83 research outputs found
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
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
. We give explicit bounds in terms of the number of parts
in the partitions, their largest part size and the smallest second
part of the three partitions. When , i.e. one of the partitions
is hook-like, the bounds are linear in , but depend exponentially on
. Moreover, similar bounds hold even when . By a separate
argument, we show that the positivity of Kronecker coefficients can be decided
in time for a bounded number of parts and without
restriction on . Related problems of computing Kronecker coefficients when
one partition is a hook, and computing characters of 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 -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 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 regarded as a
-module. Skew Howe duality
implies that the Young diagrams from each pair which
contributes to this decomposition turn out to be conjugate to each other,
i.e.~. We show that the Young diagram which corresponds
to a randomly selected irreducible component has the same
distribution as the Young diagram which consists of the boxes with entries
of a random Young tableau of rectangular shape with rows and
columns. This observation allows treatment of the asymptotic version of this
decomposition in the limit as tend to infinity.Comment: 17 pages. Version 2: change of title, section on bijective proofs
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