The UC Berkeley Representation theory and tensor categories seminar
Spring 2024 - Tuesday 2:10pm - 3:30pm, Evans 939
Organizers: Vera Serganova and Alexandra Utiralova

If you would like to be added to the seminar mailing list, contact any of the organizers.

DATE SPEAKER TITLE (click to show abstract)
January 23 Pavel Etingof, MIT
Periodic pencils of flat connections and their \(p\)-curvature Abstract: A periodic pencil of flat connections on a smooth algebraic variety \(X\) is a linear family of flat connections \(\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i\), where \(\lbrace x_i\rbrace\) are local coordinates on \(X\) and \(B_{ij}: X\to {\rm Mat}_N\) are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts \(s_j\mapsto s_j+1\) up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic \(p\), the \(p\)-curvature operators \(\lbrace C_i,1\le i\le r\rbrace\) of a periodic pencil \(\nabla\) are isospectral to the commuting endomorphisms \(C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}\), where \(B_{ij}^{(1)}\) is the Frobenius twist of \(B_{ij}\). This allows us to compute the eigenvalues of the \(p\)-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.
January 30 Alex Sherman, University of Sydney
Queer Kac-Moody algebras and d=2 Ramond Superconformal algebras Abstract: Kac-Moody (super)algebras, especially those of finite growth, have a very rich theory over the complex numbers. Determined by only a square matrix (the Cartan matrix), and including the class of simple Lie algebras (sl(n), so(n), etc.) they have deep connections to representation theory, tensor categories, physics, combinatorics, number theory, and more. The queer superalgebra, a super-generalization of gl(n), does not fit into the Kac-Moody framework due to a maximal torus which is not purely even. We remedy this by defining a new Kac-Moody construction with the most general type of torus possible in the super setting. We will then explain the classification results that we obtain, including one on finite-growth queer Kac-Moody algebras. Special appearances will be made by two d=2 superconformal algebras in the Ramond sector.
February 6 No seminar
See this SLMath workshop instead
February 13 Alexander Shapiro, University of Edinburgh
Toda and Ruijsenaars integrable systems Abstract: The phase space of the Ruijsenaars integrable system can be identified with (a Hamiltonian reduction of) the moduli space of \(GL_n\) local systems on a punctured torus. The latter admits a structure of a cluster variety. On the algebraic level, this leads to an injective homomorphism from a spherical subalgebra of the double affine Hecke algebra into the quantized algebra of global functions on the named cluster variety. From an analytic point of view, it allows for a unitary equivalence between Toda and Ruijsenaars quantum integrable systems. In particular, we will show that the eigenfunctions of Macdonald operators can be presented as a matrix coefficient of an order 4 element in the modular group evaluated on a pair of Toda eigenfunctions. Finally, we will discuss how to work on polynomial rather than analytic level, and obtain Macdonald polynomials from the Whittaker ones. During this talk we will focus on the \(n=2\) case when no Hamiltonian reduction is required. This talk is based on a joint work with P. DiFrancesco, R. Kedem, S. Khoroshkin, and G. Schrader.
February 20 Subho Chatterjee, UC Davis
Probabilities and Supergeometry: Measurement theory for dynamical discrete systems Abstract: Discrete probabilistic systems, like bits on a computer or faces of a coin, abound in nature. We propose a geometric model describing dynamics and measurement theory for such systems. Our approach is covariant with respect to choices of clocks and laboratories. The configuration space is a super phasespacetime modelled by an odd dimensional symplectic supermanifold, observables are superfunctions and states are suitable (star) squares of superfunctions. The data of an odd dimensional symplectic supermanifold canonically incorporates dynamics. We also obtain dynamical probabilities using convex polyhedral cones and find that they obey Markov-like evolution.

arxiv no.: 2311.05711
February 27 Milen Yakimov, Northeastern University
Reflective centres of module categories and quantum K-matrices Abstract: Braided monoidal categories have applications in various situations, in particular their universal R-matrices give solutions of the quantum Yang-Baxter equation and representations of braid groups of type A. There are powerful methods for constructing them: Drinfeld doubles of Hopf algebras and Drinfeld centres of monoidal categories. On the other hand, universal K-matrices, leading to solutions of the reflection equation and representations of braid groups of type B are much less well understood. We will describe a construction of reflective centers of module categories. It gives rise to braided module categories and a quantum double construction for universal K-matrices.

