—  Mathematics & Statistics

The TACO Seminar | Fall 2023

Wednesdays, 4:00–5:00. Zoom or IES 110

The TACO seminar focuses on topics in Topology, Algebra, Combinatorics, and Operators.

Organizers:  Emily Peters, Carmen Rovi

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)

Some of the talks will be on zoom. If you would like to attend a zoom talk let one of the organizers know so we can add you to the list of participants.

September 20: Anup Poudel. A comparison between diagrammatic categories

Starting from Kuperberg, presenting a (ribbon) tensor category using diagrammatic generators and relations has become a valuable resource for studying topological invariants coming from the category. In this talk, we will compare two diagrammatic categories coming from the category of representations of quantum SL(n). We will show, diagrammatically, that these two categories contain the same topological data and are categorically equivalent as ribbon categories. This lets us prove a conjecture of Le-Sikora. We will also compare the associated skein categories. The exposition will be more or less self-contained.

September 27: Chris Drupieski. The Lie superalgebra of transpositions

In this talk I will report on progress, joint with Jonathan Kujawa, to answer a series of questions originally posed by MathOverflow user WunderNatur in August 2022: Considering the group algebra of the symmetric group CS_n as a superalgebra (by considering the even permutations in to be of even superdegree and the odd permutations in to be of odd superdegree), and then in turn considering CS_n as a Lie superalgebra via the super commutator, what is the structure of CS_n as a Lie superalgebra, and what is the structure of the Lie sub-superalgebra of CS_n generated by the transpositions? The non-super versions of these questions were previously answered by Ivan Marin, with very different results.

October 4: George Seeliger. A raising operator formula for Macdonald polynomials via LLT polynomials in the elliptic Hall algebra

Macdonald polynomials are a basis of symmetric functions with coefficients in Q(q,t) exhibiting deep connections to representation theory and algebraic geometry. In particular, specific specializations of the q,t parameters recover various widely-studied bases of symmetric functions, such as Hall-Littlewood polynomials, Jack polynomials, q-Whittaker functions, and Schur functions. Central to this study is the fact that the Schur function basis expansion of the Macdonald polynomials have coefficients which are polynomials in q,t with nonnegative integer coefficients. One approach to this result lies in first expanding Macdonald polynomials into LLT polynomials via the work of Haglund-Haiman-Loehr. LLT polynomials were first introduced by Lascoux-Leclerc-Thibbon as a q-deformation of a product of Schur polynomials and have subsequently appeared in the study of Macdonald polynomials and related families. In this talk, I will explain this background and provide a new explicit "raising operator" formula for Macdonald polynomials that follows from a realization of LLT polynomials in the elliptic Hall algebra of Burban and Schiffmann, which we describe via an isomorphism between the shuffle algebra studied by Feigin and Tsymbaliuk and part of the Schiffmann algebra. This work is joint with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun.

October 11: Emily McGovern. A $Web(SL_n^{-})$ Embedding through $\tilde{A}_{n-1}$ buildings

In this talk, we describe an embedding of $Web(SL_n^{-})$, a diagrammatic monodical category, into a class of graph planar algebras. These graph planar algebras arise from affine type A buildings, a combinatorial structure developed by Tits in the 1970s. The relationship between finite projective geometries and affine buildings is key in establish the existence of an embedding functor in positive characteristic graph planar algebras.


November 15: Maxine Calle. A Equivariant Trees and Partition Complexes

Abstract: Given a finite set, the collection of partitions of this set forms a poset category under the coarsening relation. This category is directly related to a space of trees, which in turn has interesting connections to operads. But what if the finite set comes equipped with a group action? What is an "equivariant partition"? And what connection is there to equivariant trees? We will explore possible answers to these questions in this talk, based on joint projects with J. Bergner, P. Bonventre, D. Chan, A. Osorno, and M. Sarazola.

November 29: Mike Zabrocki. Representations of the quasi-partition algebras

The quasi-partition algebra is subalgebra of the partition algebra and is indexed by diagrams that have no singletons. We use the structure of the partition algebra and its representations to construct the irreducible representations of the quasi-partition algebra. This is joint work with Rosa Orellana and Nancy Wallace