The GEARS seminar

Barbara Santana Muniz (Jagiellonian University, Krakow)

Title: Atomic decomposition of affine Kazhdan–Lusztig polynomials

Abstract: The spherical Hecke algebra is a deformation of the ring of dominant weights Z[X+], which appears as the decategorification of the categories in the geometric Satake equivalence. Its associated Kazhdan–Lusztig polynomials are fundamental combinatorial objects with multiple incarnations across representation theory. Chiefly, they admit interpretation as quantum analogues of weight multiplicities, although realizing this interpretation for all types is a long-standing problem in algebraic combinatorics.

In this talk, we will introduce these Hecke algebras in accessible terms and discuss the existence of an atomic decomposition, a strengthening of the monotonicity condition on the KL polynomials that is closely related to this realization problem.

Julie Tavernier (University of Bath)

Title: When the McKay correspondence goes wild

Abstract: A theorem by Batyrev says that for a finite group G, the Euler characteristic of a crepant resolution of the quotient of affine space by the action of G is equal to the number of conjugacy classes of G. This is part of the McKay correspondence.
But does this result also hold over fields of prime characteristic?
In this talk, we explore what happens to this correspondence when |G| is divisible by the characteristic of the base field, known as the wild McKay correspondence. We attempt to recover a version of Batyrev's statement for certain metacyclic groups, and discuss the difficulties occurring in the wild case. This is joint work with Takehiko Yasuda.

Xinyang Liu (University of Newcastle)

Title: Representation Theory of Very Non-Standard Quantum 𝔰𝔬(2N)

Abstract: The classical Gelfand–Tsetlin basis yields explicit constructions associated with a chain of nested orthogonal Lie algebras. To study its quantum analogue, standard Drinfeld–Jimbo quantum groups are insufficient.

In this talk, I will introduce a quantum analogue of 𝔰𝔬(2N) within the framework of Letzter’s quantum symmetric pairs and classify its finite-dimensional representations over arbitrary fields, where q is not a root of unity. These algebras depend on an additional parameter c. Under mild restrictions on c, we show that all such representations are deformations of classical 𝔰𝔬(2N)-modules. Furthermore, we classify these representations by their highest weights and demonstrate how these weights deform the classical notion of a dominant integral weight.

This talk is based on joint work with Stefan Kolb and Martina Balagovic.

Diana Bergerova (Edinburgh)

Title: Joyce's construction of the Donaldson-Thomas sheaf

Abstract: In simple terms, the Donaldson-Thomas (DT) invariant counts stable coherent sheaves on Calabi-Yau threefolds. It turns out that this count is closely related to taking the weighted Euler characteristic of the threefold itself. In some special cases we actually can express the weights using certain perverse sheaves called the vanishing cycles sheaves. We thus arrive to a possible construction of a categorified DT invariant, i.e. a sheaf whose cohomology gives the DT invariants. In this talk, I aim to explain this idea and, in particular, which problems arise in wanting to define such an invariant.

Emmanouil Sfinarolakis (Herriot Watt)

Title: Shannon Entropy to KL Divergence: Operadic Derivations Away from the Origin

Abstract: Bradley showed that Shannon entropy is a derivation of the operad of topological simplices, and that every derivation of this type recovers a scalar multiple of Shannon entropy when evaluated at the origin. This talk explains how this picture extends to relative entropy. The positive product operad O°_{KL} = Δ° × Δ° is introduced as the natural operadic setting for pairs of probability distributions, and the chain rule for Kullback--Leibler divergence is shown to be an operadic Leibniz rule. In this setting, KL divergence becomes a scalar derivation for the natural P--expectation coefficient structure.

The main focus is what happens beyond the scalar origin trace. The talk describes the scalar coefficient bimodule, its realization inside constant endomorphisms, and the full conditional endomorphism bimodule. While the scalar theory recovers the classical KL chain rule, the operator--valued theory reveals additional structure away from the origin: the affine theory admits a rigid finite--dimensional classification, while the nonlinear theory reveals a much larger family of derivations, including familiar dispersion laws as special cases. Thus, the framework generalises Bradley’s theorem from Shannon entropy to KL divergence and uncovers a richer operator--valued theory of information and dispersion.

Willoughby Seago (Glasgow)

Title: Alternating Vertex Models and Super Symmetry

Abstract: The six vertex model is a combinatorial tool with connections to many areas of mathematics, including the theory of quantum groups and crystals, where the combinatorics of Young tableaux play a key role through the RSK correspondence. In this talk I will introduce these topics, and a generalisation of the six vertex model, Young tableaux, and RSK, to the supersymmetric setting. Time permitting, I will discuss ongoing work relating these to crystals for quantum groups of Lie superalgebras.