Semistandard Young Tableaux Polytopes Sara Solhjem Joint ...sarsolhj/GSCC.pdfSara Solhjem (NDSU)...

Semistandard Young Tableaux Polytopes

Sara SolhjemJoint work with Jessica Striker

North Dakota State University

Graduate Student Combinatorics Conference 2017

April 9, 2017

Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, 2017 1 / 37

Main Topics

1 Background

2 Polytope with shape λ and max entry n

3 Polytope of m × n sign matrices

4 Comparing P(λ, n) and P(m, n)

5 Future Connections


Main Topics

1 Background






Semistandard Young Tableaux

Definition

A Young diagram λ is a collection of boxes, or cells,arranged in left-justified rows, with a weakly decreasingnumber of boxes in each row.


Semistandard Young Tableaux

Definition

A semistandard Young tableau, denoted SSYT, is definedas a filling of a Young diagram such that the rows areweakly increasing and the columns are strictly increasing.

1 1 2 4 5 7

2 2 3

3 4 5

6


SSYT(m,n)

Definition

Let SSYT (m, n) denote the set of semistandard Youngtableaux with at most m columns and entries at most n.

1 1 2 4 5 7

2 2 3

3 4 5

6

SSYT (6, 7)


Notation with the SSYT(m,n)

1 1 2 4 5 7

2 2 3

3 4 5

6

λ is the shape using row length. → [6, 3, 3, 1]λi is the length of a row in the tableau.λ1 = 6, λ2 = 3, λ3 = 3, λ4 = 1`i is the length of a column read right to left.`0 = 0, `1 = 1, `2 = 1, `3 = 1, `4 = 3,`5 = 3, `6 = 4`(λ) is the number of rows in the tableau. → 4Sara Solhjem (NDSU) Semistandard Young Tableaux Polytopes April 9, 2017 7 / 37

m × n Sign Matrices

Definition (Aval 2008)

Sign matrices are m × n matrices with the followingproperties:

entries ∈ {−1, 0, 1}the partial sums of the columns are 0 or 1

the partial sums of the rows are always nonnegative

0 0 0 0 1 00 0 0 1 −1 00 1 1 −1 1 01 0 −1 1 −1 00 0 1 −1 0 1


Bijection with semistandard Young Tableau

Aval also showed that sign matrices are in bijection withsemistandard Young Tableau.

1 1 2 4 5

2 2 3

3 4 5

6

⇐⇒

0 0 0 0 1 00 0 0 1 −1 00 1 1 −1 1 01 0 −1 1 −1 00 0 1 −1 0 1


SSYT(λ, n)

Definition

Let SSYT (λ, n) denote the set of semistandard Youngtableaux of shape λ and entries at most n.

Proposition (Stanley 1971)

The number of SSYT of shape λ with maximum entry ofn, is given by the hook-content formula:∏

u∈λ

n + c(u)

h(u)


Sign Matrices M(λ, n)

Definition

Let M(λ, n) be the set of λ1 × n sign matrices M = (Mij)such that:

n∑j=1

Mij = `i − `i−1 for all 1 ≤ i ≤ λ1,

Call M(λ, n) the set of sign matrices of shape λ andcontent at most n.

Proposition (S. and Striker)

M(λ, n) is in explicit bijection with SSYT (λ, n).


Tableau shape λ and max entry at most n

Notice the correspondence between the shape of thetableau and the size of the matrix:

the number of rows of the matrix and the length ofthe first row of the tableau, λ1.

the number of columns in the matrix and the maxentry in the tableau.

the total sum of each row and `i − `i−1.

1 1 2 4 5

2 2 3

3 4 5

6

⇐⇒

0 0 0 0 1 00 0 0 1 −1 00 1 1 −1 1 01 0 −1 1 −1 00 0 1 −1 0 1


What is a Polytope?

A polytope can be defined in two equivalent ways:

As the convex hull of a finite set of points{x1, x2, . . . , xn}, that is, the set of all expressions of

the formn∑

i=1

µixi wheren∑

i=1

µi = 1 and all the µi are

nonnegative.

As the bounded intersection of finitely many closedhalfspaces.

Thus a polytope can be specified by a set of points or bya set of linear inequalities.


Main Topics

1 Background






The P(λ, n) Polytope

Definition

Let P(λ, n) be the polytope defined as the convex hull ofall the matrices in M(λ, n).

