- Susumu Ariki (Osaka)
Cellularity for algebras of tame representation type
ABSTRACT: The cellularity suffices to show that an algebra of finite representation type is a Brauer graph algebra whose Brauer graph is a straight line. We examine the similar property for algebras which are tame of polynomial growth. This is a joint work with R. Kase, K. Miyamoto and K. Wada.
- Thorsten Heidersdorf (Bonn)
Semisimplification of representation categories
ABSTRACT: Many categories of finite dimensional representations can be semisimplified. Formally one considers the quotient of the representation category $\mathcal{C}$ by its largest proper tensor ideal (the negligible morphisms). This has been considered by Mathieu for tilting modules in char. $p > 0$, by Andersen and many others for tilting modules of quantum groups at roots of unity and by André and Kahn for representations of non-reductive algebraic groups in char. $0$. I will talk about the case where $\mathcal{C}$ is the representation category of a supergroup $G$, notably the case where $G = \mathop{GL} (n|n)$. The negligible quotient is in this case described by a pro-reductive group. I will explain how to determine this group and discuss possible applications.
- Catharina Stroppel (Bonn) #1
Quiver Schur algebras and gradings
ABSTRACT: In this talk I will explain the construction of Quiver Schur algebras. This are gemetrically defined versions of affine Schur algebras and provide an extension of KLR algebras. As an application we will construct gradings on cyclotomic Schur algebras. We will explain the relevance of this construction to categorification.
- Ryo Fujita (Kyoto)
Coherent IC-sheaves on type $A_n$ affine Grassmannians and dual canonical basis of affine type $A_1$
ABSTRACT: The convolution ring of the equivariant $K$-theory of the $\mathop{GL}_n$ affine Grassmannian was identified with a quantum unipotent cell of the loop group $\mathop{LSL}_2$ by Cautis-Williams. In this talk, we further identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves (coherent IC-sheaves) with the dual canonical basis of the quantum unipotent cell. This is a joint work with Michael Finkelberg.
- Hankung Ko (Bonn)
Quantum polynomial functors
ABSTRACT: The (strict) polynomial functors form a monoidal category equivalent to the polynomial representations of general linear groups. This functor interpretation provides a useful way to understand $GL_n$ representation theory. I will present a quantization of the polynomial functors, joint with Valentin Buciumas. Two important insights from the classical polynomial functors are the stability property and the composition of functors. I will describe these ideas and explain what generalizes and what does not generalize.
- Tomoyuki Arakawa (Kyoto)
Chiral algebras of class $S$ and Moore-Tachikawa symplectic varieties
ABSTRACT: Recently, Braverman, Finkelberg and Nakajima have constructed a new family of the (possibly singular) symplectic varieties called the Moore-Tachikawa symplectic varieties. We upgrade this construction to the setting of vertex algebras. This not only gives a functorial construction of the genus zero chiral algebras of class $S$, that is, the vertex algebras corresponding to the theory of class $S$ in $4d$ $N = 2$ SCFTs via the $4d/2d$ duality, but also proves a conjecture in physics on the relation between the Higgs branches of $4d$ $N = 2$ SCFTs and the corresponding vertex algebras for the theory of class $S$.
- Hideya Watanabe (Tokyo Institute of Technology)
Global crystal bases of modules over a quantum symmetric pair
ABSTRACT: Quantum symmetric pairs (QSPs) appear in many areas of mathematics and physics such as representation theory, low-dimensional topology, and integrable systems. In this talk, I introduce the notion of global crystal bases for modules over a QSP of type $\mathsf{AIII}$, and show some applications in the representation theory. Time permitting, I talk about recent progress on crystal basis theory for QSPs of other types.
- Sota Asai (Nagoya)
The wall-chamber structures of the real-valued Grothendieck groups
ABSTRACT: We consider a finite-dimensional algebra $A$ over a field and the real-valued Grothendieck group of the category of finite-dimensional projective $A$-modules. The real-valued Grothendieck group can be identified with a Euclidean space, and Brüstle--Smith--Treffinger defined a wall-chamber structure of the real-valued Grothendieck group via the semistability conditions introduced by King.
In this talk, I will observe the wall-chamber structure from the point of view of the numerical torsion(-free) classes defined by Baumann--Kamnitzer--Tingley, and explain my new result that the chambers of the wall-chamber structure bijectively correspond to the 2-term silting objects of the perfect derived category.
Moreover, I will explain a combinatorial algorithm to obtain the wall-chamber structure in the case $A$ is a path algebra.
- Catharina Stroppel (Bonn) #2
Categorified coideal subalgebras and Deligne categories
ABSTRACT: In this talk I will consider the Deligne category $\mathrm{Rep} ((\delta))$ and its module category. The main point is to show that it categorifies a certain Fock space for a coideal subalgebra inside a quantum group. As an application we obtain a connection between decomposition numbers in Brauer centralizer algebras and Kazdhan-Lusztig polynomials of type $\mathsf D$. If time allows we will also mention the relevance of this construction to the representation theory of Lie superalgebras.
