# Oberseminar Darstellungstheorie, Winter Term 2016/2017

Contact:
Catharina Stroppel email
Daniel Tubbenhauer email

## Where and when?

• Seminar room 1.008, Mathematik-Zentrum, Endenicher Allee 60, 53115 Bonn, Germany
• Every (second) Friday from 14:15-15:15 (with some exceptions)
• There will be a lecture series of Dennis Gaitsgory:
• The first lecture will take place Tuesday 10.01.2016 from 16:30-18:30 in the lecture hall of the MPIM
• The second lecture will take place Wednesday 11.01.2016 from 14:00-16:00 in the Lipschitz-Saal, Mathematik-Zentrum, Endenicher Allee 60
• The third lecture will take place Friday 13.01.2016 from 12:30-14:30 in the lecture hall of the MPIM
• There is an extra website available: Lecture series home
• David Jordan's talk will not take place October the 28th and is canceled
• Emily Norton's talk is scheduled for Tuesday the 20.12.2016, 14:15-15:15, Seminarraum, Max-Planck-Institut für Mathematik (MPIM)
• Olivier Schiffmann's talk will not take place January the 20th and is canceled
• Kevin Coulembier's is scheduled for Monday the 20.02.2017, 13:15-14:15, Hörsaal, Max-Planck-Institut für Mathematik (MPIM)
• First talk: Friday 21.10.2016

## Schedule

• Friday 21.10.2016, Speaker: Inna Entova Eisenbud (Hebrew University of Jerusalem), Title: On finite-dimensional representations of the Lie superalgebra $P(n)$
• Friday 28.10.2016, Speaker: Talk canceled! David Jordan (University of Edinburgh), Title: The quantum Springer sheaf
• Friday 11.11.2016, Speaker: Andrew Hubery (Universität Bielefeld), Title: Euler characteristics of quiver Grassmannians
• Friday 18.11.2016, Speaker: Jonathan Comes (The College of Idaho), Title: Jellyfish partition categories
• Friday 25.11.2016, Speaker: Andrew Mathas (University of Sydney), Title: Alternating Hecke algebras
• Friday 02.12.2016, Speaker: Zhengfang Wang (Université Paris Diderot-Paris 7), Title: Singular Hochschild cohomology and higher algebraic structures
• Friday 16.12.2016, Speaker: Chun-Ju Lai (University of Virginia), Title: Affine Hecke algebras and quantum symmetric pairs
• Tuesday 20.12.2016, Speaker: Emily Norton (MPIM Bonn), Title: Decomposition numbers for rational Cherednik algebras
• Tuesday 10.01.2016, Speaker: Dennis Gaitsgory (Harvard University), Title: Hirzebruch-Riemann-Roch as a categorical trace I
• Wednesday 11.01.2016, Speaker: Dennis Gaitsgory (Harvard University), Title: Hirzebruch-Riemann-Roch as a categorical trace II
• Friday 13.01.2016, Speaker: Dennis Gaitsgory (Harvard University), Title: Hirzebruch-Riemann-Roch as a categorical trace III
• Friday 20.01.2017, Speaker: Talk canceled! Olivier Schiffmann (Université de Paris-Sud Orsay), Title: Cohomological Hall algebras of quivers
• Friday 03.02.2017, Speaker: Vanessa Miemietz (University of East Anglia), Title: Simple transitive $2$-representations via coalgebra $1$-morphisms
• Monday 20.02.2017, Speaker: Kevin Coulembier (University of Sydney), Title: The periplectic Brauer algebra
• Friday 24.02.2017, Speaker: Jieru Zhu (University of Oklahoma), Title: Presenting cyclotomic Schur algebras

## Abstracts

Inna Entova Eisenbud,
Title: On finite-dimensional representations of the Lie superalgebra $P(n)$

Given a supervector space $V = \mathbb{C}^{(n|n)}$ with an odd symmetric bilinear form, the periplectic Lie superalgebra $P(n)$ consists of linear transformations preserving this form. This originally appeared in the classification of classical-type Lie superalgebras due to Kac: for $n>2$, this algebra has a simple ideal of codimension 1, which is one of the two “strange” series of simple superalgebras.
In this talk, I will present some of the results we obtained concerning the category $\mathrm{Rep}(P(n))$ of finite-dimensional representations. Unlike the rest of the classical-type Lie superalgebras, the structure of this category has not been well understood until now.
I will explain the classification of blocks in this category, the combinatorics behind the Kazhdan-Lusztig coefficients, and present a categorical action of the infinite Temperley-Lieb algebra through translation functors.
This is part of a joint project with M. Balagovic, Z. Daugherty, M. Gorelik, I. Halacheva, J. Hennig , M. Seong Im, G. Letzter, E. Norton, V. Serganova, C. Stroppel.
The talk will not assume prior knowledge of Lie superalgebras.

