Representation theory seminar
Organized by the representation theory group at SIMIS
Tuesday at 4:15 pm
Dmitry Solovyev, YMSC, Tsinghua
11/04, SIMIS room TBA
Limit shapes of probability measures in representation theory of U_q(sl_2) at roots of unity
Limit shape phenomenon emerges in systems with random behavior. It manifests a formation of the most probable state, where all other macroscopically different states are exponentially improbable. In this talk, we explore such phenomena in the Grothendieck ring of the category of tilting modules for the quantum group U_q(sl_2) with divided powers, where q is an even root of unity. Considering large tensor powers of the defining representation, we describe the most probable trajectory in the main Weyl chamber with respect to the character probability measure and analyze fluctuations around this limit shape. This talk is based on arXiv:2404.03933, a joint work with A. Lachowska, O. Postnova and N. Reshetikhin.
Past talks
Jian-Rong Li, University of Vienna
Thursday 2pm! 10/23, SIMIS room 1010
Boundary q-characters of finite-dimensional representations of quantum affine symmetric pairs
Frenkel and Reshetikhin introduced q-characters for finite-dimensional representations of quantum affine algebras, providing a fundamental tool in their representation theory. Boundary q-characters for finite dimensional representations of quantum affine symmetric pairs of split types were introduced and developed by Tomasz Przezdziecki and myself. In this talk, I will present a new joint work Tomasz Przezdziecki on evaluation modules for split quantum affine symmetric pairs. By computing the action of generators in Lu and Wang’s Drinfeld-type presentation on Gelfand–Tsetlin bases, we determine the spectrum of a large commutative subalgebra arising from this presentation. This leads to an explicit formula for boundary analogues of q-characters, which we interpret combinatorially in terms of semistandard Young tableaux. Our results show that boundary q-characters share familiar features with ordinary q-characters—such as a version of the highest weight property—while also exhibiting new phenomena, including an additional symmetry.
Bart Vlaar, BIMSA
10/14, SIMIS room 1010
Towards q-characters for quantum symmetric pairs
Q-characters appeared in work by E. Frenkel and N. Reshetikhin in 1999 as a tool to study the category of finite-dimensional modules of quantum affine algebras $U_q(\hat{\mathfrak{g}})$. One way to define them is to take the partial trace of the universal R-matrix R (braided structure on the category $\mathcal{O}_q$).
Quantum symmetric pairs have been studied since the 1990s. They consist of a quantized universal enveloping algebra and a suitable coideal subalgebra. More recently it has become clear that each of them is equipped with a universal K-matrix, which defines a braided structure on a particular module category over the monoidal category $\mathcal{O}_q$, compatible with the braiding defined by R. With it one can play similar games as one can with R.
We will discuss some recent developments, possible applications and remaining obstacles. Based on joint work with Andrea Appel (University of Parma).
Ivan Sechin, BIMSA
09/23, SIMIS room 1010
Spectral duality for Gaudin systems associated with reflection equation
Two classical integrable systems are called spectrally dual if they are defined on the same phase space (or if there exists a Poisson map between their phase spaces), and the spectral curves of these two systems (given by the equation Γ(z,w) = 0) are related via the change of coordinates z and w. I will give an elementary proof of the spectral duality of two classical Gaudin models: the first one is associated with the orthogonal Lie algebra, while the second one is constructed from the reflection equation for the general linear Lie algebra. Based on the joint work with Andrii Liashyk and Mikhail Vasilev.
Lucien Hennecart, CNRS
06/23, SIMIS room 1210
Mackey formula for representations of reductive groups
In this talk, I will describe the construction of induction and restriction morphisms on the critical cohomology associated with a function on a representation of a reductive group. The induction morphisms play a key role in obtaining a cohomological integrality decomposition, which is a decomposition into finite-dimensional pieces with enumerative significance. The building blocks of this decomposition are given by the BPS cohomology. After discussing this decomposition and its geometric meaning, I will present a cohomological version of the Mackey formula that relates the induction and restriction operations. I will also present examples of the formalism in low dimensions. This can be seen as a generalization to more general stacks of the cohomological Hall algebra multiplication and its localized restriction.
Weiping Li, HKUST
06/16, SIMIS room 1210
Infinite dimensional algebra and instanton moduli spaces
Given a projective smooth surface X and its blowup surface Y, Yoshioka calculated the blowup formula relating Betti numbers of the moduli space of rank two sheaves on X with those on Y. Nakajima asked the question of a representation-theoretic interpretation of the blowup formula. In the joint work with Qingyuan Jiang and Yu Zhao, we studied the representation of an extended Clifford algebra on the cohomology of the moduli space of stable sheaves on Y and answered Nakajima’s question.
Mikhail Bershtein, SISSA Trieste
At 2pm! 06/09, SIMIS room 1210
Highest-weight vectors and three-point functions in coset decomposition
We revisit the classical Goddard Kent Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators between these vectors. This leads to relations on conformal blocks. Due to the AGT relation, these relations are equivalent to blowup relations on Nekrasov partition functions with the presence of the surface defect. These relations can be used to prove Kyiv formulas for the Painlevé tau-functions (following Nekrasov’s method).
Emile Bouaziz, Academia Sinica
05/19, SIMIS room 1210
Elliptic Constructions via DAG
I’ll talk about some recent work with Adeel Khan in which we construct elliptic versions of loop spaces from derived algebraic geometry. I’ll explain the construction, which passes through a Tannakian formalism, and sketch the resulting equivariant elliptic cohomology theory. Time permitting I will explain some expected applications to geometric representation theory.