Alternating Sign Matrices and Symmetry

1. Alternating Sign Matricesand Symmetry (or)Generalized Q*bert Games: An Introduction (or)The Problem With Lewis Carroll By Nickolas Chura

2. What to discuss… • What are Alternating Sign Matrices? • A counting problem • How are they symmetric? • Some faces of Alternating Sign Matrices

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

4. 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…

5. 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?

6. Matrices & Determinants The determinant of the 2-by-2 matrix is defined to be the number Example: The determinant of is equal to

7. 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.

8. Matrices & Determinants The Method of Condensation – Example:

9. Matrices & Determinants Why haven’t 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.

10. 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)…

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

12. 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

13. 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.

14. 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:

15. Alternating Sign Matrices Compare the growth of and n!:

16. 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.

17. 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.

18. Alternating Sign Matrices An example of Square Ice:

19. 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

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

21. 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

22. Alternating Sign Matrices Our Square Ice graph and its ASM:

23. 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.

24. 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.

25. Square Ice Our resulting graph becomes

26. 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.

27. Square Ice The 7 3-by-3 ASMs and their Square Ice blue subgraphs:

28. Square Ice Now number the external blue vertices… and call vertices joined by a path paired.

29. Square Ice Now rotate the numbers 60o anticlockwise… and the pairing gets rotated clockwise.

30. 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.

31. 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.

32. 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.

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

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

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

36. Square Ice Define functions Gkwhich switch the edge- colors of all alternating k-squares. Then define Hk = Gk OR where R switches the color of every edge in the graph. Finally, define the function G = H0O H1which is our desired bijection!

37. 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.

38. 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.

39. 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.

40. 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.

41. Blue subgraph after H1 Blue subgraph after H0 Blue subgraph The 1-vertices The 0-vertices

42. Green subgraph after H1 after H0 The 1-vertices The 0-vertices

43. 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…

44. Another problem… One method is by using Young Tableaux. Here are the partitions of the number 4.

45. 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.

46. Another problem… A plane partition of 11 cubes:

47. Another problem… But the most famous plane partition of all:

48. 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.

49. Another problem… An example of a descending plane partition: 6 6 6 4 3 3 3 2

50. 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.