Abstracts CANMEX 2023

I will report on recent joint work in progress with Takuya Murata. Motivated by work of Harada—Kaveh and the desire to construct moment map type maps to Newton-Okounkov polytopes we study maps to toric varieties. As algebro-geometric methods turn out to not be appropriate we use results of Mather involving Whitney stratifications to obtain a collapsing map. Although the result is more general our main application concerns toric degenerations where we construct a map from the general to the special fibre. Moreover, we generalize a result of Harada-Kaveh constructing an integrable system on a projective variety that admits a toric degeneration induced by the moment map on the toric variety. A first preprint of the work is available on arxiv:2210.13137.

We discuss a connection between cluster algebras, the representation theory of quantum affine algebras of type, the existence of imaginary modules and the relationship with the representation theory of the general linear group over a non-archimedean field. The talk is based on joint work with Matheus Brito.

This is based on joint work-in-progress with Colin Ingalls and Charles Paquette. Given a finite-dimensional algebra, we can associate a quiver and choose a dimension vector. Work of Alistair King shows that, if the dimension vector is unimodular, there is a moduli space of stable representations with a universal sheaf. Work of Reineke–Schröer and Hoskins–Schaffhauser shows that this does not hold for all dimension vectors; in general, we obtain a only “quasi-universal sheaf”. I will discuss this construction from several perspectives, and explain applications to the representation theory of finite-dimensional algebras.

The zesting construction is a way to describe braided or ribbon categories whose fusion rings are related by central extension of a grading group by invertible objects. At the level of string diagrams this construction has a strong topological flavor that allows one to give a physical characterization of the relationship between the categories’ associated 3D Reshetikhin-Turaev TQFTs.

In this talk we will discuss recent work that defines a more general procedure for G-crossed braided fusion categories. This new point of view reproduces the original zesting construction, places it in the context of the well established extension theory of fusion categories by finite groups, and connects zesting to 3D homotopy quantum field theory (HQFT).

Belavin-Drinfeld quantum groups are deformations of the algebra of functions of a semi-simple algebraic Poisson group whose Poisson structure is given by the classification of factorizable Lie bialgebra structures on (semi-)simple Lie algebras. This talk will survey the current state of knowledge of this subject together with a number of conjectures and future research directions.

In this talk we present the concept of m-periodic Gorenstein objects relative to a pair of classes of objects in an abelian category, as a generalization of m-strongly Gorenstein projective modules over associative rings. We mention some properties when the pair satisties certain homological conditions, like for example when the pair is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relate to these pairs are also established. This is a joint work with O. Mendoza and M. A. Pérez.

I will explain a general procedure to construct (potentially infinitely many) Newton-Okounkov bodies for a given line bundle on a minimal model of a cluster variety. This construction applies for example to line bundles on Grassmannians and flag varieties, and recovers as particular examples distinguished polytopes arising in representation theory such as Gelfand-Tsetlin polytopes, Littelmann/Berenstein-Zelevisnky string polytopes and the flow valuation Newton-Okounkov polytopes constructed by Rietsch-Williams. Time permitting, I will provide further details of our construction for the Grassmannians as, in this case, one can use either the A or the X cluster structure to construct Newton-Okounkov polytopes. Interestingly, the Euler form of the dimer algebra used by Jensen-King-Su and Baur-King-Marsh to categorify the cluster structure on the homogeneous coordinate ring of a Grassmannian can be used to provide an equivalence between the two kinds of polytopes.

In this talk, I will provide some historical background on the subject of tensor triangular geometry and support theory for representations. The origins of this subject started with the work of Alperin and Carlson in the late 1970’s and 1980’s with a significant milestone being the introduction of tensor triangular geometry by Paul Balmer in the mid 2000’s. Since this time there have been many exciting new ideas in the subject with the modern development of tensor categories.

The general noncommutative version of Balmer’s tensor triangular geometry was recently developed by Nakano, Vashaw and Yakimov. Insights from noncommutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of a monoidal triangulated category, and then to associate to a topological space–the non-commutative Balmer spectrum. Our theory works well in finite tensor categories. In this talk, I will mainly focus on applications to studying the geometry around finite-dimensional Hopf algebras.

This talk represents joint work with Kent Vashaw and Milen Yakimov.

The road between fusion categories and subfactors (of von Neumann algebras) is well-trodden in both directions; subfactors provide examples of fusion categories not seen elsewhere, and fusion category techniques can be applied in, eg, the construction and classification of subfactors. In this talk we translate the main ideas of Ocneanu’s paragroup theory into tensor categorical language; we begin by explaining why connections on bipartite graphs correspond to a pair of functors from the Temperley-Lieb-Jones categories into a graph category, and commutator data.

