Seminar
Representation theory seminar 2025 Semester 2
Organisers: Dougal Davis, Kari Vilonen, Ting Xue
Place and time: Peter Hall 162, Tuesdays, 2-3PM
Please contact one of the organisers to be added to the mailing list.
Upcoming Talks
7 October Ae Ja Yee (PennState) The Rogers–Ramanujan identities and their generalizations
Abstract: The most fascinating identities in the theory of partitions are the Rogers–Ramanujan identities, which were originally proved by Rogers and rediscovered by Ramanujan. These identities have been a great source of research in the past decades. They were reproved through many different approaches, and there also exist numerous similar identities and generalizations. In this talk, I will give a survey of the Rogers–Ramanujan identities with a few notable generalizations.
Past talks
1:30-3PM 5 Aug Matt Emerton (Chicago) A local-global compatibility for cohomology of Hilbert modular varieties
Abstract: The speaker, together with Xinwen Zhu, has proposed a conjecture that describes the cohomology of Shimura varieties with coefficients in an automorphic local system in terms of the conjectural categorical local Langlands correspondence (so this is a conjecture which depends on a conjecture!). Eugen Hellmann has proposed a formula for the values of the p-adic categorical local Langlands correspondence in the case of locally algebraic types, after inverting p. In this talk we will state, and sketch the proof of, a result in the direction of our conjecture with Zhu in the Hilbert modular case, using Hellmann’s proposal for the relevant cases of the p-adic categorical local Langlands correspondence. The main inputs in the proof are the de Rham comparison theorem between p-adic etale and p-adic de Rham cohomology, along with the elementary theory of q-expansions of Hilbert modular forms.
1:30-3PM 12 Aug Ivan Losev (Yale) Harish-Chandra center for affine Kac-Moody algebras in positive characteristic
Abstract: This talk is based on a joint work in preparation with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90’s describes the center of the universal enveloping algebra of an (untwisted) affine Kac-Moody Lie algebra at the so called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic p at an arbitrary non-critical level. Namely, we prove that the loop group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite dimensional affine space that is “p times smaller than the Feigin-Frenkel center”. In my talk I plan to introduce all necessary notions, state the result, explain motivations and examples.
1:30-3PM 13 Aug 2:30-4pm Ivan Losev (Yale) Categorical Heisenberg actions and modular representations of rational Cherednik algebras
Abstract: This is based on arXiv:2408.02485, joint with Bezrukavnikov. We construct certain functors on categories of representations of rational Cherednik algebras associated with symmetric groups in zero and large positive characteristic. Our functors are indexed by pairs of a partition and a rational number, the slope. For a given slope, the functors give an action of the positive half of the Heisenberg Lie algebra, while when the slope varies and the characteristic is positive we get a categorical action of the positive half of the elliptic Hall algebra. In this talk I will define rational Cherednik algebras and their categories of modular representations, state our main result and explain how our construction affinizes a version of the Steinberg tensor product theorem for the quantum GL_n. Time permitting, I’ll say a couple of words about the construction, a categorical representation of the positive half of the elliptic Hall algebra, etc.
1:30-3PM 19 Aug Edmund Heng (Sydney) Fusion quivers and Coxeter groups
Abstract: The study of module categories over fusion categories have focussed mostly on the semisimple ones. In this talk I will introduce the notion of fusion quivers and their representations, the categories of which form hereditary (global projective dimension 1) abelian module categories over fusion categories. This naive “one-step”
generalisation from semisimple module categories uncovers a wealth of interesting new connections to Coxeter theory. In particular, I will present a classification result in the spirit of Gabriel: the finite-representation-type fusion quivers are now classified by the Coxeter—Dynkin diagrams: these include the (crystallographic) Dynkin diagram from Lie algebras, and perhaps surprisingly, also the non-crystallographic diagrams H and I, which all together classify the finite Coxeter groups. If time allows, I will also discuss some applications to Coxeter and Artin—Tits groups.
2-3 pm 16 Sep Chenyan Wu (Melbourne) Hecke algebras and Langlands parameters for Bernstein blocks of the metaplectic groups
We study the Bernstein blocks of representations of the metaplectic groups in the p-adic setting. We show that they are equivalent to the categories of right modules over certain extended Hecke algebras with parameters in the sense of Lusztig and that these parameters can be expressed in terms of the Langlands parameters of the supercuspidal support. Via the Hecke algebras, we find that these Bernstein blocks, when suitably combined, can be expressed in terms of unipotent Bernstein blocks of quasisplit classical groups, where the representations theory and Langlands parameters are well known. This is a joint work with Volker Heiermann.
23 Sep Paul Zinn-Justin (Melbourne) Characteristic classes of matrix Schubert and positroid cells
In recent work with A. Knutson, we provided new formulae for Chern-Schwartz-MacPherson and motivic Chern classes of open Richardson varieties in terms of certain combinatorial data (pipe dreams).
In type A, a particular case, of special interest, is that of matrix Schubert cells. I will try to give a different take on these formulae by computing the CSM classes using deformations of characteristic cycles.
If time permits, I will then generalise this construction to twisted matrix positroid cells.
3 special lectures by Ae Ja Yee (PennState) Integer Partitions: Identities, Congruences, Combinatorial Statistics
11, 18, 25 Sep (Thursdays) 11-12 Peter Hall 162
The theory of partitions has as its central focus the decomposition of integers into sums of integers. This seemingly basic study leads to deep problems and extensive applications stretching from the hard hexagon model in statistical mechanics, to modeling problems in computer science, to group representation theory.
In my three lectures, I will give a brief introduction to the theory of partitions focusing on the following three topics.
Day 1: Partition Identities.
In this talk, I will introduce some basics on integer partitions such as the partition function 
Day 2: Partition Congruences.
The partition function
In this talk, I will give a brief sketch of the proof of the mod $5$ congruence and then discuss Dyson’s rank and crank statistics.
Day 3: Partition Statistics.
Integer partitions carry various interesting statistics. Of those, the most loved and studied statistics are Dyson’s rank and crank, which explain Ramanujan’s mod 5, 7 and 11 partition congruences. In my second lecture, we saw Dyson’s statistics, but there are other cranks introduced by Garvan, Kim and Stanton found other cranks, which split the set of partitions into 
the three congruences of Ramanujan. In this talk, I will discuss these cranks.