Alternating Sign Matrices and Symmetry

Alternating Sign Matricesand Symmetry

(or)

Generalized Q*bert Games: An Introduction

(or)

The Problem With Lewis Carroll

By Nickolas Chura

What to discuss…

• What are Alternating Sign Matrices?

• A counting problem

• How are they symmetric?

• Some faces of Alternating Sign Matrices

Matrices & Determinants

• A Matrix is a rectangular array of numbers.

• An example of a 3-by-3 matrix:

475.2

202

513


• The determinant of a square matrix is a number.

• The study of Matrices and Determinants has been traced back to the 2nd century BC.

• Question: How are determinants computed?

Answer: Lots of ways. Here is a lesser-known

method…

• Charles Dodgson (a.k.a. Lewis Carroll) developed a method for computing determinants called the method of condensation.


• So how does it work?

http://en.wikipedia.org/wiki/Image:LewisCarrollSelfPhoto.jpg


The determinant of the 2-by-2 matrix

dc

ba

.bcad is defined to be the number

Example: The determinant of

45

13

is equal to 7512)5)(1()4)(3(


The Method of Condensation –• Let A be an n-by-n matrix. Compute the

determinant of each connected 2-by-2 minor of A and form an (n-1)-by-(n-1) matrix from these numbers.

• Repeat this process until a 1-by-1 matrix results.• After 2 iterations, each entry in resulting

matrices must be divided by the center entry of the corresponding 3-by-3 submatrix 2 steps prior.


The Method of Condensation –

Example:

100

272

440

10

254

36

6108

2108

475

2224

1412

0232

2311

210

20


Why haven’t you heard of this method before?

475.2

202

513

Consider using condensation on our 1st example

of a matrix:

Will condensation give the correct answer of 31?

Ans: No. We will end up with 0/0.


Ignoring the problem of division by zero, if we use

condensation on a general 3-by-3 matrix:

cegcdhbfgbdfhebdfhebdiafhaei

fheiegdh

cebfbdae

ihg

fed

cba

11

We get 7 distinct terms (up to sign)…


For each term, create a 3-by-3 matrix whose entries

are the exponents of the variables in their original

positions.

cegcdhbfgbdfhebdiafhaei 1

001

010

100

010

001

100

001

100

010

010

111

010

100

001

010

010

100

001

100

010

001


Things to notice about these matrices:• The entries are 0, 1, or -1

• The columns and rows each sum to 1

• The nonzero entries of rows and columns alternate in sign

001

010

100

010

001

100

001

100

010

010

111

010

100

001

010

010

100

001

100

010

001

Definition: An Alternating Sign Matrix (or ASM) is an n-by-n matrix whose entries are each 0, 1, or

-1 with the property that the sum of each row or column is 1, and the non-zero entries in any row or column alternate in sign.

00010

01011

11100

01110

00100

10

01)1(Examples:

Any permutation matrix is an ASM.

Alternating Sign Matrices

David Robbins introduced

ASMs and studied them along

with Howard Rumsey and

William Mills in the 1980s.

They conjectured that the

number of n-by-n ASMs is

given by the formula:

1

0 )!(

)!13(n

jn jn

jA


nACompare the growth of and n!:

n n! An

1 1 1

2 2 2

3 6 7

4 24 42

5 120 429

6 720 7436

7 5040 218348

8 40320 10850216

9 362880 911835460

10 3628800 129534272700

Alternating Sign Matrices• This is an important counting problem which answers many interesting questions.• Conjecture was proved in 1996 by Doron Zeilberger.• Also in 1996, Greg Kuperberg discovered a connection to physics, leading to a simpler proof.


What did Kuperberg discover?

Physicists had been studying ASMs under a

different name: Square Ice

Square ice?

It is a 2-dimensional square lattice of water

molecules.


An example of Square Ice:

Alternating Sign MatricesSquare Ice is really a connected directed

graph:• Oxygen atoms are vertices• Hydrogen atoms are edges• An edge points toward the vertex which

it is bonded to• Require* that Hydrogens are bonded all along the sides and none top or bottom


Our example of Square Ice seen as a graph:


To change Square Ice into an ASM:• There are 6 types of internal vertices• Replace the vertices by 0, 1, or -1 according to their type


Our Square Ice graph and its ASM:

0010

1000

0111

0010

Alternating Sign MatricesTo change an ASM into Square Ice:• Replace the 1s and -1s by their vertex types.• Choose the 0-vertex type so orientations

along the horizontal and vertical paths through that vertex are unchanged.• Conclusion: ASMs and Square Ice

(with *) are in bijection.

Square Ice• Impose coordinates on our graph.

• Define the parity of a vertex (x, y) to be the parity of x + y.

• Color an edge blue if it points from an odd

to an even vertex, color green otherwise.