We construct families of algebras that are quantum analogues of the algebra of differential operators on a symmetric space of the form GL(n,F)/U(n,F), where F is a real division algebra. We then define a natural basis of Capelli operators and prove that their spectra recover the interpolation variant of Macdonald polynomials. This talks is based on joint work with G. Letzter and S. Sahi.

I will discuss how certain results in modular representation theory of finite groups can be generalized to representations of algebraic supergroups with reductive even part. In particular, we will see what is a Sylow subgroup in the supercase and prove Green correspondence for some supergroups.

Given a cluster algebra A, the Laurent phenomenon can be interpreted as saying that every cluster gives rise to an open torus, called a cluster torus, on Spec(A). The deep locus of Spec(A) consists precisely of those points that do not belong to any cluster torus. I will give a complete description of the deep locus of cluster varieties of finite type with really full rank, as well as some cluster varieties of infinite type, including the maximal positroid strata in Gr(3,n). Time permitting, I will sketch applications to homological mirror symmetry of Grassmannians. This is joint work with Marco Castronovo, Mikhail Gorsky and David Speyer.

The study of p-DG algebras arises naturally when trying to category quantum groups at prime roots of unity. We will review this framework and construct some categorical representations for the case of sl(2). This is joint work with Mikhail Khovanov and You Qi.

One of celebrated theorems of Maurice Auslander about artin algebras, describes the correspondence between {algebras A of finite representation type} and {algebras B, with gl.dim.B ≤ 2 ≤ dom.dim.B}. The second class of algebras are now called Auslander algebras.
Higher Auslander algebras were introduced by O. Iyama as {algebras C, with gl.dim.C ≤ k ≤ dom.dim.C}. It was shown that there is correspondence between {higher representation finite algebras} and {higher Auslander algebras}.

I will talk about recent work of Emre Sen who had created several approaches for constructing new families of higher Auslander algebras and, as a consequence, he obtained new higher representation finite algebras. In addition to that Emre, Shijie Zhu and I, have a joint paper in which we have another method of constructing higher Auslander algebras in the family of Nakayama algebras.

Higher Auslander algebras, higher representation finite algebras and higher cluster tilting modules are still not well understood, even in the families of well known classes of algebras, so creating families of such will be particularly useful.

Irreducible representations of the symmetric group are parameterized by Young diagrams and their bases are further parameterized by standard tableaux on Young diagrams. There is a similar story for a class of representations of the double affine Hecke algebra (DAHA) taking periodic tableaux, or for the rational Cherednik algebra (a.k.a. rational DAHA) with appropriate modifications. This construction of the basis makes use of an alternate presentation of the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra. We make use of this presentation and associated combinatorics to give a geometric construction of the irreducible RCA module parameterized by the trivial (one row) Young diagram.

In particular, we use the geometry of the parabolic Hilbert scheme of points on a plane curve singularity. The basis of equivariant homology that comes from torus fixed points becomes our “tableau” basis.

This is joint work with Eugene Gorsky and José Simental.

Affine Grassmannians play an important role in geometric representation theory, perhaps most notably through the Geometric Satake Equivalence. A natural but difficult question is to ask whether this theory can be extended to include “affine Grassmannians” of Kac-Moody types. Work of Braverman-Finkelberg suggests that, rather than try to define the entire affine Grassmannian in this generality, one should instead focus on the simpler problem of defining so-called affine Grassmannian slices (these slices should be finite-dimensional varieties).

Recent developments from mathematical physics suggest a solution to this problem: use the theory of Coulomb branches, as developed by Braverman-Finkelberg-Nakajima, to *define* slices in the affine Grassmannian for Kac-Moody types.

In this talk we will discuss one piece of evidence that this Coulomb branch construction is reasonable: namely, that the resulting varieties embed into one another in a natural way. This result is based on joint work with Dinakar Muthiah.

Positroid strata of Grassmannian are known to carry cluster structures, which are oftendescribed by reduced plabic graphs. However, there are only a limited number of equivalence moves and mutations that can be performed on reduced plabic graphs. By applying an iterative sequence of T-shifts to a reduced plabic graph, we are able to construct a Legendrian weave, which is much more flexible. Moreover, Legendrian weaves allow us to give a topological interpretation of the cluster structure on positroid strata. As an application, we prove that the Muller-Speyer twist map coincides with the Donaldson-Thomas transformation on positroid strata, and this result, in turn, implies that the source labeled seed and the target labeled seed are quasi-cluster equivalent.