Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Markov chains for promotion and nonabelian
sandpile models
or
The power of R-trivial monoids
Anne Schilling
Department of Mathematics, UC Davis
based on
A. Ayyer, S. Klee, A. Schilling, J. Alg. Comb. 39 (2014)
A. Ayyer, A. Schilling, B. Steinberg, N. Thiéry, Comm. Math. Phys. 335 (2015)
A. Ayyer, A. Schilling, B. Steinberg, N. Thiéry, Int. J. Alg. & Comp. 25 (2015)
A. Ayyer, A. Schilling, N. Thiéry, Exp. Math. 26 (2017)
Institut Henri Poincaré, Paris, February 20, 2017
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outline
Promotion Markov Chains:
Markov chains on linear extensions of finite posets via
promotion operators:
Nice stationary distributions!
Integer eigenvalues and nice multiplicities for rooted forests
(generalizations of derangements)
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Outline
Promotion Markov Chains:
Markov chains on linear extensions of finite posets via
promotion operators:
Nice stationary distributions!
Integer eigenvalues and nice multiplicities for rooted forests
(generalizations of derangements)
Directed Nonabelian Sandpile Models:
Grain toppling on arborescences:
Nice stationary distributions and wreath product interpretation.
Integer eigenvalues and nice multiplicities!
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Outline
Promotion Markov Chains:
Markov chains on linear extensions of finite posets via
promotion operators:
Nice stationary distributions!
Integer eigenvalues and nice multiplicities for rooted forests
(generalizations of derangements)
Directed Nonabelian Sandpile Models:
Grain toppling on arborescences:
Nice stationary distributions and wreath product interpretation.
Integer eigenvalues and nice multiplicities!
Other Markov chains:
Further examples with nice eigenvalues and multiplicities:
Walk on reduced words of longest element of Coxeter group
Toom models
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Outline
Promotion Markov Chains:
Markov chains on linear extensions of finite posets via
promotion operators:
Nice stationary distributions!
Integer eigenvalues and nice multiplicities for rooted forests
(generalizations of derangements)
Directed Nonabelian Sandpile Models:
Grain toppling on arborescences:
Nice stationary distributions and wreath product interpretation.
Integer eigenvalues and nice multiplicities!
Other Markov chains:
Further examples with nice eigenvalues and multiplicities:
Walk on reduced words of longest element of Coxeter group
Toom models
Representation Theory of Monoids:
Use the representation theory of R-trivial monoids.
Promotion Markov chains
Nonabelian sandpile model
Outline
1
Promotion Markov chains
2
Nonabelian sandpile model
3
Representation theory of monoids
4
Outlook
Representation theory of monoids
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Posets
P - a partially ordered set with order ≺
|P| = n, “naturally” labeled by integers in [n]
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Posets
P - a partially ordered set with order ≺
|P| = n, “naturally” labeled by integers in [n]
L(P) - linear extensions of P, ways of arranging elements of
P in a line respecting the order
L(P) = {π ∈ Sn : i ≺ j ⇒ πi−1 < πj−1 } 3 e
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Posets
P - a partially ordered set with order ≺
|P| = n, “naturally” labeled by integers in [n]
L(P) - linear extensions of P, ways of arranging elements of
P in a line respecting the order
L(P) = {π ∈ Sn : i ≺ j ⇒ πi−1 < πj−1 } 3 e
4
3
, L(P) = {1234, 1243, 1423, 2134, 2143}.
Eg: P =
1
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
3
4
2
1
8
Representation theory of monoids
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
9
8
7
6
5
−→
3
4
3
4
2
1
2
1
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
9
8
7
6
5
4
−→
3
4
3
2
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
9
8
7
6
5
−→
3
4
2
1
4
2
3
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
9
8
7
5
−→
3
4
6
4
2
1
2
3
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
8
7
9
5
−→
3
4
6
4
2
1
2
3
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
10
8
7
9
5
−→
3
4
6
4
2
1
2
3
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Promotion (aka jeu de taquin)
[Schützenberger ’72]
9
7
6
5
9
8
6
8
4
∂
−→
3
4
5
3
2
1
1
2
7
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Algebraic formulation in terms of transposition
Define τi on L(P)


π1 · · · πi−1 πi+1 πi · · · πn
τi π =


π1 · · · πn
if πi and πi+1 are not
comparable in P,
otherwise.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Algebraic formulation in terms of transposition
Define τi on L(P)


π1 · · · πi−1 πi+1 πi · · · πn
τi π =


π1 · · · πn
if πi and πi+1 are not
comparable in P,
otherwise.
Then define a generalized promotion operator [Haiman ’92,
Malvenuto & Reutenauer ’94]
∂j = τn−1 τn−2 · · · τj .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Algebraic formulation in terms of transposition
Define τi on L(P)