Square Ice

Our resulting graph becomes

Square IceFacts about the 2-colored graph:

• Exterior edges alternate in color.• Monochromatic components are either paths connecting exterior vertices or they are

cycles.• The graph is determined by either the blue

or green subgraph.

Square Ice

The 7 3-by-3 ASMs and their Square Ice

blue subgraphs:

010

111

010

001

010

100

010

001

100

001

100

010

100

001

010

010

100

001

100

010

001

Square Ice

Now number the external blue vertices…

and call vertices joined by a path paired.

Square Ice

Now rotate the numbers 60o anticlockwise…

and the pairing gets rotated clockwise.

Square Ice

The pairing of these graphs is (2,3)(4,5)(6,1).

But after rotation, it becomes (1,2)(3,4)(5,6).

But we already had graphs with this pairing…

so there were 2 before and 2 after rotating.

Square IceTheorem.

Let A(pb,pg,L) be the set of ASMs with blue

pairing pb, green pairing pg, and total number

of cycles L. If p/b is pb rotated clockwise and

p/g is pg rotated anticlockwise, then the sets

A(pb,pg,L) and A(p/b,p/

g,L) are in bijection.

We will construct this bijection in stages.

There is a more general property here:

Square IceThe parity of a square in the graph is the

parity of its lower left or upper right vertex.

Here are the 1-squares…and here are the 0-squares.We refer to a square of parity k as a

k-square.

Square IceCall a square alternating if its 4 sides

alternate in color around the square.

Here are the alternating squares.

Square IceCall a vertex k-fixed if its incident blue edges

are on different k-squares.

These are the 1-fixed vertices.

Square IceCall a vertex k-fixed if its incident blue edges

are on different k-squares.

These are the 0-fixed vertices.

Square IceDefine functions Gk which switch the edge-

colors of all alternating k-squares.

Then define Hk = Gk O R where R switches

the color of every edge in the graph.

Finally, define the function G = H0 O H1 which

is our desired bijection!

Square IceWhat needs to be shown?

• The functions Hk send paths to paths and

cycles to cycles.

Method: Show that k-fixed vertices are k-

fixed and connected before and after Hk.

Determine what happens on the edges of the graph to paths after Hk. Characterize

paths and cycles by their k-fixed vertices.

Square IceWhat needs to be shown?• Show that the total number of cycles is unchanged.• The blue pairing rotates clockwise and the

green pairing rotates anticlockwise.• Show bijectivity of G.

Square Ice• Finally, reflection over the line y = x composed with either Hk will rotate pairings

and preserve the total number of cycles.

• Conclude that D2n is a symmetry group on

ASMs.

Now a large example…

000000010000000

000001010010000

000011100010010

000100100010000

001100010000000

000110110001000

000000001000000

011000011101111

000001000000000

000011000001000

001011101100010

101011100000000

000001000000000

001000000000000

000000000100000• Take a 15-by-15 ASM and look at its blue

subgraph.• Consider a path and see how the functions

H1 and H0 preserve the 1- and 0-vertices.

• Repeat for the green subgraph.

Blue subgraph after H1Blue subgraph after H0Blue subgraph

The

1-vertices

The

0-vertices

Green subgraph after H1after H0

The

1-vertices

The

0-vertices

Another problem…Recall: An integer partition is a way to write

a positive integer as a sum of other positive

integers.

Example: The number 4 can be written as

4, 3+1, 2+2, 2+1+1, and 1+1+1+1.

This can be shown with a diagram…

Another problem…

One method is by using Young Tableaux.

Here are the partitions of the number 4.

Another problem…Mathematicians Percival MacMahon, Basil

Gordon, Donald Knuth, and others researched

a 3-D generalization of integer partitions.

Enter: Plane partitions

A plane partition is an assemblage of unit

cubes pushed into a corner.

Another problem…

A plane partition of 11 cubes:

Another problem…

But the most famous plane partition of all:

Another problem…A descending plane partition of order n is

a 2-dimensional array of positive integers

less than or equal to n such that the left-

hand edges are successively indented, there

is weak decrease across rows and strict

decrease down columns, and the number of

entries in a row is strictly less than the

largest entry in that row.

Another problem…

An example of a descending plane partition:

6 6 6 4 3 3 3 2

Another problem…Theorem:

The number of descending plane partitions

with largest part less than or equal to r

equals the number of n-by-n ASMs.

Even more problems!More counting problems are tied up in

counting ASMs.

Example: Jig saw puzzles

(see the poster)

Conclusion

For people who like to count, ASMs are

where it’s at.

References• The book Proofs and Confirmations by

David Bressoud• How the Alternating Sign Matrix Conjecture

Was Solved, by James Propp• A Large Dihedral Symmetry of the Set of

Alternating Sign Matrices, by Benjamin

Wieland

Thank You

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Alternating Sign Matrices and Symmetry