Research Interests

My research lies at the intersection of representation theory and algebraic combinatorics, exploring the connections between abstract algebraic structures and rich combinatorial patterns. I often use combinatorial methods to address problems in representation theory and apply representation-theoretic techniques to gain deeper insights into combinatorics.

Two fundamental problems in the representation theory of groups and algebras are the tensor product decomposition and the restriction problem, which seek explicit formulas for specific nonnegative integers known as multiplicities. For general linear groups GL_n(C), the multiplicities in tensor product decompositions are elegantly described by the Littlewood–Richardson rule, first stated in 1934 and proven in the 1970s. This result uncovers profound connections between representation theory, combinatorics, and algebraic geometry. In contrast, the multiplicities for symmetric groups S_n, known as the Kronecker problem (introduced by Murnaghan in 1938), still remain a mysterious open problem (#P−hard in terms of computational complexity).  The restriction problem examines how representations of a group are restricted to its subgroups. For GL_n(C) and its subgroup S_n, this is one of the most significant open questions in algebraic combinatorics.

Addressing these problems is a significant focus of my research. Collaborating with peers, I introduced a new diagram algebra based on multiset partitions. Our algebra adapts the well-known theme of Schur–Weyl duality as it extends the classical partition algebra of Paul Martin and Vaughan Jones (1990), initially developed in connection with statistical physics. Then, we give two innovative approaches to attack the restriction problem: moments of character polynomials and polynomial induction. These methods have resolved the restriction problem in several significant cases. These works received notable interest from experts, including Mike Zabrocki, Rosa Orellana, Anne Schilling, and Paul Martin, the inventor of partition algebra.

Apart from the theme above, I have explored many beautiful topics in algebraic combinatorics, for example,

• combinatorics of q, t-Catalan polynomials,

• Crystal structure (Masaki Kashiwara) for the Burge correspondence, a variant of the Robinson–Schensted–Knuth (RSK) correspondence,

• Quasi-Steinberg characters for complex reflection groups, motivated by a question of Dipendra Prasad.

My current research interests include

• investigating the asymptotical behavior of character table sums of a finite Coxeter group,

• classifying multiplicity-free tensor products and revealing their connection to simultaneous conjugacy classes, and Kronecker-Hecke algebra for any finite groups, and

• exploring the poset structure on integer partitions induced by the relation: monomial-positivity of difference of Schur polynomials. We adapted the notion of immersion of representations (due to Prasad and Raghunathan) for GL_n(C).