Alternating Sign Matrices and Symmetry

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Alternating Sign Matricesand Symmetry
(or)Generalized Qbert Games An Introduction
(or)The Problem With Lewis Carroll
By Nickolas Chura
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What to discuss

• What are Alternating Sign Matrices?
• A counting problem
• How are they symmetric?
• Some faces of Alternating Sign Matrices

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Matrices Determinants

• A Matrix is a rectangular array of numbers.
• An example of a 3-by-3 matrix

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Matrices Determinants

• 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
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Matrices Determinants

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

• So how does it work?

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Matrices Determinants

• The determinant of the 2-by-2 matrix

is defined to be the number
Example The determinant of
is equal to
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Matrices Determinants

• 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.

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Matrices Determinants

• The Method of Condensation
• Example

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Matrices Determinants

• Why havent you heard of this method before?

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.
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Matrices Determinants

• Ignoring the problem of division by zero, if we
  use
• condensation on a general 3-by-3 matrix

We get 7 distinct terms (up to sign)
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Matrices Determinants

• For each term, create a 3-by-3 matrix whose
  entries
• are the exponents of the variables in their
  original
• positions.

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Matrices Determinants

• 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

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• 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.

Examples
Any permutation matrix is an ASM.
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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
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Alternating Sign Matrices
Compare the growth of and n!
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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.

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Alternating Sign Matrices
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.
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Alternating Sign Matrices
An example of Square Ice
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Alternating Sign Matrices

• Square 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

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Alternating Sign Matrices
Our example of Square Ice seen as a graph
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Alternating Sign Matrices

• 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

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Alternating Sign Matrices
Our Square Ice graph and its ASM
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Alternating Sign Matrices

• To 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.

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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.

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Square Ice
Our resulting graph becomes
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Square Ice

• Facts 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.

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Square Ice
The 7 3-by-3 ASMs and their Square Ice blue
subgraphs
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Square Ice
Now number the external blue vertices
and call vertices joined by a path paired.
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Square Ice
Now rotate the numbers 60o anticlockwise
and the pairing gets rotated clockwise.
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Square Ice
The pairing of these graphs is (2,3)(4,5)(6,1).
But we already had graphs with this pairing
But after rotation, it becomes (1,2)(3,4)(5,6).
so there were 2 before and 2 after rotating.
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Square Ice
Theorem. 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.
There is a more general property here
We will construct this bijection in stages.
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Square Ice

The 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.
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Square Ice

Call a square alternating if its 4
sides alternate in color around the square.
Here are the alternating squares.
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Square Ice

Call a vertex k-fixed if its incident blue
edges are on different k-squares.
These are the 1-fixed vertices.
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Square Ice

Call a vertex k-fixed if its incident blue
edges are on different k-squares.
These are the 0-fixed vertices.
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Square Ice

Define 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!
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Square Ice

• What 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.

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Square Ice

• What 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.

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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.

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Now a large example

• 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.

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Blue subgraph after H1
Blue subgraph after H0
Blue subgraph
The 1-vertices
The 0-vertices
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Green subgraph
after H1
after H0
The 1-vertices
The 0-vertices
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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, 31, 22, 211, and 1111.
• This can be shown with a diagram

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Another problem

One method is by using Young Tableaux.
Here are the partitions of the number 4.
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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.
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Another problem
A plane partition of 11 cubes
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Another problem
But the most famous plane partition of all
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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.
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Another problem
An example of a descending plane partition
6 6 6 4 3
3 3
2
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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.
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Even more problems!
More counting problems are tied up in counting
ASMs. Example Jig saw puzzles
(see the poster)
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Conclusion
For people who like to count, ASMs are where its
at.
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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

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Thank You