π1 · · · πi−1 πi+1 πi · · · πn
τi π =


π1 · · · πn
if πi and πi+1 are not
comparable in P,
otherwise.
Then define a generalized promotion operator [Haiman ’92,
Malvenuto & Reutenauer ’94]
∂j = τn−1 τn−2 · · · τj .
The case j = 1 is the previous promotion operator.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Algebraic formulation on previous example
9
7
6
5
3
4
2
1
8
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
7
6
5
τ
1
−→
3
4
3
4
2
1
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
7
6
5
τ τ
2 1
−→
3
4
3
4
2
1
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
7
6
5
τ3 τ2 τ1
−→
3
4
4
3
2
1
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
7
6
4
τ4 τ3 τ2 τ1
−→
3
4
5
3
2
1
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
7
6
4
τ 5 τ 4 τ 3 τ 2 τ1
−→
3
4
5
3
2
1
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
6
7
4
τ6 τ 5 τ 4 τ 3 τ 2 τ1
−→
3
4
5
3
2
1
1
2
8
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
6
8
4
τ7 τ 6 τ 5 τ 4 τ 3 τ2 τ1
−→
3
4
5
3
2
1
1
2
7
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Algebraic formulation on previous example
9
7
9
6
5
8
6
8
4
τ8 τ7 τ 6 τ 5 τ 4 τ 3 τ2 τ1
−→
3
4
5
3
2
1
1
2
7
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
The Directed Graph
Given P, let G be the graph whose vertex set is L(P)
There is an edge π → π 0 if π 0 = ∂j π for some j.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
The Directed Graph
Given P, let G be the graph whose vertex set is L(P)
There is an edge π → π 0 if π 0 = ∂j π for some j.
Lemma
G is strongly connected.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
The Directed Graph
Given P, let G be the graph whose vertex set is L(P)
There is an edge π → π 0 if π 0 = ∂j π for some j.
Lemma
G is strongly connected.
∂1
1234
∂1
∂1
∂2
∂3
∂2
2143
1423
∂2
∂3
∂2
1243
∂1
2134
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Uniform Promotion Graph
We will define two Markov chains on this underlying graph.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Uniform Promotion Graph
We will define two Markov chains on this underlying graph.
Uniform promotion graph
The edge π → π 0 , where π 0 = ∂j π has weight xj .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Uniform Promotion Graph
We will define two Markov chains on this underlying graph.
Uniform promotion graph
The edge π → π 0 , where π 0 = ∂j π has weight xj .
Probability distribution:
Theorem (AKS 2014)
The stationary distribution of the Markov chain is uniform.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Uniform Promotion Graph
We will define two Markov chains on this underlying graph.
Uniform promotion graph
The edge π → π 0 , where π 0 = ∂j π has weight xj .
Probability distribution:
Theorem (AKS 2014)
The stationary distribution of the Markov chain is uniform.
Follows from the fact that ∂ik = ∂i for large enough k.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Example
x1
1234
x1
x1
x2
x3
x2
2143
1423
x2
x3
x2
1243
x1
2134
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Promotion Graph
The edge π → π 0 , where π 0 = ∂j π has weight xπ(j) .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Promotion Graph
The edge π → π 0 , where π 0 = ∂j π has weight xπ(j) .
Stationary distribution:
Theorem (AKS 2014)
The stationary state weight w (π) of the linear extension π ∈ L(P)
for the continuous time Markov chain for the promotion graph is
given by
n
Y
x1 + · · · + xi
w (π) =
,
xπ1 + · · · + xπi
i=1
assuming w (e) = 1.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Promotion Graph
The edge π → π 0 , where π 0 = ∂j π has weight xπ(j) .
Stationary distribution:
Theorem (AKS 2014)
The stationary state weight w (π) of the linear extension π ∈ L(P)
for the continuous time Markov chain for the promotion graph is
given by
n
Y
x1 + · · · + xi
w (π) =
,
xπ1 + · · · + xπi
i=1
assuming w (e) = 1.
Note that w (π) is independent of P.
Proved by induction.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Example
x1
1234
x2
x1
x3 x2
x4
2143
x4
1423
x2
x1
1243
x1
x3 x4
2134
x2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Example (continued)
The transition matrix

