Lozenge tilings with free boundaries

We study lozenge tilings of a domain with partially free boundary. In particular, we consider a trapezoidal domain (half hexagon), s.t. the horizontal lozenges on the long side can intersect it anywhere to protrude halfway across. We show that the positions of the horizontal lozenges near the opposite flat vertical boundary have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (the GUE-corners/minors process). We also prove the existence of a limit shape of the height function, which is also a vertically symmetric plane partition. Both behaviors are shown to coincide with those of the corresponding doubled fixed-boundary hexagonal domain. We also consider domains where the different sides converge to $\infty$ at different rates and recover again the GUE-corners process near the boundary.Comment: 27 pages, 4 figures; version 2-- typos fixed, improved proofs and computations, incorporated referee's comments. To appear in Letters in Mathematical Physic

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.Comment: Accepted to Advances in Applied Mathematics. Final versio

On the complexity of computing Kronecker coefficients

We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction on $M$. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of $S_n$ are also considered.Comment: v3: incorporated referee's comments; accepted to Computational Complexit

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.Comment: Published at http://dx.doi.org/10.1214/14-AOP955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial. Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.Comment: 20 page

Skew Howe duality and random rectangular Young tableaux

We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(\lambda,\mu)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$\mu=\lambda'$. We show that the Young diagram $\lambda$ which corresponds to a randomly selected irreducible component $(\lambda,\lambda')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular shape with $m$ rows and $n$ columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as $m,n,p\to\infty$ tend to infinity.Comment: 17 pages. Version 2: change of title, section on bijective proofs improve