Talk canceled! David Jordan,
Title: The quantum Springer sheaf

The Springer resolution, and resulting Springer sheaf lies at the heart of geometric representation theory, via its relation to the Beilinson-Bernstein theorem, and Lusztig's theory of character sheaves.
Most traditionally, the Springer sheaf is constructed geometrically, in the language of Borel-Moore homology and perverse sheaves; however Hotta and Kashiwara gave it a purely algebraic reincarnation in the language of equivariant $D(\mathfrak{g})$-modules.
In this talk, I'll explain some joint work with Monica Vazirani, to define and compute the so-called quantum Springer sheaf, which plays an analogous role for the quantum group $\boldsymbol{\mathrm{U}}_q(\mathfrak{g})$. Our main tool to study these is a “genus one” Schur-Weyl duality type functor to representations of the double affine Hecke algebra.

Andrew Hubery,
Title: Euler characteristics of quiver Grassmannians

We show how to define an Euler characteristic for the Grassmannian of submodules of a rigid module, for any finite-dimensional hereditary algebra over a finite field. Moreover, we prove that this number is positive whenever the Grassmannian is non-empty, generalising the known case of quiver Grassmannians. As an application, this shows that, for any symmetrisable acyclic cluster algebra, the Laurent expansion of a cluster variable always has non-negative coefficients.

Jonathan Comes,
Title: Jellyfish partition categories

Partition diagrams can be used to construct a category $P(n)$ which admits a full functor to the category of finite dimensional representations of the symmetric group $S_n$. On the level of endomorphism algebras this is the so-called Schur-Weyl duality between partition algebras and the symmetric group. I will use a generalization of partition categories to construct a category $JP(n)$ which admits a full functor to the category of finite dimensional representations of the alternating group $A_n$. In this case, unlike the former, the functor is also faithful.

Andrew Mathas,
Title: Alternating Hecke algebras

I will give a survey talk about the alternating Hecke algebras, which are certain deformations of the groups algebras of the alternating groups. I will start by describing their semisimple representation theory, and their character theory, and will finish with an introduction to their graded representation theory. The talk will aim to be self-contained.

Zhengfang Wang,
Title: Singular Hochschild cohomology and higher algebraic structures

We define the singular Hochschild cohomology which is a generalization of Hochschild cohomology motivated from the investigation on singularities of algebraic varieties.
In this talk, we construct a singular version of Hochschild cochain complex to compute the singular Hochschild cohomology. We prove that there is a Gerstenhaber algebra structure in the singular Hochschild cohomology and this structure has a prop interpretation. We discuss the singular version of Deligne conjecture, which is related to the higher algebraic structures (for instance, the $B_{\infty}$-algebra and homotopy Gerstenhaber algebra structures). The Batalin-Vilkovisky (BV) algebra structure is also constructed in the case of symmetric Frobenius algebras. If time allows, we will talk about some results of a joint-work in progress with Manuel Rivera in the application of singular Hochschild cohomology to string topology.

Chun-Ju Lai,
Title: Affine Hecke algebras and quantum symmetric pairs

In an influential work of Beilinson, Lusztig and MacPherson, they provide a construction for (idempotented) quantum groups of type $A$ together with its canonical basis. Although a geometric method via partial flags and dimension counting is applied, it can also be approached using Hecke algebras and combinatorics.
In this talk I will focus on the Hecke algebraic approach and present our work on a generalization to affine type $C$, which produces favorable bases for $q$-Schur algebras and certain coideal subalgebras of quantum groups of affine type $A$. We further show that these algebras are examples of quantum symmetric pairs, which are quantization of symmetric pairs consisting of a Lie algebra and its fixed-point subalgebra associated to an involution.
This is a joint work with Z. Fan, Y. Li, L. Luo, and W. Wang.

