Fall 2025 Seminar

Title:  The Delannoy category and its diagrammatics

Abstract:  N.Harman and A.Snowden discovered a semisimple monoidal pivotal category, which they called the Delannoy category, where composition of morphisms is given by computing the compact Euler characteristic of subspaces of the Euclidean space described by inequalities on the coordinates. In the talk we will explain a diagrammatic description of their category, following a joint work with N.Snyder. The number of simple objects in the Delannoy category grows exponentially, but a suitable monoidal subcategory has the Grothendieck ring isomorphic to the ring of integer-valued one-variable polynomials. That subcategory can be viewed as a categorification of the latter ring.

Title:  Springer Fibres, Parabolic Induction & Stacking Maps

Abstract:  Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in algebraic geometry, representation theory and combinatorics. In this talk, I will describe how we can use the combinatorics of (bi)tableaux to describe their geometry in low dimensions, in particular, giving a description of their irreducible components. I will also describe a process of parabolic induction coming the nilpotent cone of a Levi subalgebra. This is joint work with Lewis Topley, and separately with Mee Seong Im and Arik Wilbert.

Title:  Semisimplifying categorical Heisenberg actions, diagrammatics, and periodic equivalences

Abstract:  Semisimplification functors on tensor categories underlie many powerful constructions in representation theory, including the Frobenius functor in modular representation theory and the Duflo-Serganova functor from Lie superalgebras.  I will introduce a systematic approach to applying semisimplification functors to categories defined over positive characteristic, with the most important examples being representation categories and degenerate categorical Heisenberg actions.  In the latter case, the functors will always define (non-exact!) morphisms of categorical actions, and in particular cases categorify a certain element of the mod-p centre of affine sl_p.   These functors admit a convenient diagrammatic description which clarifies their properties.  Finally, I will explain how these functors naturally globalize known equivalences of subcategories of representations of S_n, originally due to Henke-Koenig.

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Have a great break!