This is me in front of my poster at MAAGC 2019.
Hi, I am Jiayuan Wang [CV]. I received my Ph.D. at The George Washington University Department of Mathematics in Spring 2022. My adviser is Joel Brewster Lewis. Starting Fall 2022, I am a Hsiung Visiting Assistant Professor at Lehigh University Department of Mathematics, under the guidance of Mark Skandera.
Research
My research is in combinatorics. I like factorization problems in complex reflection groups. Currently, I'm studying symmetric functions (in particular, the LLT polynomials) and totally nonnegative matrices (in particular, the nonnegativity properties of differences of products of permanents).
Complex reflection groups are complex versions of real reflection groups, also known as Coxeter groups. I have a paper with my adviser on the Hurwitz action in complex reflection groups (Combinatorial Theory, 2(1)). An extended abstract is published in Séminaire Lotharingien de Combinatoire, 85B, proceedings of the 33rd International Conference on "Formal Power Series and Algebraic Combinatorics" (FPSAC 2021), with an accompanying poster. I gave a talk [preseminar, seminar] on this project at the Discrete Math Seminar at UMass Amherst. There is also a fifteen-minute presentation at the Graduate Student Combinatorics Conference (GSCC 2021).
In a finite real reflection group, the reflection length of each element is equal to the codimension of its fixed space. However, in complex reflection groups, this is not always the case. In addition, for certain elements, the two functions (reflection length and codimension of fixed space) give rise to two different partial orders on the group. In this preprint with my adviser, we characterized elements in the combinatorial family of complex reflection groups for which the intervals below the element in these two posets coincide.
In a finite Coxeter group, fully commutative elements have been thoroughly studied. My paper on fully commutative elements in complex reflection groups (Discrete Mathematics, 346(12)) extended this study to the complex setting. This work is inspired by John Stembridge's paper, Some combinatorial aspects of reduced words in finite Coxeter groups and Kyu-Hwan Lee's talk, Fully commutative elements of complex reflection groups [slides] at Sage Days FPSAC 2019. There is a twenty-five-minute presentation I gave at the fourteenth annual Binghamton University Graduate Combinatorics, Algebra and Topology Conference 2021 (BUGCAT Conference). Here are the slides based on the talk I gave at AMS Contributed Paper Session (Lie Theory, Group Theory and Related Topics) during JMM 2022.
LLT polynomials are related to chromatic symmetric functions via a plethystic substitution. There are many open questions regarding these families of symmetric functions, including the e-positivity for the chromatic symmetric functions and the q-statistic for the Schur expansion of the LLT polynomials. Applying the plethystic substitution to the chromatic symmetric functions, we were able to define LLT-analogs of induced sign and induced trivial characters from the trace space for the Hecke algebra. Using immanant-like generating functions, we can combinatorially describe these characters. See the poster on our recent results here.
Contact Info
Email: jiw922 if-you're-not-a-computer-you-know-what-goes-here lehigh.edu