Permutations, Card Shuffling, and Representation Theorysaliola/maths/talks/slides/cumc-2013.pdfPermutations Card Shuﬄing Representation theory Permutations, Card Shuﬄing, and Representation

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


Permutations

and

Increasing Subsequences


Permutations : Definitions and Notations



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




Example : σ(1) = 3 σ(2) = 1 σ(3) = 2




Example : σ(1) = 3 σ(2) = 1 σ(3) = 2

2-line notation :

(

1 2 33 1 2

)




Example : σ(1) = 3 σ(2) = 1 σ(3) = 2

2-line notation :

(

1 2 33 1 2

)

1-line notation : 3 1 2




Example : σ(1) = 3 σ(2) = 1 σ(3) = 2

2-line notation :

(

1 2 33 1 2

)

1-line notation : 3 1 2

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


increasing subsequences

Definition

inck(σ) = #

{


of length k in σ

}



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123

increasing

subsequences

of length 2

inc2(σ)



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123

increasing 12

subsequences

of length 2

inc2(σ)



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123

increasing 12

subsequences 1 3

of length 2

inc2(σ)



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123

increasing 12

subsequences 1 3

of length 2 23

inc2(σ)



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123

increasing 12

subsequences 1 3

of length 2 23

inc2(σ) 3



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132

increasing 12 13

subsequences 1 3

of length 2 23

inc2(σ) 3



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132

increasing 12 13

subsequences 1 3 1 2

of length 2 23

inc2(σ) 3



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132

increasing 12 13

subsequences 1 3 1 2

of length 2 23

inc2(σ) 3 2



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132 213

increasing 12 13 2 3

subsequences 1 3 1 2 13

of length 2 23

inc2(σ) 3 2 2



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132 213 231

increasing 12 13 2 3 23


of length 2 23

inc2(σ) 3 2 2 1



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132 213 231 312

increasing 12 13 2 3 23 12


of length 2 23

inc2(σ) 3 2 2 1 1



Definition

inck(σ) = #

{


of length k in σ

}

Example

σ 123 132 213 231 312 321

increasing 12 13 2 3 23 12


of length 2 23

inc2(σ) 3 2 2 1 1 0


Matrix of increasing k-subsequences

Incn,k =

τ...

σ · · · inck(τ−1σ) · · ·

...


Inc3,1 =[

inc1(τ−1σ)

]

123 132 213 231 312 321

123

132

213

231

312

321


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


Inc3,2 =[

inc2(τ−1σ)

]

123 132 213 231 312 321

123

132

213

231

312

321


Inc3,2 =[

inc2(τ−1σ)

]

123 132 213 231 312 321

123 3

132 2

213 2

231 1

312 1

321 0


Inc3,2 =[

inc2(τ−1σ)

]

123 132 213 231 312 321

123 3 2

132 2 3

213 2 1

231 1 0

312 1 2

321 0 1


Inc3,2 =[

inc2(τ−1σ)

]

123 132 213 231 312 321

123 3 2 2 1 1 0

132 2 3 1 0 2 1

213 2 1 3 2 0 1

231 1 0 2 3 1 2

312 1 2 0 1 3 2

321 0 1 1 2 2 3


Inc3,3 =[

inc3(τ−1σ)

]

123 132 213 231 312 321

123 1 0 0 0 0 0

132 0 1 0 0 0 0

213 0 0 1 0 0 0

231 0 0 0 1 0 0

312 0 0 0 0 1 0

321 0 0 0 0 0 1


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 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4


