Greta Panova

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 theO(n = 1) dense loop model.

Strict unimodality of q-binomial coefficients

We prove the strict unimodality of the q-binomial coefficients (nk)q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of Sn representations.

Lozenge tilings with free boundary

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

Matrices with restricted entries and

{\infty}

-analogues of permutations

∞ at different rates and recover again the GUE-corners process near the boundary.

Hook formulas for skew shapes I. q-analogues and bijections

The geometric complexity theory program is an approach to separate algebraic complexity classes, more precisely to show the superpolynomial growth of the determinantal complexity dc(perm) of the permanent polynomial. Mulmuley and Sohoni showed that the vanishing behaviour of rectangular Kronecker coefficients could in principle be used to show some lower bounds on dc(perm) and they conjectured that superpolynomial lower bounds on dc(perm) could be shown in this way. In this paper we disprove this conjecture by Mulmuley and Sohoni, i.e., we prove that the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial lower bounds on dc(perm).

Asymptotics of the number of standard Young tableaux of skew shape

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are

No Occurrence Obstructions in Geometric Complexity Theory

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