x4
x2 + x3

 0

 0
x1
this time is given by

x4 x1 + x4
0
0
x3
0
x2
0 

x2 x2 + x3
0
x2 

x1
0
x4
x1 + x4 
0
0
x1 + x3
x3
Notice that row sums are no longer one.
The stationary distribution is
x1 + x2 + x3 (x1 + x2 )(x1 + x2 + x3 )
1,
,
,
x1 + x2 + x4 (x1 + x2 )(x1 + x2 + x4 )
x1
,
x2
x1 (x1 + x2 + x3 )
x2 (x1 + x2 + x4 )
.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Special Posets
A rooted tree is a connected poset, where each node has at
most one successor.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Special Posets
A rooted tree is a connected poset, where each node has at
most one successor.
A rooted forest is a union of rooted trees.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Special Posets
A rooted tree is a connected poset, where each node has at
most one successor.
A rooted forest is a union of rooted trees.
A chain is a totally ordered set.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Special Posets
A rooted tree is a connected poset, where each node has at
most one successor.
A rooted forest is a union of rooted trees.
A chain is a totally ordered set.
A union of chains is also a rooted forest.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
More terminology
An upper set S in P is a subset of [n] such that if x ∈ S and
y x, then also y ∈ S.
Let L be the lattice (by inclusion) of upper sets in P.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
More terminology
An upper set S in P is a subset of [n] such that if x ∈ S and
y x, then also y ∈ S.
Let L be the lattice (by inclusion) of upper sets in P.
µ(x, y ) is the Möbius function for
[x, y ] := {z ∈ L | x z y }
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
More terminology
An upper set S in P is a subset of [n] such that if x ∈ S and
y x, then also y ∈ S.
Let L be the lattice (by inclusion) of upper sets in P.
µ(x, y ) is the Möbius function for
[x, y ] := {z ∈ L | x z y }
f ([y , 1̂]) is the number of maximal chains in the interval [y , 1̂].
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
More terminology
An upper set S in P is a subset of [n] such that if x ∈ S and
y x, then also y ∈ S.
Let L be the lattice (by inclusion) of upper sets in P.
µ(x, y ) is the Möbius function for
[x, y ] := {z ∈ L | x z y }
f ([y , 1̂]) is the number of maximal chains in the interval [y , 1̂].
Brown defined, for each element x ∈ L, a derangement
number dx
X
dx =
µ(x, y )f ([y , 1̂]) .
y x
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Spectrum of the Transition Matrix
Theorem (AKS 2014)
Let P be a rooted forest, M the transition matrix of the promotion
graph. Then
Y
(λ − xS )dS ,
det(M − λ1) =
S⊆[n]
S upper set in P
P
where xS = i∈S xi and dS is the derangement number in the
lattice L (by inclusion) of upper sets in P.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Spectrum of the Transition Matrix
Theorem (AKS 2014)
Let P be a rooted forest, M the transition matrix of the promotion
graph. Then
Y
(λ − xS )dS ,
det(M − λ1) =
S⊆[n]
S upper set in P
P
where xS = i∈S xi and dS is the derangement number in the
lattice L (by inclusion) of upper sets in P.
In other words, for each upper set S ⊆ [n], there is
an eigenvalue xS
with multiplicity dS .
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Running Example
2
P=
1
3
L(P) = {123, 132, 312}
Upper sets: φ, {2}, {3}, {2, 3}, {1, 2}, {1, 2, 3}
Eigenvalues of M: x1 + x2 + x3 , x2 , 0.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A special case: the Tsetlin library
The Tsetlin library:
n books on a shelf
B1 B2 · · · Bn
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A special case: the Tsetlin library
The Tsetlin library:
n books on a shelf
B1 B2 · · · Bn
The probability of choosing book Bi is xi .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A special case: the Tsetlin library
The Tsetlin library:
n books on a shelf
B1 B2 · · · Bn
The probability of choosing book Bi is xi .
Once the book is chosen, it is moved to the back:
B1 B2 · · · Bi · · · Bn
→
B1 B2 · · · Bn Bi
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A special case: the Tsetlin library
The Tsetlin library:
n books on a shelf
B1 B2 · · · Bn
The probability of choosing book Bi is xi .
Once the book is chosen, it is moved to the back:
B1 B2 · · · Bi · · · Bn
→
B1 B2 · · · Bn Bi
Same as promotion Markov chain on the antichain!
In this case L(P) = Sn .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A Markov chain on permutations
Let π ∈ Sn be a permutation.
The stationary distribution of the Tsetlin library is given by
[Hendricks ’72]
P(π) =
n
Y
i=1
xπ1
xπi
.
+ · · · + x πi
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A Markov chain on permutations
Let π ∈ Sn be a permutation.
The stationary distribution of the Tsetlin library is given by
[Hendricks ’72]
P(π) =
n
Y
i=1
xπ1
xπi
.
+ · · · + x πi
A derangement is a permutation without fixed points.
Let dm be the number of derangements in Sm .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
A Markov chain on permutations
Let π ∈ Sn be a permutation.
The stationary distribution of the Tsetlin library is given by
[Hendricks ’72]
P(π) =
n
Y
i=1
xπ1
xπi
.
+ · · · + x πi
A derangement is a permutation without fixed points.
Let dm be the number of derangements in Sm .
Let Mn be the transition matrix. Then [Phatarfod ’91]
Y
det(Mn − λ1) =
(λ − xS )d[n]\|S|
S⊂[n]
where xS =
X
i∈S
xi .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Example
The case of n = 3:
 

123
x3 x3 0 0 x3 0
x2 x2 x2 0 0 0   132
 

 0 0 x3 x3 0 x3   213
 

M3 = 
 
x1 0 x1 x1 0 0   231
 0 0 0 x2 x2 x2   312
321
0 x1 0 0 x1 x1








Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Example
The case of n = 3:
 

123
x3 x3 0 0 x3 0
x2 x2 x2 0 0 0   132
 

 0 0 x3 x3 0 x3   213
 

M3 = 
 
x1 0 x1 x1 0 0   231
 0 0 0 x2 x2 x2   312
321
0 x1 0 0 x1 x1
P(231) =
x3 x1
(x2 + x3 )(x1 + x2 + x3 )








Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Example
The case of n = 3:
 

123
x3 x3 0 0 x3 0
x2 x2 x2 0 0 0   132
 

 0 0 x3 x3 0 x3   213
 

M3 = 
 
x1 0 x1 x1 0 0   231
 0 0 0 x2 x2 x2   312
321
0 x1 0 0 x1 x1
P(231) =
x3 x1
(x2 + x3 )(x1 + x2 + x3 )
Eigenvalues: 1, x3 , x2 , x1 and 0 twice.








Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Generalizations
Umpteen generalizations!
Different moves, more shelves.
Infinite libraries.
Hyperplane arrangements [Bidigare, Hanlon, Rockmore ’99]
Left regular bands (monoids) [Brown ’00]
R-trivial monoids [Steinberg ’06]
Outlook
Promotion Markov chains
Nonabelian sandpile model
Outline
1
Promotion Markov chains
2
Nonabelian sandpile model
3
Representation theory of monoids
4
Outlook
Representation theory of monoids
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Abelian sandpile models / chip-firing games
A graph G
Configuration: distribution of grains of sand at each site
Grains fall in at random
Grains topple to the neighbor sites
Grains fall off at sinks
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Abelian sandpile models / chip-firing games
A graph G
Configuration: distribution of grains of sand at each site
Grains fall in at random
Grains topple to the neighbor sites
Grains fall off at sinks
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Abelian sandpile models / chip-firing games
A graph G
Configuration: distribution of grains of sand at each site
Grains fall in at random
Grains topple to the neighbor sites
Grains fall off at sinks
Prototypical model for the phenomenon of self-organized
criticality, like a heap of sand
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Arborescences or upward rooted trees
Arborescence T : exactly one directed path from any vertex to
the root r
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Arborescences or upward rooted trees
Arborescence T : exactly one directed path from any vertex to
the root r
Set of leaves L: vertices with in-degree zero.
j
h
d
a
k
f
b
g
c
r
Figure: An arborescence with leaves at a, g, h, j, k.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Configurations
Threshold Tv : maximal number of grains at vertex v ∈ V .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Configurations
Threshold Tv : maximal number of grains at vertex v ∈ V .
Configuration space:
Ω(T ) = {(tv )v ∈V | 0 ≤ tv ≤ Tv }.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Configurations
Threshold Tv : maximal number of grains at vertex v ∈ V .
Configuration space:
Ω(T ) = {(tv )v ∈V | 0 ≤ tv ≤ Tv }.
Variable tv : the number of grains of sand at v ∈ V .
0
1
1
1
2
0
2
0
1
1
Figure: A configuration with all thresholds 2.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov Chains
We define two stochastic processes on these arborescences.
In both, sand grains enter at the leaves, ...
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov Chains
We define two stochastic processes on these arborescences.
In both, sand grains enter at the leaves, ...
..., topple along the vertices, ...
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov Chains
We define two stochastic processes on these arborescences.
In both, sand grains enter at the leaves, ...
..., topple along the vertices, ...
..., and exit at the root.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Markov Chains
We define two stochastic processes on these arborescences.
In both, sand grains enter at the leaves, ...
..., topple along the vertices, ...
..., and exit at the root.
Unlike in the abelian sandpile model, sand grains only enter at
leaves.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Markov Chains
We define two stochastic processes on these arborescences.
In both, sand grains enter at the leaves, ...
..., topple along the vertices, ...
..., and exit at the root.
Unlike in the abelian sandpile model, sand grains only enter at
leaves.
The operators defining the entrance of sand grains are the
same in both models.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Source Operator
Path to root: vertex v ∈ V
v ↓ = (v = v0 → v1 → · · · → va = r).
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Source Operator
Path to root: vertex v ∈ V
v ↓ = (v = v0 → v1 → · · · → va = r).
Source operator: leaf ` ∈ L
σ` : Ω(T ) → Ω(T )
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Source Operator
Path to root: vertex v ∈ V
v ↓ = (v = v0 → v1 → · · · → va = r).
Source operator: leaf ` ∈ L
σ` : Ω(T ) → Ω(T )
Follow the path `↓ from ` to the root r
Add a grain to the first vertex along the way that has not yet
reached its threshold, if such a vertex exists.
If no such vertex exists, then the grain is interpreted to have
left the tree at the root and σ` (t) = t.
Outlook
Promotion Markov chains
Nonabelian sandpile model
0
2
1
σj :
1
2
0
2
1
1
Representation theory of monoids
2
2
2
2
2
1
2
7→
1
1
2
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
0
2
1
Representation theory of monoids
0
2
1
2
σg :
2
1
2
1
2
7→
1
1
2
2
1
1
2
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Topple operators
Definition (Trickle-down sandpile model)
θv : Ω(T ) → Ω(T )
θv moves one grain from v ∈ V to the first available site along v ↓ .
If no such site exists, the grain exits the system.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Topple operators
Definition (Trickle-down sandpile model)
θv : Ω(T ) → Ω(T )
θv moves one grain from v ∈ V to the first available site along v ↓ .
If no such site exists, the grain exits the system.
Definition (Landslide sandpile model)
τv : Ω(T ) → Ω(T )
τv moves all grains from v ∈ V to the first available sites along v ↓ .
Grains remaining after the root exit the system.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Topple operators
Definition (Trickle-down sandpile model)
θv : Ω(T ) → Ω(T )
θv moves one grain from v ∈ V to the first available site along v ↓ .
If no such site exists, the grain exits the system.
Definition (Landslide sandpile model)
τv : Ω(T ) → Ω(T )
τv moves all grains from v ∈ V to the first available sites along v ↓ .
Grains remaining after the root exit the system.
Remark
If tv = 0 (no grain at site v ), then θv (t) = τv (t) = t.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Toppling in the Trickle-down sandpile model
0
2
1
2
1
0
2
θk :
2
2
1
2
2
7→
1
1
2
2
1
1
2
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Toppling in the Trickle-down sandpile model
0
2
1
2
1
0
2
θg :
2
2
1
2
1
7→
1
1
2
2
1
1
2
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Toppling in the Landslide sandpile model
0
2
1
2
1
0
2
τk :
2
2
0
2
2
7→
1
1
2
2
1
2
2
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Toppling in the Landslide sandpile model
0
2
1
2
1
0
2
τg :
2
2
1
2
0
7→
1
1
2
2
1
1
2
2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov Chains
The Trickle-down and Landslide models are discrete-time
Markov chains on Ω(T ).
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov Chains
The Trickle-down and Landslide models are discrete-time
Markov chains on Ω(T ).
Probability distribution: {xv , y` | v ∈ V , ` ∈ L}
xv : probability of choosing the topple operator θv (resp. τv )
y` : probability of choosing the source operator σ`
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov Chains
The Trickle-down and Landslide models are discrete-time
Markov chains on Ω(T ).
Probability distribution: {xv , y` | v ∈ V , ` ∈ L}
xv : probability of choosing the topple operator θv (resp. τv )
y` : probability of choosing the source operator σ`
We assume that
1
0 < xv , y` ≤ 1
2
X
v ∈V
xv +
X
`∈L
y` = 1
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Remarks
Threshold Tv = 1: If Tv = 1 for all v ∈ V , then the
Trickle-down and Landslide sandpile models are equivalent.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Remarks
Threshold Tv = 1: If Tv = 1 for all v ∈ V , then the
Trickle-down and Landslide sandpile models are equivalent.
Recursive definition: Both models can be defined recursively
by successively removing leaves.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Remarks
Threshold Tv = 1: If Tv = 1 for all v ∈ V , then the
Trickle-down and Landslide sandpile models are equivalent.
Recursive definition: Both models can be defined recursively
by successively removing leaves.
Sources on all vertices: Allow source operators at all vertices,
not just leaves!
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Ergodicity
Proposition (ASST 2015)
Gθ : directed graph with
vertex set Ω(T )
edges given by σ` for ` ∈ L and θv for v ∈ V .
Then Gθ is strongly connected and hence the Trickle-down sandpile
model is ergodic.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Ergodicity
Proposition (ASST 2015)
Gθ : directed graph with
vertex set Ω(T )
edges given by σ` for ` ∈ L and θv for v ∈ V .
Then Gθ is strongly connected and hence the Trickle-down sandpile
model is ergodic.
Proposition (ASST 2015)
Gτ : directed graph with
vertex set Ω(T )
edges given by σ` for ` ∈ L and τv for v ∈ V .
Then Gτ is strongly connected and hence the Landslide sandpile
model is ergodic.
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chains on a line with thresholds 1
σ1
111
σ1
σ1
τ3
τ3
110
σ1
010
σ1
τ1
τ1
011
τ2
τ1
σ1
τ3
τ2
τ1
τ2
τ3
τ3
101
τ2
σ1
τ2
τ3
τ1
100
τ2
001
τ1
τ1
σ1
τ3
000
τ3
τ2
τ2
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Trickle-down sandpile model: Stationary distribution
Lv := {` ∈ L | v is a vertex of `↓ }
P
Yv := `∈Lv y`
For 0 ≤ h ≤ Tv
Y h x Tv −h
ρv (h) := PTvv v T −i
v
i
i=0 Yv xv
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Trickle-down sandpile model: Stationary distribution
Lv := {` ∈ L | v is a vertex of `↓ }
P
Yv := `∈Lv y`
For 0 ≤ h ≤ Tv
Y h x Tv −h
ρv (h) := PTvv v T −i
v
i
i=0 Yv xv
Theorem (ASST 2015)
The stationary distribution of the Trickle-down sandpile Markov
chain defined on Gθ is given by the product measure
Y
P(t) =
ρv (tv ).
v ∈V
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Stationary distribution
µv (h) :=

