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My research interests are in the area of algebraic and enumerative combinatorics and its applications to statistical mechanics, representation theory and algebra.


Papers


Kronecker products, characters, partitions, and the tensor square conjectures; with Igor Pak and Ernesto Vallejo.
We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups S_n contain all irreducibles as their constituents. Our main result is that they contain representations corresponding to hooks and two row Young diagrams. For that, we develop a new sufficient condition for the positivity of Kronecker coefficients in terms of characters, and use combinatorics of rim hook tableaux combined with known results on unimodality of certain partition functions. We also present connections and speculations on random characters of S_n.


Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory; with V. Gorin.
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.


Schur times Schubert via the Fomin-Kirillov algebra; with K. Meszaros, A. Postnikov.
The aim of this paper is to extend the Pieri formula using the Fomin-Kirillov quadratic algebra. We focus on multiplication of any Schubert polynomial \mathfrak{S}_w by a Schur polynomial s_{\lambda}. We derive combinatorial expressions for the expansion coefficients for certain special partitions \lambda, including hooks and the 2 by 2 box. We achieve this by proving special cases of the nonnegativity conjecture of Fomin and Kirillov. This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.


Dyck tilings, linear extensions, descents, and inversions; joint with J.S. Kim, K.Meszaros, D.B. Wilson.
extended abstract in Discrete Math and Theoretical Computer Science, FPSAC 2012 Proceedings; full version submitted.
Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give two bijections between "cover-inclusive" Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy" between the upper and lower boundary of the tiling to descents of the linear extension.


Combinatorial applications of symmetric function theory to certain classes of permutations and truncated tableaux; PhD dissertation, Harvard, April 2011.
My dissertation includes 3 of the papers below (with some modifications) Tableaux and plane partitions of truncated shapes, Separable permutations and Greene's theorem and Bijective enumeration of permutations starting with a longest increasing subsequence. The Background chapter is a review of the theory of symmetric functions and is all one needs to know to understand these papers.


Tableaux and plane partitions of truncated shapes; version 2, submitted.
Extended abstract at FPSAC 2011, Discrete Mathematics and Theoretical Computer Science proceedings AO, 2011, p.753-764
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.

Matrices with restricted entries and q-analogues of permutations; joint with J.Lewis, R.Liu, A.Morales, S.Sam, Y.Zhang.
Extended abstract at FPSAC 2011, DMTCS, proceedings AO, 2011, p.645-656
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 $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

Separable permutations and Greene's theorem; joint with A.Crites, G.Warrington.
We study the shape of separable (3142 and 2413-avoiding) permutations under RSK in light of Greene's theorem. We show that if the shape of a separable permutation $\sigma$ is $\lambda=(\lambda_1,...,\lambda_k)$, then $\sigma$ has $k$ disjoint increasing subsequences of lengths $\lambda_1,...,\lambda_k$. As a corollary, we prove that if $\sigma$ is a separable subsequence of a word $w$, then the shape of $\sigma$ is contained in the shape of $w$ as Young diagrams. These facts are also used to exhibit lower bounds on the length of words containing certain separable permutations as patterns.

Factorization of banded permutations , Proceedings of the AMS, to appear.
We prove a conjecture of Gilbert Strang stating that a banded permutation of bandwidth $w$ can be represented as a product of at most $2w-1$ permutations of bandwidth 1.

Bijective enumeration of permutations starting with a longest increasing subsequence, Discrete Mathematics and Theoretical Computer Science (2010)
We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial calculations in recent work of Adriano Garsia and Alain Goupil. We give two `elementary' bijective proofs of this result and of its $q$-analogue, one proof using the RSK correspondence and one only permutations.

Polynomiality of some hook-length statistics, The Ramanujan Journal, to appear.
We prove an equation conjectured by Okada regarding hook-lengths of partitions, namely that $$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}(h_u^2 - i^2) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{r+1} \prod_{j=0}^{r} (n-j),$$ where $f_{\lambda}$ is the number of standard Young tableaux of shape $\lambda$ and $h_u$ is the hook length of the square $u$ of $\lambda$. We also obtain other similar formulas.

Lie Algebras, Representation theory and GL_n
Minor thesis towards Ph.D. requirements, advisor Joseph Harris.

Talks

Random tilings workshop, SUNY Stony Brook, February 2013.

UCLA Probability Seminar, January 2013.

UCLA Combinatorics Seminar, December 2012.

FPSAC, poster presentation, Nagoya, Japan, August 2012.

Workshop on convex polytopes, RIMS Kyoto, Japan, July 2012.

MSRI Postdoc Seminar, Berkeley, March 2012.

UC Berkeley Combinatorics Seminar, March 2012.

Counting tricks with symmetric functions, MSRI Evans Lecture Series, UC Berkeley, February 2012.

Dyck tilings, linear extensions and descents, UCLA Combinatorics Seminar, December 2011

Formal Power Series and Algebraic Combinatorics, Reykjavik, June 2011

Invited talk (and visitor) at Microsoft Research Theory Group, Redmond WA, May 2011

Invited talk at AMS Eastern Regional Meeting, College of the Holy Cross, Apr 2011.

University of Pennsilvania Combinatorics Seminar, Jan 2011.

Truncated tableaux and plane partitions, MIT Combinatorics Seminar.

Separable permutations, Robinson-Schensted and shortest containing supersequence, talk at Permutation Patterns 2010

Bijective enumeration of permutations starting with a longest increasing subsequence , poster presentation at Formal Power Series and Algebraic Combinatorics 2010

Bijective enumeration of permutations starting with a longest increasing subsequence , invited talk at Joint Math Meeting 2010, Special Session on Permutations.