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 05 6 4 3 5 4 4 5 3 2 4 3 3 2 2 1 1 0 4 3 3 2 2 15 4 6 5 3 4 4 3 3 2 2 1 5 4 4 3 3 2 2 3 1 0 2 14 3 5 6 4 5 3 2 2 1 1 0 4 5 3 2 4 3 3 4 2 1 3 24 5 3 4 6 5 3 4 2 1 3 2 2 3 1 0 2 1 5 4 4 3 3 23 4 4 5 5 6 2 3 1 0 2 1 3 4 2 1 3 2 4 5 3 2 4 35 4 4 3 3 2 6 5 5 4 4 3 3 2 4 3 1 2 2 1 3 2 0 14 5 3 2 4 3 5 6 4 3 5 4 2 1 3 2 0 1 3 2 4 3 1 24 3 3 2 2 1 5 4 6 5 3 4 4 3 5 4 2 3 1 0 2 3 1 23 2 2 1 1 0 4 3 5 6 4 5 3 2 4 5 3 4 2 1 3 4 2 33 4 2 1 3 2 4 5 3 4 6 5 1 0 2 3 1 2 4 3 5 4 2 32 3 1 0 2 1 3 4 4 5 5 6 2 1 3 4 2 3 3 2 4 5 3 44 3 5 4 2 3 3 2 4 3 1 2 6 5 5 4 4 3 1 2 0 1 3 23 2 4 5 3 4 2 1 3 2 0 1 5 6 4 3 5 4 2 3 1 2 4 33 2 4 3 1 2 4 3 5 4 2 3 5 4 6 5 3 4 0 1 1 2 2 32 1 3 2 0 1 3 2 4 5 3 4 4 3 5 6 4 5 1 2 2 3 3 42 1 3 4 2 3 1 0 2 3 1 2 4 5 3 4 6 5 3 4 2 3 5 41 0 2 3 1 2 2 1 3 4 2 3 3 4 4 5 5 6 2 3 3 4 4 53 4 2 3 5 4 2 3 1 2 4 3 1 2 0 1 3 2 6 5 5 4 4 32 3 3 4 4 5 1 2 0 1 3 2 2 3 1 2 4 3 5 6 4 3 5 42 3 1 2 4 3 3 4 2 3 5 4 0 1 1 2 2 3 5 4 6 5 3 41 2 0 1 3 2 2 3 3 4 4 5 1 2 2 3 3 4 4 3 5 6 4 51 2 2 3 3 4 0 1 1 2 2 3 3 4 2 3 5 4 4 5 3 4 6 50 1 1 2 2 3 1 2 2 3 3 4 2 3 3 4 4 5 3 4 4 5 5 6


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 02 4 1 0 2 1 0 2 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 02 1 4 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 01 0 2 4 1 2 1 0 0 0 0 0 0 2 0 0 1 1 0 1 0 0 0 01 2 0 1 4 2 0 1 0 0 0 0 0 1 0 0 0 0 2 0 1 1 0 00 1 1 2 2 4 0 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 12 0 1 1 0 0 4 2 2 1 1 0 0 0 1 0 0 0 0 0 1 0 0 00 2 0 0 1 1 2 4 1 0 2 1 0 0 1 0 0 0 0 0 1 0 0 01 0 0 0 0 0 2 1 4 2 0 1 1 1 2 0 0 0 0 0 0 1 0 01 0 0 0 0 0 1 0 2 4 1 2 0 0 0 2 1 1 0 0 0 1 0 00 1 0 0 0 0 1 2 0 1 4 2 0 0 0 1 0 0 1 1 2 0 0 00 1 0 0 0 0 0 1 1 2 2 4 0 0 0 1 0 0 0 0 0 2 1 11 1 2 0 0 0 0 0 1 0 0 0 4 2 2 1 1 0 0 0 0 0 1 00 0 0 2 1 1 0 0 1 0 0 0 2 4 1 0 2 1 0 0 0 0 1 00 0 1 0 0 0 1 1 2 0 0 0 2 1 4 2 0 1 0 0 0 0 0 10 0 1 0 0 0 0 0 0 2 1 1 1 0 2 4 1 2 0 0 0 0 0 10 0 0 1 0 0 0 0 0 1 0 0 1 2 0 1 4 2 1 1 0 0 2 00 0 0 1 0 0 0 0 0 1 0 0 0 1 1 2 2 4 0 0 1 1 0 21 1 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 4 2 2 1 1 00 0 1 1 0 2 0 0 0 0 1 0 0 0 0 0 1 0 2 4 1 0 2 10 0 0 0 1 0 1 1 0 0 2 0 0 0 0 0 0 1 2 1 4 2 0 10 0 0 0 1 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 2 4 1 20 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 2 0 1 2 0 1 4 20 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 2 0 1 1 2 2 4


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 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1


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








1. Incn,i Incn,j = Incn,j Incn,i2. the eigenvalues are non-negative integers








1. Incn,i Incn,j = Incn,j Incn,i2. the eigenvalues are non-negative integers

• Questions : Is this true ? Why ? What are these integers ?


Some Surprises


Some Surprises

they do commute !

◦ 2011 : we gave an enumerative, inductive proof


Some Surprises

they do commute !


eigenvalues ?

◦ 2013 : recent work with Ton Dieker : explicit formulas


Some Surprises

they do commute !


eigenvalues ?


additional families of intriguing matrices :

◦ second family of matrices with similar properties(obtained by replacing inck with another permutation statistic)


Some Surprises

they do commute !


eigenvalues ?