P([2, 2], 3) ={µ1

(1 1 00 0 0

)+ µ2

(1 0 10 0 0

)+ µ3

(0 1 11 0 0

)+ µ4

(0 1 10 0 0

)+ µ5

(1 0 10 1 −1

)+ µ6

(0 1 11 −1 0

)}where

6∑i=1

µi = 1.


The P(λ, n) Polytope

Theorem (S. and Striker)

The dimension of the P(λ, n) polytope is λ1(n − 1) for1 ≤ `(λ) < n. When `(λ) = n the dimension is(λ1 − λn)(n − 1).

1 1 2 4 5

2 2 3

3 4 5

5 6

⇐⇒

0 0 0 0 1 00 0 0 1 −1 00 1 1 −1 1 01 0 −1 1 −1 10 0 1 −1 1 −1


Vertex Theorem


Each of the SSYT forms a vertex in the P(λ, n) polytope.

c01 c02 . . . c0n

c11

cm1

c12 c1n

cm2 cmn. . .cm3

c03

r10 r11 r12 r1n

r20 r21 r22 r2n

. .

rm0 rm1 rm2 rmn

.

...

Xm1 Xm2 Xmn

X11 X12 X1n

X21 X22 X2n

The graph of partial sums is in bijection with P(λ, n).


Partial sum graphs of all six vertices in P([2, 2], 3)

1

1 1

1 1

2

0 0

0

0

1 1 0

0 0 0

0 0 0

0

0

2

0

1 1

1

1 1

10

0

0

0

01 1

1

1

1

1 0 1

0 0 0

0 0 0

0

0

2

00 0

0 1 1

1 0

0 0 0

0

0

2

0

(a) (b) (c)

(d) (e)

0 1

1

1 1

1 1

1

0

0

0

0

0

1

1

1

11

1

0

1 0 1

0 1 -1

0 0 0

0

0

2

0

0 1 1

1 -1 0

0 0 0

0

0

2

0

0

0

0

0

0

1

1

1 1

1

0 1 1

0 0 0

0 0 0

0

0

2

0

(f)

1 1

-1


Sketch of the vertex proof

[1 0 10 1 −1

]0 1

1

1 1

1 1

1

0

0

1 0 1

0 1 -1

0 0 0

0

0

2

0

The hyperplane that separates this vertex of theP([2, 2], 3) polytope from the other five is the following:

X11 + X13 + (X11 + X21) + (X12 + X22) =2X11 + X12 + X13 + X21 + X22 = 3.5


Sketch of the vertex proof

Now using the entries of each matrix plugged into our equation:2X11 + X12 + X13 + X21 + X22 =

Graph (a): X11 = 1,X12 = 1,X13 = 0,X21 = 0,X22 = 0,X23 = 0 → 2 + 1 + 0 + 0 + 0 = 3;

Graph (b): X11 = 1,X12 = 0,X13 = 1,X21 = 0,X22 = 0,X23 = 0 → 2 + 0 + 1 + 0 + 0 = 3;

Graph (c): X11 = 0,X12 = 1,X13 = 1,X21 = 1,X22 = 0,X23 = -1 → 0 + 1 + 1 + 1 + 0 = 3;

Graph (d): X11 = 0,X12 = 1,X13 = 1,X21 = 0,X22 = 0,X23 = 0 → 0 + 1 + 1 + 0 + 0 = 2;

Graph (e): X11 = 1,X12 = 0,X13 = 1,X21 = 0,X22 = 1,X23 = -1 → 2 + 0 + 1 + 0 + 1 = 4;

Graph (f): X11 = 0,X12 = 1,X13 = 1,X21 = 1,X22 =-1,X23 = 0 → 0 + 1 + 1 + 1 + (-1) = 2.

You can see that our vertex is on one side of2X11 + X12 + X13 + X21 + X22 = 3.5 and the other five are on the other.


