10,282 research outputs found
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 be a finite Weyl group and be the corresponding affine
Weyl group. We show that a large element in , randomly generated by
(reduced) multiplication by simple generators, almost surely has one of
-specific shapes. Equivalently, a reduced random walk in the regions of
the affine Coxeter arrangement asymptotically approaches one of -many
directions. The coordinates of this direction, together with the probabilities
of each direction can be calculated via a Markov chain on . Our results,
applied to type , show that a large random -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
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