Minutes from REACH, 4/2/02 - Gabriel D. Carroll Reminders: -> Seth takes notes Thursday -> Roberto will discuss relation between continuous and discrete Hirota equations Thursday -> David will see if he has notes about the alternate Somos-4 -> Jim will work on a T-shirt design -> Gregg will inform owner of the Somos sequence site that it needs to be updated (the one that Roberto talked about; not Jim's Somos sequence site) Details: Seth will take notes on Thursday. David and Gregg will be available for next week. David is writing up a general proof of crosses and wrenches. He is not including a discussion of coordinate changes such as those we have used to obtain the Somos graphs; this is to be investigated separately. Gabriel is writing up a proof of Kuo condensation for suitable induced subgraphs of square lattices with some horizontal edges deleted. Hopefully it will explicitly provide a general combinatorial model for Gale-Robinson sequences. Jim asks about hexagonal lattices. David comments that they are problematic because they correspond to initial conditions of slope 1, which the light cone consequently misses. The fact that Kuo is able to treat them in his article is therefore puzzling. Jim asks whether quadratic recurrences other than the Somos ones, such as those in the article by David Gale provided at the beginning of the semester, can be lifted to higher dimensions so as to yield combinatorial interpretations. This could be worth examining. Roberto agrees to prepare, for Thursday, a discussion of how our recurrences (the discrete Hirota equation) relate to their continuous analogues (wave equations). Our recurrences are related to cluster algebras on one hand, Grassmannians of infinite-dimensional spaces (used to solve wave equations) on the other. Gregg suggests examining directly the relation between cluster algebras and Grassmannians, especially since the latter can be parametrized by polynomial constraints for which the machinery of cluster algebras seems applicable. Seth presents the Somos-5 graphs. He is trying more generally to fit our graphs into a hex-trap lattice - a modified hexagonal lattice where some hexagons are "split" into two trapezoids by horizontal edges; the split hexagons occur periodically. The tilings of the Lowell House bathroom floors provide inspiration here. The lattices corresponding to Somos-k (k = 2, 3, 4, 5) fit a regular pattern, and sure enough, as is discovered midway through the meeting, the next one appears to represent the three-term version of Somos-6 (i.e. s_(n-3)*s_(n+3) = s_(n-2)*s_(n+2) + s_(n-1)*s_(n+1) ). Gregg notices that his spines (which satisfy the recurrence obtained by replacing one right-hand term of the original recurrence by 1) constitute the diagonals of these graphs; it seems reasonable that the other spines (which come from replacing the other term) should correspond to the diagonals in the other direction - a sort of "duality." Someone should flesh this out. David comments that he and Dennis have a Mathematica notebook that computes the initial conditions corresponding to a given Gale-Robinson recurrence under the requisite change of variables. It's only in rough shape, though. David will check whether he has information in his notes about the alternate Somos-4 recurrence, which he may have used as a test case for the aforementioned notebook. It will be of interest to make sure we understand the details here. Jim asks whether it would be fruitful to work with the subgraphs induced by arbitrary regions of arbitrary hex-trap arrays and see what happens. A hex-trap lattice has certain symmetries, which are two-dimensional affine transformations. It might be worthwhile to work out how these relate to the three-dimensional transformations used to change variables in the corresponding recurrences. Jim asks a question about the face-variables version of Somos-3 (which we obtain by simply writing down the octahedron recurrence, without any reindexing, and using suitable initial conditions): The variables split naturally into three levels. If we replace the bottom variables by 1, each resulting term of a Somos-3 polynomial corresponds to something like an ASM. The same thing happens if we replace the top variables by 1. We seem to get the same variant of ASMs both ways. Why? A helpful way of thinking of this is to consider the analogous situation with Somos-2, which is just Dodgson condensation. Here there are two layers of initial conditions, and we get two different families of polynomials, depending on which layer we set equal to 1. These polynomials are the same, except for a shift of indices and a reversing of the signs on all the exponents. Somos-3 seems to have the same kind of symmetry. Can we understand the same pattern (if it occurs) for Somos-4 and Somos-5? There may be an easy algebraic proof, obtained by simply running the Somos sequences backwards and using symmetry. Jim also suggests working with the face-variables version of the cube recurrence. He suggests that the picture we want to obtain may be algebraic rather than combinatorial. For example, compatible pairs of ASMs are described by suitable linear inequalities. The objects counted by the cube recurrence (which can be read off from the exponents of the terms of the face-variable polynomials generated by the recurrence) may similarly be best described as integer arrays governed by some linear constraints. For more on this, see two of Jim's posts to the bilinear forum: one posted on Nov 15, 2000 and the other posted on Aug 28, 2001. (The latter refers to the recurrence as the "hexagon recurrence", but Jim thinks the term "cube recurrence" is better.) Some discussion of T-shirt designs occurs. We would like to find nice ways of illustrating the matchings of Somos graphs. Seth suggests dualizing the hexagonal lattice to obtain a triangular grid; the matchings then become tilings with lozenges and bowties, where the bowties can occur only in certain positions. Gabriel suggests dualizing the square lattice to obtain a square grid, tiled by dominoes, with some walls that a domino cannot cross. Jim will produce some random tilings using vaxrandom to see if there are interesting boundary effects. Meanwhile he encourages others to experiment with different visual representations. Roberto consults a website about Somos sequences, which, we discover, is out of date (e.g. it claims that integrality is unproven for Somos-7) -- Gregg will track down the owner and ask him/her to update it. The site tells us that, empirically, the minimum value of n such that the n-th term of Somos-k is not an integer is 2k+1 (k = 8, 9) and 2k (k = 10, 11, 12). It could be worthwhile to prove that integrality always fails at position 2k for k > 9, but then again, it's probably not very interesting. Roberto uses Ehrhart reciprocity to define Somos-k for k < 0. We expect it will just be the usual Somos recurrence, with some sign changes thrown in. Gregg has been looking a little at "random walks" (or not so random ones) in Somos cluster algebras to see if we obtain any interesting sequences. Nothing has come up yet. He also suggests trying to use cluster-algebra windows to prove that, if sufficiently many consecutive terms of a Somos-k sequence are all integers, so is the next. Roberto suggests making a table of the n-th term of Somos-k (indexed by n and k) and looking for patterns. Jim thinks this could be a cave. There are three programs to count matchings of subgraphs of square grids. One is ren (short for "urban renewal"). Another is vaxmaple, which runs as a front end to Maple. Yet another is vaxmacs, which runs in Emacs and has the advantage of offering an interactive environment - you can modify the graph and compute its number of matchings at any time, as well as other nice features. It is decided that we will wait until April 15 for the next round of Monday meetings.