Research
My research interests lie in the field of Algebraic Combinatorics. Some of my favorite combinatorial objects include:
the Kronecker coefficients
vacillating tableaux
My research interests lie in the field of Algebraic Combinatorics. Some of my favorite combinatorial objects include:
the Kronecker coefficients
vacillating tableaux
On the Correspondence Between Integer Sequences and Vacillating Tableaux (with Zhanar Berikkyzy, Pamela E. Harris, Anna Pun and Catherine Yan), submitted (2024).
A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape lambda and length 2k. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map $DI_n^k$ for general integers $n$ and $k$. In particular, we characterize the integer sequences $\bsy{i}$ whose corresponding shape, $\lambda$, in the image $DI_n^k(i)$, satisfies $\lambda_1 = n$ or $\lambda_1 = n -k$.
The Newton polytope of the Kronecker product (with Greta Panova), submitted (2023).
We study the Kronecker product of two Schur functions $s_\lambda\ast s_\mu$, defined as the image of the characteristic map of the product of two $S_n$ irreducible characters. We prove special cases of a conjecture of Monical--Tokcan--Yong that its monomial expansion has a saturated Newton polytope. Our proofs employ the Horn inequalities for positivity of Littlewood-Richardson coefficients and imply necessary conditions for the positivity of Kronecker coefficients.
On the Kronecker product of Schur functions of square shapes, submitted (2023).
Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of irreducible representations associated with square Young diagrams. We give a formula for computing Kronecker coefficients, which are indexed by two square partitions and a three-row partition, specifically one with a short second row and the smallest part equal to 1. We also prove the positivity of square Kronecker coefficients for particular families of partitions, including three-row partitions and near-hooks.
Combinatorial identities for vacillating tableaux (with Zhanar Berikkyzy, Pamela E. Harris, Anna Pun and Catherine Yan), to appear in Integers (2023).
Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions. In this paper, we further investigate the enumeration of vacillating tableaux and derive multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux, simplified vacillating tableaux, and limiting vacillating tableaux.
On the limiting vacillating tableaux for integer sequences (with Zhanar Berikkyzy, Pamela E. Harris, Anna Pun and Catherine Yan), to appear in Journal of Combinatorics (2022).
A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$.
Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair consisting of a standard Young tableau and a vacillating tableau. In this paper, we show that for a given integer sequence $\bsy{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(\bsy{i})$ become stable when $n \rightarrow \infty$; the limit is called the limiting vacillating tableau for $\bsy{i}$. We give a characterization of the set of limiting vacillating tableaux and present explicit formulas that enumerate those vacillating tableaux.
The Newton polytope of the Kronecker product, FPSAC 2024, Bochum, Germany. July 2024. (poster, extended abstract)
UCLA Student Combinatorics Seminar, Los Angeles, CA. May 2024.
USC Combinatorics Seminar, Los Angeles, CA. Feb 2024.
UC Davis ADM Seminar, Davis, CA. Jan 2024.
SoCal Discrete Math Symposium 2023, Los Angeles, CA. Nov 2023.
UCLA Combinatorics Forum, Los Angeles, CA. Oct 2023.
The Kronecker product of Schur functions, USC Combinatorics Seminar, Los Angeles, CA. Mar 2023. (video)
On the limiting vacillating tableaux for integer sequences, JMM 2023, AMS Special Session on Research Community in Algebraic Combinatorics, Boston, MA. Jan 2023.
Oregon State University Algebra and Number Theory Seminar, Corvallis, OR. Nov 2022. (video)