Yvh xv



h+1

 (Yv + xv )
if h < Tv
YvTv
(Yv + xv )Tv
if h = Tv





Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Stationary distribution
µv (h) :=

Yvh xv



h+1

 (Yv + xv )
if h < Tv
YvTv
(Yv + xv )Tv
if h = Tv





Theorem (AST 2015)
Let Tv = 1 for all v ∈ V , v 6= r and Tr = m for some positive
integer m. Then the stationary distribution of the Landslide
sandpile model defined on Gτ is given by the product measure
Y
P(t) =
µv (tv ).
v ∈V
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Spectrum
For subsets S ⊆ V and `↓ the set of vertices on path from ` to r:
X
X
yS =
y` and xS =
xv .
`∈L,`↓ ⊆S
v ∈S
Transition matrix for Landslide sandpile model Mτ
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Spectrum
For subsets S ⊆ V and `↓ the set of vertices on path from ` to r:
X
X
yS =
y` and xS =
xv .
v ∈S
`∈L,`↓ ⊆S
Transition matrix for Landslide sandpile model Mτ
Theorem (ASST 2015)
The characteristic polynomial of Mτ is given by
Y
det(Mτ − λ1) =
(λ − (yS + xS ))TS c ,
S⊆V
where S c = V \ S and TS =
Q
v ∈S
Tv .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Spectrum
For subsets S ⊆ V and `↓ the set of vertices on path from ` to r:
X
X
yS =
y` and xS =
xv .
v ∈S
`∈L,`↓ ⊆S
Transition matrix for Landslide sandpile model Mτ
Theorem (ASST 2015)
The characteristic polynomial of Mτ is given by
Y
det(Mτ − λ1) =
(λ − (yS + xS ))TS c ,
S⊆V
where S c = V \ S and TS =
Eigenvalues: yS + xS
Multiplicities: TS c
Q
v ∈S
Tv .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Mixing time
Rate of convergence: Total variation distance from stationarity
after k steps ||P k − π||.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Mixing time
Rate of convergence: Total variation distance from stationarity
after k steps ||P k − π||.
Define p := min{xv | v ∈ V } and n := |V |.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Mixing time
Rate of convergence: Total variation distance from stationarity
after k steps ||P k − π||.
Define p := min{xv | v ∈ V } and n := |V |.
Theorem (ASST 2015)
The rate of convergence is bounded by
(kp − (n − 1))2
k
||P − π|| ≤ exp −
2kp
as long as k ≥ (n − 1)/p.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Mixing time
Rate of convergence: Total variation distance from stationarity
after k steps ||P k − π||.
Define p := min{xv | v ∈ V } and n := |V |.
Theorem (ASST 2015)
The rate of convergence is bounded by
(kp − (n − 1))2
k
||P − π|| ≤ exp −
2kp
as long as k ≥ (n − 1)/p.
Mixing time: k such that ||P k − π|| ≤ e −c
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Landslide sandpile model: Mixing time
Rate of convergence: Total variation distance from stationarity
after k steps ||P k − π||.
Define p := min{xv | v ∈ V } and n := |V |.
Theorem (ASST 2015)
The rate of convergence is bounded by
(kp − (n − 1))2
k
||P − π|| ≤ exp −
2kp
as long as k ≥ (n − 1)/p.
Mixing time: k such that ||P k − π|| ≤ e −c
Mixing time is at most 2(n+c−1)
.
p
Outlook
Promotion Markov chains
Nonabelian sandpile model
Outline
1
Promotion Markov chains
2
Nonabelian sandpile model
3
Representation theory of monoids
4
Outlook
Representation theory of monoids
Outlook
Promotion Markov chains
Nonabelian sandpile model
Markov chain on reduced words
W = hsi | i ∈ I i finite Coxeter group
Representation theory of monoids
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chain on reduced words
W = hsi | i ∈ I i finite Coxeter group
Example
W = Sn symmetric group, si simple transpositions for 1 ≤ i < n.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chain on reduced words
W = hsi | i ∈ I i finite Coxeter group
Example
W = Sn symmetric group, si simple transpositions for 1 ≤ i < n.
w0 longest element in W .
R = set of reduced words of w0 .
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chain on reduced words
W = hsi | i ∈ I i finite Coxeter group
Example
W = Sn symmetric group, si simple transpositions for 1 ≤ i < n.
w0 longest element in W .
