

  ρ(213) =  

· · 1 · · · · · · · 1 · 1 · · · · · · · · · · 1 · 1 · · · · · · · 1 · ·

  ρ(231) =  

· · · 1 · · · · · · · 1 · 1 · · · · · · · · 1 · 1 · · · · · · · 1 · · ·

  ρ(312) =  

· · · · 1 · · · 1 · · · · · · · · 1 1 · · · · · · · · 1 · · · 1 · · · ·

  ρ(321) =  

· · · · · 1 · · · 1 · · · · · · 1 · · 1 · · · · · · 1 · · · 1 · · · · ·

 

slide-139
SLIDE 139

Permutations Card Shuffling Representation theory

Regular representation of S3

ρ(123) =  

1 · · · · · · 1 · · · · · · 1 · · · · · · 1 · · · · · · 1 · · · · · · 1

  ρ(132) =  

· 1 · · · · 1 · · · · · · · · 1 · · · · 1 · · · · · · · · 1 · · · · 1 ·

  ρ(213) =  

· · 1 · · · · · · · 1 · 1 · · · · · · · · · · 1 · 1 · · · · · · · 1 · ·

  ρ(231) =  

· · · 1 · · · · · · · 1 · 1 · · · · · · · · 1 · 1 · · · · · · · 1 · · ·

  ρ(312) =  

· · · · 1 · · · 1 · · · · · · · · 1 1 · · · · · · · · 1 · · · 1 · · · ·

  ρ(321) =  

· · · · · 1 · · · 1 · · · · · · 1 · · 1 · · · · · · 1 · · · 1 · · · · ·

  3ρ(123) + 2ρ(132) + 2ρ(213) + 1ρ(231) + 1ρ(312) =  

3 2 2 1 1 0 2 3 1 0 2 1 2 1 3 2 0 1 1 0 2 3 1 2 1 2 0 1 3 2 0 1 1 2 2 3

 

slide-140
SLIDE 140

Permutations Card Shuffling Representation theory

Incn,k and the regular representation of Sn Theorem (Reiner–S–Welker 2011)

Incn,k =

  • σ∈Sn

inck(σ) reg(σ)

where reg is the regular representation of Sn.

slide-141
SLIDE 141

Permutations Card Shuffling Representation theory

Calculation of eigenvalues via irreducibles

(decompose the regular representation into irreducible representations)

3·123 2 · 132 2 · 213 1 · 231 1 · 312

slide-142
SLIDE 142

Permutations Card Shuffling Representation theory

Calculation of eigenvalues via irreducibles

(decompose the regular representation into irreducible representations)

3·123 2 · 132 2 · 213 1 · 231 1 · 312 trivial : 3 ·

  • 1
  • + 2 ·
  • 1
  • + 2 ·
  • 1
  • + 1 ·
  • 1
  • + 1 ·
  • 1
  • =
  • 9
slide-143
SLIDE 143

Permutations Card Shuffling Representation theory

Calculation of eigenvalues via irreducibles

(decompose the regular representation into irreducible representations)

3·123 2 · 132 2 · 213 1 · 231 1 · 312 sign : 3 ·

  • 1
  • + 2 ·
  • −1
  • + 2 ·
  • −1
  • + 1 ·
  • 1
  • + 1 ·
  • 1
  • =
  • 1
slide-144
SLIDE 144

Permutations Card Shuffling Representation theory

Calculation of eigenvalues via irreducibles

(decompose the regular representation into irreducible representations)

3·123 2 · 132 2 · 213 1 · 231 1 · 312 χ : 3·[ 1 0

0 1 ] + 2·[ 1 0 1 −1 ] + 2·

0 −1

−1 0

  • + 1·

−1 1

−1 0

  • + 1·

0 −1

1 −1

  • =

4 −2

slide-145
SLIDE 145

Permutations Card Shuffling Representation theory

Calculation of eigenvalues via irreducibles

(decompose the regular representation into irreducible representations)

3·123 2 · 132 2 · 213 1 · 231 1 · 312 χ : 3·[ 1 0

0 1 ] + 2·[ 1 0 1 −1 ] + 2·

0 −1

−1 0

  • + 1·

−1 1

−1 0

  • + 1·

0 −1

1 −1

  • =

4 −2

  • eigs(Inc3,2) = {9, 1, 4, 4, 0, 0}
slide-146
SLIDE 146

Permutations Card Shuffling Representation theory

Final Remarks

  • This research was facilitated by computer exploration with the
  • math. software Sage (sagemath.org) and Sage-Combinat :
  • to test the conjectures
  • to compute the eigenvalues of the matrices
  • to construct irreducible representations of Sn
  • to decompose the eigenspaces into irreducible representations
  • to search for other transition matrices with these properties
  • to provide ideas for proofs
  • Second family of random walks with similar properties.

Our analysis combines what we’ve seen today :

  • BHR theory of random walks to analyze the kernels ; and
  • representation theory to analyze the other eigenspaces.