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


Card Shuffling


random-to-random shuffle

deck of cards :

σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ9



deck of cards :

σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ9

remove a card at random :

σ3σ1 σ2 ↑ σ4 σ5 σ6 σ7 σ8 σ9



deck of cards :

σ1 σ2 σ3 σ4 σ5 σ6 σ7 σ8 σ9

remove a card at random :

σ3σ1 σ2 ↑ σ4 σ5 σ6 σ7 σ8 σ9

insert the card at random :

↓σ1 σ2 σ4 σ5 σ6 σ7 σ3 σ8 σ9


Transition matrix of the random-to-random shuffleentries : probability of going from σ to τ using one shuffle

123 132 213 231 312 321

123

132

213

231

312

321



123 132 213 231 312 321

123 39132

213

231

312

321

3 ways to obtain 123 from 123 :

1 2 3 1 2 3 1 2 3



123 132 213 231 312 321

123 3929

132

213

231

312

321


1 2 3 1 2 3



123 132 213 231 312 321

123 3929

29

132

213

231

312

321


1 2 3 1 2 3



123 132 213 231 312 321

123 3929

29

19

132

213

231

312

321

1 way to obtain 231 from 123 :

1 2 3



123 132 213 231 312 321

123 3929

29

19

19

132

213

231

312

321

1 way to obtain 312 from 123 :

1 2 3



123 132 213 231 312 321

123 3929

29

19

19

09

132

213

231

312

321



123 132 213 231 312 321

123 3 2 2 1 1 0

132 2 3 1 0 2 1

213 2 1 3 2 0 1

231 1 0 2 3 1 2

312 1 2 0 1 3 2

321 0 1 1 2 2 3

×1

9



123 132 213 231 312 321

123 3 2 2 1 1 0

132 2 3 1 0 2 1

213 2 1 3 2 0 1

231 1 0 2 3 1 2

312 1 2 0 1 3 2

321 0 1 1 2 2 3

×1

9

Incn,n−1(renorm.)= random-to-random shuffle


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 ~πs.t. ~π T = ~π

rate of convergence~v Tn −→ ~π

←→ governed byeigenvalues of T


random walks on the chambers

of a hyperplane arrangement


faces of a hyperplane arrangement

a set of hyperplanes partitions Rn into faces :

b




bb

the origin




b

rays emanating from the origin




b

chambers cut out by the hyperplanes


product of two faces

xy :=

{

the face first encountered after a small

movement along a line from x toward y

x

y



xy :=

{



x

y

b

b



xy :=

{



xy

x

y

b

b


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






v5 < v1 < v3 < v2 < v4

• so chambers correspond to permutations :

v5 < v1 < v3 < v2 < v4 ←→[

5, 1, 3, 2, 4]






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}]



combinatorial description :

faces ↔ ordered set partitions of {1, . . . , n} :[

{2, 3}, {4}, {1, 5}]

6=[

{4}, {1, 5}, {2, 3}]

chambers ↔ partitions into singletons[

{2}, {3}, {4}, {1}, {5}]

product ↔ intersection of sets in the partition


Product of set compositions



[

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}{6}{3}{2}]



[

w

{2, 5}{1, 3, 4, 6}]

·[

w

{4}{1}{5}{6}{3}{2}]

=[

{2, 5} ∩ {4}



[

w

{2, 5}{1, 3, 4, 6}]

·[

w

{4}{1}{5}{6}{3}{2}]

=[

∅



[

w

{2, 5}{1, 3, 4, 6}]

·[

w

{4}{1}{5}{6}{3}{2}]

=[



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}

w

{1}{5}{6}{3}{2}]

=[

{2, 5} ∩ {1}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}

w

{1}{5}{6}{3}{2}]

=[



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}

w

{5}{6}{3}{2}]

=[

{2, 5} ∩ {5}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}

w

{5}{6}{3}{2}]

=[

{5}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}

w

{6}{3}{2}]

=[

{5}{2, 5} ∩ {6}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}

w

{6}{3}{2}]

=[

{5}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}{6}

w

{3}{2}]

=[

{5}{2, 5} ∩ {3}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}{6}{3}

w

{2}]

=[

{5}{2, 5} ∩ {2}



[

w

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}{6}{3}

w

{2}]

=[

{5}{2}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

w

{4}{1}{5}{6}{3}{2}]

=[

{5}{2}{1, 3, 4, 6} ∩ {4}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

w

{4}{1}{5}{6}{3}{2}]

=[

{5}{2}{4}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}

w

{1}{5}{6}{3}{2}]