Emily Norton,
Title: Decomposition numbers for rational Cherednik algebras

In a highest weight category, “decomposition numbers” refers to multiplicities of simple objects in standard objects. I will describe results that are known about decomposition numbers for Category $\mathcal{O}$ of a rational Cherednik algebra, and mention some questions that remain open.

Dennis Gaitsgory,
Title: Hirzebruch-Riemann-Roch as a categorical trace

Let $X$ be a smooth proper scheme over a field of characteristic $0$, and let $E$ be a vector bundle on $X$. The classical Hirzebruch-Riemann-Roch says that the Euler characteristic of the cohomology $H^*(X,E)$ equals $\int_X \mathrm{ch}(E) \mathrm{Td}(X)$. Thus, HRR is an equality of numbers, i.e., elements of a set. In these talks, we will explain a proof of HRR that uses the hierarchy $\{2\text{-categories}\} \rightarrow \{1\text{-categories}\} \rightarrow \{\text{Vector spaces}\} \rightarrow \{\text{Numbers}\}$. I.e., the origin of HRR will be $2$-categorical. The procedure by which we go down from $2$-categories to numbers is that of “categorical trace”.
However, in order to carry out our program, we will need to venture into the world of higher categories: the $2$-category we will be working with consists of DG-categories, the latter being higher categorical objects. And the process of calculation of the categorical trace will involve derived algebraic geometry: the key geometric player will be the self-intersection of the diagonal of $X$, a.k.a. the inertia (derived) scheme of $X$.
So, this series of talks can be regarded as providing a motivation for studying higher category theory and derived algebraic geometry: we will use them in order to prove an equality of numbers. That said, we will try to make these talks self-contained, and so some necessary background will be supplied.

Talk canceled! Olivier Schiffmann,
Title: Cohomological Hall algebras of quivers

We will report on joint work with T. Bozec and E. Vasserot. We consider the cohomological Hall algebra of the preprojective algebra of an arbitrary quiver and prove that is generated by a simple family of elements. This algebra, which strictly contains the Kac-Moody algebra associated to the quiver, acts faithfully on the cohomology of Nakajima quiver varieties, and its graded character is given by the (full) Kac polynomial. For some general reasons, it is a (deformed) Borcherds-Kac-Moody algebra and its Borcherds-Cartan datum encodes the dimension of the spaces of cuspidal elements in the Hall algebra of the quiver (over a finite field). We show that this number of cuspidals is a polynomial in the size of the finite field, and we offer some speculation as to a possible geometric interpretation of that polynomial.

Vanessa Miemietz,
Title: Simple transitive $2$-representations via coalgebra $1$-morphisms

I will explain how to obtain every simple transitive $2$-representation of a fiat $2$-category via some (co)algebra $1$-morphism in its abelianisation.

Kevin Coulembier,
Title: The periplectic Brauer algebra

An analogue of the Brauer algebra was introduced by Moon to study invariant theory for the periplectic Lie superalgebra. This algebra is not cellular in the sense of Graham and Lehrer, but has an interesting structure of a standardly based algebra, in the sense of Du and Rui. Starting from this structure we derive the Cartan decomposition matrix and study cohomology (including construction of quasi-hereditary $1$-covers) for this algebra. As an application we obtain the block decomposition of the category of integrable modules over the periplectic Lie supergroup.

Jieru Zhu,
Title: Presenting cyclotomic Schur algebras

A classical result states that the action of $\mathfrak{gl}(V)$ and the symmetric group on $d$ letters mutually centralize each other on the $d$-fold tensor of $V$. If $V$ admits an action by $\mathbb{Z}/r\mathbb{Z}$, it induces an action of the wreath product of $\mathbb{Z}/r\mathbb{Z}$ and the symmetric group on $d$ letters. A Levi Lie subalgebra $\mathfrak{g}$ of $\mathfrak{gl}(V)$ gives the full centralizer of this action, and we further showed a presentation for the cyclotomic Schur algebra as a quotient of the enveloping algebra of $\mathfrak{g}$. This also provides a PBW type basis and a second presentation with idempotent generators. These results extend to the quantum setting and yield similar presentations and a basis for the the cyclotomic q-Schur algebra. When $r=2$, they become presentations for the Type B hyperoctahedral Schur algebra defined by Richard Green.