R = set of reduced words of w0 .
Markov chain:
w ∈R
Define ∂i : R → R by prepending i to w and removing the
leftmost letter in w that makes iw non-reduced.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chain on reduced words
W = hsi | i ∈ I i finite Coxeter group
Example
W = Sn symmetric group, si simple transpositions for 1 ≤ i < n.
w0 longest element in W .
R = set of reduced words of w0 .
Markov chain:
w ∈R
Define ∂i : R → R by prepending i to w and removing the
leftmost letter in w that makes iw non-reduced.
Example
w = 231231 ∈ R for S4 . Then ∂1 (w ) = 123121 since
123123 = 121323 = 212323 is not reduced!
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Methods from the representation theory of monoids
Our models have exceptionally nice eigenvalues.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Methods from the representation theory of monoids
Our models have exceptionally nice eigenvalues.
In fact many examples of Markov chains have similar behaviors:
Promotion Markov chain
Nonabelian directed sandpile models
Toom models
Walks on longest words of finite Coxeter groups
Free tree monoid
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Methods from the representation theory of monoids
Our models have exceptionally nice eigenvalues.
In fact many examples of Markov chains have similar behaviors:
Promotion Markov chain
Nonabelian directed sandpile models
Toom models
Walks on longest words of finite Coxeter groups
Free tree monoid
Is there some uniform explanation?
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Methods from the representation theory of monoids
Our models have exceptionally nice eigenvalues.
In fact many examples of Markov chains have similar behaviors:
Promotion Markov chain
Nonabelian directed sandpile models
Toom models
Walks on longest words of finite Coxeter groups
Free tree monoid
Is there some uniform explanation?
Yes!
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Approach: monoids and representation theory
A monoid M is a set with an associative product and an identity.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Approach: monoids and representation theory
A monoid M is a set with an associative product and an identity.
Definition (Transition monoid of a Markov chain / automaton)
mi transition operators of the Markov chain
E.g.:
σ` and τv for the Landslide sandpile model
Monoid: (M, ◦) = hmi i
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Approach: monoids and representation theory
A monoid M is a set with an associative product and an identity.
Definition (Transition monoid of a Markov chain / automaton)
mi transition operators of the Markov chain
E.g.:
σ` and τv for the Landslide sandpile model
Monoid: (M, ◦) = hmi i
Alternatively from transition matrix M of Markov chain:
mi = Mxi =1;x1 =···=xi−1 =xi+1 =···=xn =0
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
The left Cayley graph for the 1D sandpile model
id
τ3
τ3
τ2
τ3
τ2
τ1
τ2 τ3
τ1
τ1
σ1
τ2
τ2
τ2
τ3
τ3
τ1
σ1
τ1 τ3
τ1
τ3
τ3 τ1
τ1
σ1 τ3
τ2 τ1
τ1
τ2
σ1
τ3 σ1
τ1
τ2
σ1
τ1 τ2
τ1
τ3 τ2
τ1
σ1
σ1
σ1
τ2
σ1
σ1 τ2 τ3
τ2
τ2 σ1
τ3
τ2
τ2
τ3 σ1 τ2
τ2
τ1
τ1
τ3 τ2
σ1
τ3
τ3
τ3
τ2
τ1 τ3
σ1 τ3
τ3 τ1 τ2
σ1 τ2
τ2
σ1
τ3
τ2
τ1
τ2
σ1 τ3
τ1
τ3 σ1
τ2τ2
τ2 τ3 τ1
τ1 τ2 τ3
τ2 τ3 τ1 τ2
τ3
τ2
τ1
σ1 τ
2
τ2 τ3 σ1
τ1
σ1
σ1
τ1
τ2
τ1
τ1 τ3 σ1
111
τ1 σ1
σ1
τ1
τ1
τ1
σ1
σ1
τ1
σ1
τ3
τ3
110
σ1
τ2
σ1
σ1 σ1
σ1
τ3
010
σ1 τ3 σ1
τ1
τ1
101
τ3 τ
3
τ3
τ3 τ3
τ1
τ1
σ1
τ3
τ1
τ2
τ2
τ3
τ1 σ1 σ1
110
τ3
τ3
τ1
τ1
τ2
τ2
τ2
σ1 τ1
τ1
τ1
τ1
τ2
τ3
000
τ1
τ3
τ2 σ1 σ1
101
τ1
σ1
τ3
τ3 τ2 σ1
001
τ2
τ2
σ1
σ1
001
σ1
τ3
τ2
τ3
τ2
τ3 τ1 σ1
010
σ1 τ3
τ1
τ3
τ2
σ1
τ2
τ3
τ3
τ2
τ2 τ1 σ1
100
τ1
σ1
τ2
τ2
τ3
τ3 τ2 τ1
000
τ3
τ1
τ2 τ3
σ1
σ1
τ2
τ3 σ1 σ1
011
τ3
100
σ1 τ3 σ1 τ2
σ1
τ2
τ2
τ3
τ1
σ1
σ1
σ1 σ1 σ1
111
τ2
τ3
σ1
τ1
σ1
τ3
σ1
τ1 τ3 σ1 τ2
σ1
σ1
011
τ2
τ1
σ1
σ1
τ1
σ1
τ2
τ1
σ1
τ1
τ2 τ3 σ1 τ2
τ1
τ2
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
The right Cayley graph for the 1D sandpile model
id
σ1
τ1
τ3
σ1
τ1
τ2
τ1
τ2