=[

{5}{2}{4}{1, 3, 4, 6} ∩ {1}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}

w

{1}{5}{6}{3}{2}]

=[

{5}{2}{4}{1}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}{1}

w

{5}{6}{3}{2}]

=[

{5}{2}{4}{1}{1, 3, 4, 6} ∩ {5}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}{1}{5}

w

{6}{3}{2}]

=[

{5}{2}{4}{1}{1, 3, 4, 6} ∩ {6}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}{1}{5}

w

{6}{3}{2}]

=[

{5}{2}{4}{1}{6}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}{1}{5}{6}

w

{3}{2}]

=[

{5}{2}{4}{1}{6}{1, 3, 4, 6} ∩ {3}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}{1}{5}{6}

w

{3}{2}]

=[

{5}{2}{4}{1}{6}{3}



[

{2, 5}

w

{1, 3, 4, 6}]

·[

{4}{1}{5}{6}{3}

w

{2}]

=[

{5}{2}{4}{1}{6}{3}{1, 3, 4, 6} ∩ {2}



[

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}{6}{3}{2}]

=[

{5}{2}{4}{1}{6}{3}]



[

{2, 5}{1, 3, 4, 6}]

·[

{4}{1}{5}{6}{3}{2}]

=[

{5}{2}{4}{1}{6}{3}]

y

y


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.






and move to y × c.

Example :

[1, 2, 3, 4, 5, 6]






and move to y × c.

Example :

[{2, 5}{1, 3, 4, 6}]× [1, 2, 3, 4, 5, 6]






and move to y × c.

Example :

[{2, 5}{1, 3, 4, 6}]× [1, 2, 3, 4, 5, 6] = [2, 5, 1, 3, 4, 6]






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]






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]






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]






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]






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]






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]






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 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



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









Extended by Brown (2000) :

◦ extended results to larger class of examples

◦ used algebraic techniques and representation theory

Others :

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


Incn,k and hyperplane chamber walks

Theorem (Reiner–S–Welker 2011)

Incn,k = Tk Ttk

where Tk is the transition matrix of a random

walk on the braid arrangement.


Incn,k and hyperplane chamber walks


Incn,k = Tk Ttk

where Tk is the transition matrix of a random

walk on the braid arrangement.

Consequently, ker(Incn,k) = ker(Tk)


Representation theory


Representation Theory

A representation of a group G is a group homomorphism

ρ : G −→ Matd




ρ : G −→ Matd

That is,

• ρ(g) is a (d× d)–matrix




ρ : G −→ Matd

That is,


• ρ(gh) = ρ(g)ρ(h)




ρ : G −→ Matd

That is,


• ρ(gh) = ρ(g)ρ(h)

Examples :

• trivial representation :

ρ(g) = [1]


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 · · ·· · · · · 11 · · · · ·· · · 1 · ·· 1 · · · ·

ρ(321) =

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


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 · · ·· · · · · 11 · · · · ·· · · 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 02 3 1 0 2 12 1 3 2 0 11 0 2 3 1 21 2 0 1 3 20 1 1 2 2 3


Incn,k and the regular representation of Sn


Incn,k =∑

σ∈Sn

inck(σ) reg(σ)

where reg is the regular representation of Sn.


Calculation of eigenvalues via irreducibles(decompose the regular representation into irreducible representations)

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



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

trivial :

3 ·[

1]

+ 2 ·[

1]

+ 2 ·[

1]

+ 1 ·[

1]

+ 1 ·[

1]

=[

9]



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

sign :

3 ·[

1]

+ 2 ·[

−1]

+ 2 ·[

−1]

+ 1 ·[

1]

+ 1 ·[

1]

=[

1]



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

χ :

3·[ 1 00 1 ] + 2·[1 01 −1 ] + 2·

[

0 −1−1 0

]

+ 1·[

−1 1−1 0

]

+ 1·[

0 −11 −1

]

=

[

4 −20 0

]



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

χ :

3·[ 1 00 1 ] + 2·[1 01 −1 ] + 2·

[

0 −1−1 0

]

+ 1·[

−1 1−1 0

]

+ 1·[

0 −11 −1

]

=

[

4 −20 0

]

eigs(Inc3,2) = {9, 1, 4, 4, 0, 0}


Final Remarks

• This research was facilitated by computer exploration with themath. 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.

PermutationsCard ShufflingRepresentation theory

Documents

Permutations, Card Shuffling, and Representation Theorysaliola/maths/talks/slides/cumc-2013.pdfPermutations Card Shuﬄing Representation theory Permutations, Card Shuﬄing, and Representation