TECHNICAL EXPLANATION OF THE 1998 TILINGS RESEARCH GROUP SHIRT (dated 5/8/98)

Both the front and back of the shirts depict alternating-sign matrices in graphical form.

For a discussion of alternating-sign matrices and the square ice model, see pages 5-6 and 27 of "Alternating-Sign Matrices and Domino Tilings" by Noam Elkies, Greg Kuperberg, Michael Larsen, and myself. (See http://jamespropp.org/shirt.html for links to these and other Web-pages mentioned here.)

An alternating-sign matrix is an n-by-n array of 0's, +1's, and -1's such that the non-zero entries in each row or column alternate in sign, beginning and ending with a +1:

```                               0  1  0  0
1 -1  0  1
0  1  0  0
0  0  1  0
```

In the square ice model, the edges of a finite n-by-n square grid like

```                               |  |  |  |
--o--o--o--o--
|  |  |  |
--o--o--o--o--
|  |  |  |
--o--o--o--o--
|  |  |  |
--o--o--o--o--
|  |  |  |
```
are oriented so that at each vertex there are two arrows pointing in and two arrows pointing out. Along the boundaries, we require that the edges point in along the top and bottom and out along the left and right sides. It is not hard to show that states of the square ice model satisfying these boundary conditions are in one-to-one correspondence with alternating-sign matrices of order n. On the other hand, it wasn't until the middle 1990's that enumerative combinatorialists were able to verify their conjecture (first formulated in the early 1980's) that the number of configurations is equal to
```	                    1!4!7!...(3n-2)!
--------------------
(n)!(n+1)!...(2n-1)!
```
This result was first proved by Doron Zeilberger and then given a different proof by Greg Kuperberg. For pointers into the literature on this problem, see Kuperberg's article "Another proof of the alternating sign matrix conjecture". But I digress.

You can turn states of the square ice model into tilings as follows: Put a dot in the middle of each square, and connect two dots by a 90-degree circular arc that bulges in the direction of the arrow that it crosses. Then each vertex (with two incoming arrows and two outgoing arrows) gets replaced by a region bounded by four 90-degree circular arcs, two of which bulge inward and two of which bulge outward. We call such a region a GASKET if the two inward bulges are opposite each other and a BASKET if they are adjacent.

In this way, an alternating-sign matrix corresponds bijectively to a way of tiling a certain region (basically a large square bounded by inward bulges along the top and bottom and outward bulges along the left and right) with gaskets and baskets. The front of the shirt depicts the gaskets-and-baskets tiling corresponding to a random alternating-sign matrix of order 50. "Random" means that all possibilities are equally likely to occur as outputs of the algorithm that was used. This algorithm, called "coupling from the past", may be of independent interest to you; see "Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics" (by David Wilson and myself) or "Generating Random Elements of a Finite Distributive Lattice" (by myself) or "Coupling from the Past: A User's Guide" (by David Wilson and myself) if you want to know more about this aspect of things.

In the picture, the two orientations of gaskets and the four orientations of baskets are all given distinct colors (six in all).

One thing to notice is that a randomly-chosen alternating-sign matrix tends to have its randomness concentrated in the middle. That is, nearly every alternating-sign matrix has a non-homogeneous spatial structure that is not at all part of the original definition. There are reasons to believe that, as n (the size of the matrix) goes to infinity, the boundary between the central chaotic region and the outer "frozen" regions converges to a perfect circle; see "Random Domino Tilings and the Arctic Circle Theorem" (by William Jockusch, Peter Shor and myself) and "Local Statistics for Random Domino Tilings of the Aztec Diamond" (by Henry Cohn, Noam Elkies, and myself) for an example of this phenomenon in the context of tilings of Aztec diamonds by dominoes, and "The Shape of a Typical Boxed Plane Partition" (by Henry Cohn, Michael Larsen and myself) for an example of this phenomenon in the context of tilings of hexagons by rhombuses. But it is still too soon to predict confidently that the limit shape is exactly circular. Considering how hard it was simply to COUNT alternating-sign matrices of order n, I suspect it will be quite a while before anyone does a rigorous analysis of the shape of this boundary.

Another way to depict the square-ice state is to alternately color the vertices black and white, and then to select just those edges that are oriented from a black vertex to a white vertex. This set of edges contains exactly half of the edges on the boundary of the region, along with many edges in the interior (exactly half of the interior edges, in fact) which link up those boundary edges to form paths of various lengths (along with other edges that form closed loops in the interior).

The back of the shirt depicts the family of paths and closed loops that arises in this way, using the same alternating-sign matrix that gave rise to the picture on the front of the shirt.

There are various combinatorial mysteries associated with this way of looking at an alternating-sign matrix as a family of paths joining up points on the boundary of a square. My students and I are currently working hard at explaining the enigmatic enumerative patterns we have observed.

The members of the Tilings Research Group during 1997-1998 were: Jim Propp, David Wilson, Henry Cohn, Ben Raphael, Karen Acquista, Matthew Blum, Carl Bosley, Constantin Chiscanu, Edward Early, Nicholas Eriksson, David Farris, Lukas Fidkowski, Marisa Gioioso, Harald Helfgott, Eric Kuo, Yvonne Lai, Annie Oreskovich, Vis Taraz, Ben Wieland, Lauren Williams, Jason Woolever, and Juwell Wu.