τ1
τ2
σ1
σ1 τ2
τ3
τ3
τ2
τ2
τ1
σ1
σ1 τ3
τ1
τ3
τ1
τ3 σ1
σ1
τ2
τ2
τ2
σ1
τ2
τ3
τ2 σ1
τ3
τ1
τ2 τ3
τ3 σ1 τ2
τ1
τ1
τ2 τ3 σ1
σ1
τ3
τ2
σ1
τ2
σ1
τ2
τ3
σ1
σ1
σ1
σ1
τ1 τ2 τ
3
τ3 σ1 σ1
011
σ1
τ2 σ1 σ1
101
τ1 τ2 τ
3
τ2 τ1
τ3
τ2 τ3 σ1 τ2
τ2
τ3
σ1
σ1
3
τ3
τ3 τ2
τ1 τ2 τ3
τ2
τ2
τ3 τ1
τ1
τ3
τ1
τ2
τ1 σ1
τ3
σ1
τ1
σ1
τ1
τ2
τ3
τ1 τ3 σ1
τ3
τ2
τ1
τ3
τ2
τ2
τ3
τ2 τ3 τ1 τ2
τ1
σ1
τ1
σ1
τ3 τ2 τ1
000
τ1 τ2 τ
3
This graph is acyclic: R-triviality
τ1
σ1
τ1 τ2 τ
3
τ3 τ1 τ2
τ2
τ3
τ1
σ1
τ1 τ3
τ3
τ1
τ2
τ1
τ2 τ3 τ1
τ3 τ2 σ1
001
τ1 τ2 τ
τ1
τ3
τ1
τ2
σ1
τ1
τ3
σ1
τ2
σ1
τ1
σ1
σ1 σ1 σ1
111
τ3
τ1
τ1
τ2
τ1
τ3
τ1 τ2
τ3
τ2τ3
σ1
σ1 τ3 σ1
σ1 τ3 σ1 τ2
σ1 τ2 τ3
τ3
τ2
τ3
τ2
τ1
σ1
σ1 σ1
τ3
σ1
σ1
τ1 τ3 σ1 τ2
τ2
τ3
σ1
σ1
σ1
τ2 τ1 σ1
100
σ1
τ3 τ1 σ1
010
τ1 τ2 τ
3
τ2
τ3
τ1
σ1
σ1
σ1
τ1 σ1 σ1
110
τ1 τ2 τ
3
σ1
τ1 τ2 τ
3
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
The right Cayley graph for the 1D sandpile model
id
σ1
τ1
τ3
σ1
τ1
τ2
τ1
τ2
τ1
τ2
σ1
σ1 τ2
τ3
τ3
τ2
τ2
τ1
σ1
σ1 τ3
τ1
τ3
τ1
τ3 σ1
σ1
τ2
τ2
τ2
σ1
τ2
τ3
τ2 σ1
τ3
τ1
τ2 τ3
τ3 σ1 τ2
τ1
τ1
τ2 τ3 σ1
σ1
τ3
τ2
σ1
τ2
σ1
τ2
τ3
σ1
σ1
σ1
σ1
τ1 τ2 τ
3
τ3 σ1 σ1
011
σ1
τ2 τ1
τ3
τ2 τ3 σ1 τ2
τ2
τ3
τ2 σ1 σ1
101
τ1 τ2 τ
3
σ1
σ1
3
τ3 τ2
τ2
τ2
τ1 τ2 τ3
τ3 τ1
τ1
τ3
τ1
τ2
τ1 σ1
τ3
σ1
τ1
σ1
τ1 τ3 σ1
τ3
τ2
τ1
τ2
τ3
τ1
τ3
τ2
τ2
τ2 τ3 τ1 τ2
τ1
σ1
τ1
σ1
τ3 τ2 τ1
000
τ1 τ2 τ
3
τ1
σ1
τ1 τ2 τ
3
τ3 τ1 τ2
τ2
τ3
τ1
σ1
τ1 τ3
τ3
τ1
τ2
τ3
This graph is acyclic: R-triviality
Not too deep
τ3
τ1
τ2 τ3 τ1
τ3 τ2 σ1
001
τ1 τ2 τ
τ1
τ3
τ1
τ2
σ1
τ1
τ3
σ1
τ2
σ1
τ1
σ1
σ1 σ1 σ1
111
τ3
τ1
τ1
τ2
τ1
τ3
τ1 τ2
τ3
τ2τ3
σ1
σ1 τ3 σ1
σ1 τ3 σ1 τ2
σ1 τ2 τ3
τ3
τ2
τ3
τ2
τ1
σ1
σ1 σ1
τ3
σ1
σ1
τ1 τ3 σ1 τ2
τ2
τ3
σ1
σ1
σ1
τ2 τ1 σ1
100
σ1
τ3 τ1 σ1
010
τ1 τ2 τ
3
τ2
τ3
τ1
σ1
σ1
σ1
τ1 σ1 σ1
110
τ1 τ2 τ
3
σ1
τ1 τ2 τ
3
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
The right Cayley graph for the 1D sandpile model
id
σ1
τ1
τ3
σ1
τ1
τ2
τ1
τ2
τ1
τ2
σ1
σ1 τ2
τ3
τ3
τ2
τ2
τ1
σ1
σ1 τ3
τ1
τ3
τ1
τ3 σ1
σ1
τ2
τ2
τ2
σ1
τ2
τ3
τ2 σ1
τ3
τ1
τ2 τ3
τ3 σ1 τ2
τ1
τ1
τ2 τ3 σ1
σ1
τ3
τ2
σ1
τ2
σ1
τ2
τ3
σ1
σ1
σ1
σ1
τ1 τ2 τ
3
τ3 σ1 σ1
011
σ1
τ2 σ1 σ1
101
τ1 τ2 τ
3
τ2 τ1
τ3
τ2 τ3 σ1 τ2
τ2
τ3
σ1
σ1
3
τ3
τ3 τ2
τ1 τ2 τ3
τ2
τ2
τ3 τ1
τ1
τ3
τ1
τ2
τ1 σ1
τ3
σ1
τ1
σ1
τ1
τ2
τ3
τ1 τ3 σ1
τ3
τ2
τ1
τ3
τ2
τ2
τ3
τ2 τ3 τ1 τ2
τ1
σ1
τ1
σ1
τ3 τ2 τ1
000
τ1 τ2 τ
3
τ1
σ1
τ1 τ2 τ
3
τ3 τ1 τ2
τ2
τ3
τ1
σ1
τ1 τ3
τ3
τ1
τ2
τ1
τ2 τ3 τ1
τ3 τ2 σ1
001
τ1 τ2 τ
τ1
τ3
τ1
τ2
σ1
τ1
τ3
σ1
τ2
σ1
τ1
σ1
σ1 σ1 σ1
111
τ3
τ1
τ1
τ2
τ1
τ3
τ1 τ2
τ3
τ2τ3
σ1
σ1 τ3 σ1
σ1 τ3 σ1 τ2
σ1 τ2 τ3
τ3
τ2
τ3
τ2
τ1
σ1
σ1 σ1
τ3
σ1
σ1
τ1 τ3 σ1 τ2
τ2
τ3
σ1
σ1
σ1
τ2 τ1 σ1
100
σ1
τ3 τ1 σ1
010
τ1 τ2 τ
3
τ2
τ3
τ1
σ1
σ1
σ1
τ1 σ1 σ1
110
τ1 τ2 τ
This graph is acyclic: R-triviality
Not too deep =⇒ bound on the rates of convergence
3
σ1
τ1 τ2 τ
3
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Definitions: Green relations
Left and right preorders on M:
x ≤R y
if
y ∈ xM
x ≤L y
if
y ∈ Mx
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Definitions: Green relations
Left and right preorders on M:
x ≤R y
if
y ∈ xM
x ≤L y
if
y ∈ Mx
Equivalence classes on M:
xRy
if
y M = xM
xL y
if
My = Mx
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Definitions: Green relations
Left and right preorders on M:
x ≤R y
if
y ∈ xM
x ≤L y
if
y ∈ Mx
Equivalence classes on M:
xRy
if
y M = xM
xL y
if
My = Mx
Definition
M is R-trivial (L -trivial) if all R-classes (L -classes) are
singletons. Equivalently, if the preorders are partial orders.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
R-trivial monoid for promotion example
100
010
001
2
000
110
001
2
3
111
000
000
1
3 2 1
1
000
100
011
3
3
000
111
000
3 2 1
2
1
000
000
111
3 2 1
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Strategy
Method
Show that M is R-trivial
⇒ matrix representation can be triangularized
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Strategy
Method
Show that M is R-trivial
⇒ matrix representation can be triangularized
Eigenvalues indexed by a lattice of subsets of the generators
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Strategy
Method
