arXiv subjects

Victor L. Zhang

Publications and source records attributed to Victor L. Zhang.

3 recordsLinked to original sources

arXiv2026

Lusztig-Vogan categories of equal rank 2

Lusztig-Vogan categories are categorifications of the principal block of the Lusztig-Vogan module over the Hecke algebra, which captures information about characters of irreducible admissible representations of a real reductive group. Lusztig-Vogan categories can be constructed as module categories over Soergel bimodules. In this paper, we describe the structure of the rank 2 Lusztig-Vogan categories corresponding to equal rank real groups. More precisely, we classify indecomposable objects and describe the action of generating Soergel bimodules, recovering the $W$-graph of the underlying Lusztig-Vogan module. We also provide an algorithm which completes this procedure for arbitrary finite rank Lusztig-Vogan categories, including those which do not correspond to a real reductive group.

math.RT↗

arXiv2026

On detection probabilities of link invariants

We prove that, for many standard link invariants, both the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. In fact, almost every such link belongs to an invariant fiber whose size is itself exponential in n. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website gives a much more detailed view of the data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.

math.GT↗

arXiv2025

Semisimplifying categorical Heisenberg actions and periodic equivalences

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the cyclic group Brauer construction and categorify the Glauberman correspondence. In the setting of degenerate categorical Heisenberg actions, we obtain a rich collection of functors which commute with the categorical action. Applied to well-known categorifications of the basic representation and Fock space, our functors give explicit realizations of periodic equivalences for polynomial functors and symmetric groups first studied by Henke-Koenig. This allows us to globalize the equivalences of Henke-Koenig by symmetric monoidal functors. We apply these results to deduce branching properties of certain modular representations of $S_n$.

math.RT↗