## Tentative Schedule

Time | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|

8:30-9:00 | Breakfast | Breakfast | Breakfast | Breakfast | Breakfast |

9:00-9:50 | Registration | Serganova | Labardini Fragoso | Walton | Pelaez |

10:05-10:55 | Kleshchev | Gautam | Clark | Toledano Laredo | Kharchenko |

11:00-11:30 | Coffee | Coffee | Coffee | Coffee | Coffee |

11:30-12:20 | Wang | Chari | Montgomery | Vazirani | Huisgen-Zimmerman |

12:30-2:00 | Lunch | Lunch | Lunch | Lunch | Lunch |

2:00-2:50 | Sussan | Vallejo | Leclerc | Rouquier | McNamara |

3:00-3:30 | Coffee | Reception Poster Session KAP 410 |
Coffee | Coffee | Coffee |

3:30-4:20 | Jimenez Rolland | Saenz Valadez | Friedlander | ||

4:40-5:30 | Mendoza Hernandez | Jasso | |||

7:00 | Dinner University Club |

## Titles and Abstracts

## Monday May 30th

### Alexander Kleshchev (University of Oregon)

RoCK blocks of symmetric groups and Hecke algebras

We present a joint result with Anton Evseev, which describes every block of a symmetric group up to derived equivalence as a certain Turner double algebra. Turner doubles are Schur-algebra-like 'local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. This description was conjectured by Will Turner. It relies on the work of Chuang-Kessar and Chuang-Rouquier. (RoCK=Rouquier+Chuang+Kessar). Key idea is a connection with Khovanov-Lauda-Rouquier algebras and their semicuspidal representations.

### Weiqiang Wang (University of Virginia)

Categorification of quantum symmetric pairs

A quantum coideal subalgebra of quantum group of type A admits a geometric realization and a canonical basis with positivity property, according to our recent work with Yiqiang Li, Bao, and Kujawa. In this talk, we will explain some of the backgrouds, introduce a 2-category a la Khovanov-Lauda-Rouquier which categorifies the coideal algebra, and its 2-representations via flag varieties and category O of type B . This is a joint work with Bao, Shan and Webster.

### Joshua Sussan (CUNY)

\(p\)-DG algebras and quantum \(\mathfrak{sl}(2)\)

We motivate the notion of a \(p\)-DG algebra from the perspective of categorifying the WRT \(3\)-manifold invariant. We then study a specific \(p\)-DG algebra coming from the \(2\)-representation theory of quantum \(\mathfrak{sl}(2)\).

### Rita Jiménez Rolland (CCMM-UNAM)

FIW-modules and point-counting over finite fields

In this talk we consider some families of varieties with actions of certain finite reflection groups – varieties such as the hyperplane complements or complex flag manifolds associated to these groups. Beautiful results of Grothendieck–Lefschetz and Lehrer relate the topology of these complex varieties with point-counting over finite fields. Church, Ellenberg and Farb noticed that their representation stability results on the cohomology groups corresponds to asymptotic stability for "polynomial" statistics on the varieties over finite fields. Our goal is to discuss this correspondence and describe what is the underlying algebraic structure of these families' cohomology rings that makes the formulas convergent. We prove that asymptotic stability holds in general for subquotients of FIW–algebras finitely generated in degree at most one, a result that is in some sense sharp. This is joint work with Jennifer Wilson.

## Tuesday May 31st

### Vera Serganova (UC Berkeley)

Finite-dimensional representations of the Lie superalgebra \(P(n)\)

The Lie superalgebra \(P(n)\) is one of the classical Lie superalgebras introduced by Kac in 1977. It preserves a non-degenerate even symmetric form on a super vector space. For a long time the problem of finding irreducible characters of \(P(n)\) remained open while the similar problems had been solved for all other simple Lie superalgebras. I will present some new results (joint with Entova-Aizenbud, Gorelik and others) which shed some light on the problem.

### Sachin Gautam (Perimeter Institute)

Elliptic Quantum Groups

In 1995 G. Felder introduced an elliptic \(R\)–matrix, which quantizes the classical dynamical \(r\)–matrix arising from the study of conformal blocks on elliptic curves. The elliptic \(R\)–matrix satisfies a dynamical analog of the Yang–Baxter equation and can be used to define the elliptic quantum group of \(\mathfrak{sl}_n\) in the same vein as the usual \(R\)–matrices gives rise to quantum groups via the \(RTT\) formalism of Faddeev, Reshetikhin and Takhtajan. In this talk I will explain Felder’s definition and present its generalization to the case of arbitrary Kac–Moody Lie algebras analogous to the Drinfeld’s new presentation of Yangians and quantum loop algebras. I will also present a method of constructing representations of the elliptic quantum group using \(q\)–difference equations. Our construction gives rise to a classification of irreducible representations of the elliptic quantum group, which is reminiscent of the Drinfeld’s classification of irreducible representations of Yangians. This talk is based on a joint work with V. Toledano Laredo.