Show that M is R-trivial
⇒ matrix representation can be triangularized
Eigenvalues indexed by a lattice of subsets of the generators
Multiplicities from Möbius inversion on the lattice
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Strategy
Method
Show that M is R-trivial
⇒ matrix representation can be triangularized
Eigenvalues indexed by a lattice of subsets of the generators
Multiplicities from Möbius inversion on the lattice
Representation theory point of view
Simple modules are of dimension 1
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Strategy
Method
Show that M is R-trivial
⇒ matrix representation can be triangularized
Eigenvalues indexed by a lattice of subsets of the generators
Multiplicities from Möbius inversion on the lattice
Representation theory point of view
Simple modules are of dimension 1
Compute the character of a transformation module
(counting fixed points)
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Strategy
Method
Show that M is R-trivial
⇒ matrix representation can be triangularized
Eigenvalues indexed by a lattice of subsets of the generators
Multiplicities from Möbius inversion on the lattice
Representation theory point of view
Simple modules are of dimension 1
Compute the character of a transformation module
(counting fixed points)
Recover the composition factors using the character table
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chains and Representation Theory
The idea of decomposing the configuration space is not new!
Using representation theory of groups
Diaconis et al.
Nice combinatorics (symmetric functions, ...)
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chains and Representation Theory
The idea of decomposing the configuration space is not new!
Using representation theory of groups
Diaconis et al.
Nice combinatorics (symmetric functions, ...)
Using representation theory of right regular band
Tsetlin library, Hyperplane arrangements, ...
Bidigare, Hanlon, Rockmore ’99, Brown ’00, Saliola, ...
Revived the interest for representation theory of monoids
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chains and Representation Theory
The idea of decomposing the configuration space is not new!
Using representation theory of groups
Diaconis et al.
Nice combinatorics (symmetric functions, ...)
Using representation theory of right regular band
Tsetlin library, Hyperplane arrangements, ...
Bidigare, Hanlon, Rockmore ’99, Brown ’00, Saliola, ...
Revived the interest for representation theory of monoids
Using representation of R-trivial monoids?
Steinberg ’06, ...
Not semi-simple.
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Markov chains and Representation Theory
The idea of decomposing the configuration space is not new!
Using representation theory of groups
Diaconis et al.
Nice combinatorics (symmetric functions, ...)
Using representation theory of right regular band
Tsetlin library, Hyperplane arrangements, ...
Bidigare, Hanlon, Rockmore ’99, Brown ’00, Saliola, ...
Revived the interest for representation theory of monoids
Using representation of R-trivial monoids?
Steinberg ’06, ...
Not semi-simple. But simple modules of dimension 1!
Nice combinatorics
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
R-trivial machinery:
Multitude of models fits this setting!
Outlook
Promotion Markov chains
Nonabelian sandpile model
Representation theory of monoids
Outlook
Outlook
R-trivial machinery:
Multitude of models fits this setting!
Mixing time for linear extensions [AST17]:
Counting linear extensions is an important problem in practice.
Define random-to-random promotion operator on posets via
τi τi+1 · · · τj .
Explicit conjecture for second largest eigenvalue for
random-to-random shuffling on posets.
Conjectured mixing time is O(n2 log n) (as opposed to
Bubley-Dyer’s result of O(n3 log n))
Promotion Markov chains
Nonabelian sandpile model
Thank you !
Representation theory of monoids
Outlook