Inequality description of P(λ, n)


P(λ, n) consists of all λ1 × n real matrices X = (Xij) such that:

0 ≤i ′∑

i=1

Xij ≤ 1 for all 1 ≤ i ′ ≤ λ1, 1 ≤ j ≤ n

0 ≤j ′∑

j=1

Xij for all 1 ≤ j ′ ≤ n, 1 ≤ i ≤ λ1

n∑j=1

Xij = `i − `i−1 for all 1 ≤ i ≤ λ1


Sketch of the proof of the inequality description

.9 0 .3 .80 .1 .6 −.70 .9 −.1 .2

0

0

0

0

0 00

0

.9 0 .3 .8

0 .1 .6 -.7

0 .9 -.1 .2

.9

.9

.9

.9

.9

.9

.9

1.22

0

0

0

.3 .8

.1 .7

.1 .1

.8

.81

1

.3

Left: A matrix in P([3, 3, 1], 4); Right: An open circuitgraph


Sketch of the proof of the inequality description

.9 0 .3 .80 .1 .6 −.70 .9 −.1 .2

=

.7

.1 + .7

1 0 .2 .80 .1 .7 −.80 .9 −.1 .2

+.1

.1 + .7

.2 0 1 .80 .1 −.1 00 .9 −.1 .2

The decomposition of the previous matrix.


Enumeration of P(λ, n)

Definition

A facet is a face of the polytope that is dimension oneless than the polytope.

Conjecture

The formula for the number of facets of P(λ, n) forrectangular and square tableaux is 3nλ1 − n − 5λ1 + 5.


Main Topics

1 Background






P(m, n) Polytope

Definition

Let P(m, n) be the polytope defined as the convex hull ofall m × n sign matrices.


The vertices of P(m, n) are the sign matrices of sizem × n.


Inequality description of P(m, n)


P(m, n) consists of all m× n real matrices X = {xij} suchthat:

0 ≤i ′∑i=1

xij ≤ 1 for all 1 ≤ i ′ ≤ m, 1 ≤ j ≤ n.

0 ≤j ′∑j=1

xij for all 1 ≤ i ≤ m, 1 ≤ j ′ ≤ n.

Thus all of the partial sums of the columns are between 0and 1 and the partial sums of all the rows are nonnegative.


Enumerations


The dimension of P(m, n) is mn for all m > 1.


There are n(3m − 1)− 2(m − 2) facets in the P(m, n)polytope.


Main Topics

1 Background






Compare P(λ, n) and P(m, n)

What is different between P(λ, n) and P(m, n)?

P(λ, n) has a fixed shape of tableau which give a signmatrix of size λ1 × n. Then these certain signmatrices are used for the convex hull of P(λ, n).

P(m, n) is the convex hull of all m × n sign matrices.They correspond to all tableaux that fit into an m × nbox.



(0, 0, 0) (1, 0, 0)

(0, 1, 0)

(0, 0, 1)

(1, 1, 0)

(1, 1, 1)(0, 1, 1)

(1, 0, 1)

P(1, 3)



(0, 0, 0) (1, 0, 0)

(0, 1, 0)

(0, 0, 1)

(1, 1, 0)

(1, 1, 1)(0, 1, 1)

(1, 0, 1)

1

2

3

32

31

3

12

21

∅P(1, 3) and the corresponding tableaux.



(0, 0, 0) (1, 0, 0)

(0, 1, 0)

(0, 0, 1)

(1, 1, 0)

(1, 1, 1)(0, 1, 1)

(1, 0, 1)

1

2

3

32

31

3

12

21

∅

P ([1], 3)

P ([1, 1], 3) P ([1, 1, 1], 3)

P ([ ], 3)

P(1, 3) and the corresponding tableaux.


Main Topics

1 Background






Comparing the ASM and Birkhoff Polytopes

Birkhoff ASMn P(m, n)

Dimension (n − 1)2 (n − 1)2 mn

Inequality rows and columns sum to 1 0 ≤ partial columns ≤ 1Description entries ≥ 0 partial sums ≥ 0 row partials ≥ 0

Vertices n!n−1∏j=0

(3j + 1)!

(n + j)!

∏1≤i≤j≤n

m + i + j − 1

i + j − 1

Facets n2 4[(n − 2)2 + 1] 3mn − n − 2(m − 1)


What is next?

Are there any connections to these polytopes?

Transportation Polytopes

Gelfand-Tsetlin Polytope

Other things with these polytopes?

Describe the face lattice

Explore more with P(m, n) vs. P(n,m) vs. P(λ, n).

Look into the symmetry and other possible geometry


Thanks!


Semistandard Young Tableaux Polytopes Sara Solhjem Joint ...sarsolhj/GSCC.pdfSara Solhjem (NDSU)...

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