This is a joint work with Robert Laugwitz and Chelsea Walton.
March 5 Sarah Witherspoon, Texas A&M University
Varieties for tensor categories Abstract: We will give a brief survey of three types of topological spaces associated to objects in certain tensor categories, some in more limited settings than others, namely the Balmer support, the cohomological support, and the rank varieties. These are important tools in representation theory, and are best understood in the classical setting of finite group representations. Much is known more generally, while there are still many questions. We will include some recent results, focusing particularly on the question of how the spaces associated to a tensor product of objects are related to those of the objects.
March 12 Dmitry Kaledin, HSE University
Tensor categories and the Grothendieck construction Abstract: Tensor categories of various types are usual described via explicit associativity and/or commutativity isomorphisms that have to satisfy higher constraints. This is rather heavy technically and not very canonical. I am going to advertise an alternative packaging for the same data based on a purely categorical gadget called the Grothendieck construction. If time permits, I will also try to show how this leads to a very transparent understanding of deformations.
March 19 Ben Davison, University of Edinburgh
R matrices and shuffle algebras Abstract: For any quiver Q there is an associated shuffle algebra structure on the ring of symmetric functions, which can be realised as the cohomological Hall algebra of the category of representations of the quiver. Mimicking Green's coproduct for finitary Hall algebras, one obtains also a kind of localised coproduct, along with an easy construction of an R-matrix for the resulting (localised) Hopf algebra. This provides a very flexible framework for producing new quantum groups, using the formalism of Faddeev, Reshetikhin and Takhtadzhyan, after a choice of favoured modules for the shuffle algebra. I will explain what is known: that using a specific variant of this construction we obtain the new Yangians defined by Maulik and Okounkov (via recent joint work with Botta). I will also present some conjectures regarding the algebra we obtain this way for certain other choices of cohomological Hall algebras and modules over them. The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. Is e buidheann carthannais a th’ ann an Oilthigh Dhùn Èideann, clàraichte an Alba, àireamh clàraidh SC005336.
March 26 No seminar
Spring break
April 2 Julia Pevtsova, University of Washington
Are categories of modular representations of finite groups (locally) regular? Abstract: For a symmetric tensor triangular category T we suggest a notion of “local regularity” which agrees with the classical concept for a derived category D(A) for a commutative ring A. We study this property for stable categories of a finite group scheme (in positive characteristic) and discover that it is related to many other, such as dualizability and various finiteness conditions. I’ll review some fundamental principles of tensor triangular geometry and the stratification of the stable category of a finite group G over a field of positive characteristic. Then we specialize to the fibers of the stable category at homogeneous prime ideals p in the cohomology ring and show that they are regular.

Joint work with D. Benson, S. Iyengar, H. Krause.
April 9 Iva Halacheva, Northeastern University
Bethe algebras, cacti, and crystals Abstract: The Bethe subalgebras of the Yangian Y(gl(n)) form a family of maximal commutative subalgebras indexed by points of the Deligne-Mumford compactification of the moduli space M(0,n+2). When considering a point C in the real locus of this parameter space, the corresponding Bethe subalgebra B(C) acts with simple spectrum on a given tame representation of Y(gl(n)). This results in an unramified covering, whose fiber over C is the set of eigenlines for the action of B(C). I will discuss the identification of each fiber with a collection of Gelfand-Tsetlin keystone patterns, which carry a gl(n)-crystal structure, as well as the monodromy action realized by a type of cactus group. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.
April 16 Yuri Berest, Cornell University
Representation homology of spaces and the strong Macdonald conjectures Abstract: In 1982, I. G. Macdonald published a series of beautiful combinatorial conjectures related to classical root systems (or equivalently, compact Lie groups). These conjectures were in the focus of research in representation theory and geometry for over 30 years. In the early 1990s, B. Feigin and P. Hanlon proposed a homological refinement of the Macdonald conjectures (nowadays known as the strong Macdonald conjectures) that were studied by many mathematicians and eventually settled by S. Fishel, I. Grojnowski and C. Teleman in 2008. In this talk, I will give a topological interpretation of the strong Macdonald conjectures and present a series of new conjectures suggested by topology that remain wide open.
April 23 Jonathan Kujawa, Oregon State University
The Lie superalgebra generated by transpositions Abstract: The symmetric group has been the object of study since forever. Nevertheless, there are still new things to say. Using the commutator, you can view the group algebra of the symmetric group as a Lie algebra. In 2003, Marin described this Lie algebra and the subalgebra generated by the transpositions. Since the symmetric group naturally splits into even and odd permutations, you can also ask about the graded version of the commutator. This makes the group algebra into a Lie superalgebra. In 2023, Chris Drupieski and I obtained the super analogue of Marin’s results.
April 29 (Monday) 12-1PM, 891 Evans. Alberto Elduque, University of Zaragoza
From the Albert Algebra to Kac's Jordan superalgebra via tensor categories Abstract: Kevin McCrimmon proved the surprising result that Kac's ten-dimensional simple Jordan superalgebra satisfies the super version of the Cayley-Hamilton equation of degree 3 in characteristic 5. This was fundamental in the discovery of a new simple Lie superalgebra in characteristic 5. In this talk it will be shown that over fields of characteristic 5, Kac's superalgebra may be obtained from the Albert algebra, i.e. the Jordan algebra of 3 by 3 hermitian matrices over the octonions, by looking at it as an algebra in the tensor category of representations of the cyclic group of order 5 and passing to the associated semisimple category: the Verlinde category.
Note the unusual place and time.
April 30 Alistair Savage, University of Ottawa
Diagrammatics for real supergroups Abstract: We introduce diagrammatic monoidal supercategories controlling the representation theory of real forms of the general linear, orthosymplectic, periplectic, and isomeric supergroups. As a consequence, we obtain first fundamental theorems for these real supergroups and equivalences between monoidal supercategories of tensor supermodules over the real forms of a complex supergroup. This is joint work with Saima Samchuck-Schnarch.
Fall 2023 website
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