Dimers, webs, and positroids

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally nonnegative Grassmannian. Considering pairing probabilities for the double-dimer model gives rise to Grassmann analogues of Rhoades and Skandera's Temperley-Lieb immanants. The same problem for the (probably novel) triple-dimer model gives rise to the combinatorics of Kuperberg's webs and Grassmann analogues of Pylyavskyy's web immanants. This draws a connection between the square move of plabic graphs (or urban renewal of planar bipartite graphs), and Kuperberg's square reduction of webs. Our results also suggest that canonical-like bases might be applied to the dimer model. We furthermore show that these functions on the Grassmannian are compatible with restriction to positroid varieties. Namely, our construction gives bases for the degree two and degree three components of the homogeneous coordinate ring of a positroid variety that are compatible with the cyclic group action.Comment: 25 page

Growth diagrams, Domino insertion and Sign-imbalance

We study some properties of domino insertion, focusing on aspects related to Fomin's growth diagrams. We give a self-contained proof of the semistandard domino-Schensted correspondence given by Shimozono and White, bypassing the connections with mixed insertion entirely. The correspondence is extended to the case of a nonempty 2-core and we give two dual domino-Schensted correspondences. We use our results to settle Stanley's `2^{n/2}' conjecture on sign-imbalance and to generalise the domino generating series of Kirillov, Lascoux, Leclerc and Thibon.Comment: 24 page

The shape of a random affine Weyl group element and random core partitions

Let $W$ be a finite Weyl group and ${\hat{W}}$ be the corresponding affine Weyl group. We show that a large element in ${\hat{W}}$, randomly generated by (reduced) multiplication by simple generators, almost surely has one of $|W|$-specific shapes. Equivalently, a reduced random walk in the regions of the affine Coxeter arrangement asymptotically approaches one of $|W|$-many directions. The coordinates of this direction, together with the probabilities of each direction can be calculated via a Markov chain on $W$. Our results, applied to type $\tilde{A}_{n-1}$, show that a large random $n$-core obtained from the natural growth process has a limiting shape which is a piecewise-linear graph. In this case, our random process is a periodic analogue of TASEP, and our limiting shapes can be compared with Rost's theorem on the limiting shape of TASEP.Comment: Published at http://dx.doi.org/10.1214/14-AOP915 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The uncrossing partial order on matchings is Eulerian

We prove that the partial order on the set of matchings of 2n points on a circle, given by resolving crossings, is an Eulerian poset.Comment: 6 page