### Vyjayanthi Chari (UC Riverside)

Character formulae and tensor products of prime representations of quantum affine \(\mathfrak{sl}_{n+1}\)

We study the family of prime representations of quantum affine \(\mathfrak{sl}_{n+1}\) introduced in the work of Hernandez and Leclerc. These are defined by using an \(A_n\)-quiver; in the case of the sink-source quiver and the monotonic quiver they proved that the associated subcategory of finite–dimensional representations of the quantum affine algebra was a monoidal categorification of a cluster algebra with the prime representations corresponding to cluster variables. In this talk we shall work with an arbitrary quiver and give a necessary and sufficient condition in terms of Drinfeld polynomials for a tensor product of prime representations to be irreducible. We also state precisely the ''exchange relations'' in the case when a tensor product is reducible; in other words we describe the Jordan-Holder series of the tensor product. As a consequence of our results we write an explicit formula for the character of a prime representation as an alternating linear combination of characters of the local Weyl modules for quantum affine algebras. In the case of the sink source and the monotonic quiver we give a second character formula; in the language of cluster algebras this formula express an arbitrary cluster variable in terms of the original seed. Equivalently, our formula gives the character of a prime representation in terms of the fundamental representations and the Kirillov-Reshetikhin modules. The talk is based on joint work with Matheus Brito.

### Ernesto Vallejo (CCMM-UNAM)

Matrices and Kronecker products

A Kronecker coefficient is the multiplicity of an irreducible complex representation of the symmetric group in the tensor product of other two irreducible representations. In this talk we apply a pair of notions from discrete tomography about integer matrices (uniqueness and additivity) to the computation of some Kronecker coefficients and to obtain several of their stability properties. Following this line of thought we then introduce a new family of polytopes as a tool for computing Kronecker coefficients.

## Wednesday June 1st

### Daniel Labardini Fragoso (IM-UNAM)

Surfaces with orbifold points and species with potential

Felikson-Shapiro-Tumarkin have shown that surfaces with marked points and orbifold points of order 2 give rise to cluster algebras in a natural way, by associating skew-symmetrizable matrices to their triangulations in such a way that flips and matrix mutations are compatible with each other. In this talk, based on joint work in progress with Jan Geuenich, I will present a construction of species and potentials for the triangulations of unpunctured surfaces with marked points and order-2 orbifold points, construction under which flips turn out to be compatible with certain version of mutations of species with potential.

### Sean Clark (Max Plank/Northeastern University)

Quantum enveloping \(\mathfrak{gl}(m|1)\) and canonical bases

The Lusztig-Kashiwara canonical bases of quantum enveloping Lie algebras are remarkable constructions with connections to many different areas of mathematics, notably including higher representation theory. Recently, some progress has been made toward constructing canonical bases for various families of Lie superalgebras using a variety of approaches. In this talk, I will describe a new construction of canonical bases for the half of the standard quantum enveloping algebra of \(\mathfrak{gl}(m|1)\), as well as the polynomial irreducible modules and Kac modules, via crystal bases and braid operators. I will also remark on how these methods generalize to some other cases.

### Susan Montgomery (University of Southern California)

Frobenius-Schur indicators for Hopf algebras and fusion categories

We survey recent results on the values of the FS indicators for specific Hopf algebras, such as bismash products, and for certain near groups; these are fusion categories arising in von Neumann algebras.

### Bernard Leclerc (Université de Caen)

Cluster algebras and infinite-dimensional representations of Borel
subalgebras of quantum affine algebras.

Hernandez and Jimbo have recently introduced a category \(\mathcal{O}\) of representations of a Borel subalgebra \(U_q(\mathfrak{b})\) of a quantum affine algebra \(U_q(\mathfrak{g})\). We consider some monoidal subcategories \(\mathcal{O}^+\) and \(\mathcal{O}^-\) of \(\mathcal{O}\) and show that they have a cluster structure with an initial seed given by the set of all prefundamental representations. This is a joint work with David Hernandez.

### Edith Corina Sáenz Valadez (FC-UNAM)

Standard modules and stratifying systems

In the context of quasi-hereditary algebras the concept of standard modules is very relevant. In the first part of this talk we will define these modules, present their main properties and establish their relation with quasi-hereditary algebras. In the second part, we will establish how the properties of the standard modules motivated the concept of stratifying system.

### Octavio Mendoza Hernández (IM-UNAM)

Homological systems in triangulated categories

In this talk, we introduce and develope the notion of homological systems for triangulated categories. These systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in triangulated categories. One of the consequences we get is that, attached to an homological system \(\Theta,\) there are two standardly stratified algebras \(A\) and \(B,\) which are derived equivalent. That is, the category \(\mathcal{F}(\Theta)\), of the \(\Theta\)-filtered objects in a triangulated category \(T\), admits in a very natural way a structure of an exact category, and moreover there are triangulated equivalences between the bounded derived category of the exact category \(\mathcal{F}(\Theta)\) and the bounded derived categories associated to the standardly stratified algebras \(A\) and \(B\). Some of the obtained results can be seen also under the light of the cotorsion pairs in the sense of Iyama-Nakaoka-Yoshino. We recall that cotorsion pairs are studied extensively in relation with cluster tilting categories, t-structures and co-t-structures.

## Thursday June 2nd

### Chelsea Walton (Temple University)

No Quantum Symmetry

I will discuss recent work on the lack of finite dimensional Hopf actions on (quantizations of) commutative domains. This includes results on Hopf actions on Weyl algebras, universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The starting point for these results was joint work with P. Etingof on semisimple Hopf actions on commutative domains (arXiv:1301.4161), and continued in joint work with J. Cuadra and P. Etingof on finite dimensional Hopf actions on Weyl algebras (arxiv:1409.1644, arXiv:1509.01165) and with P. Etingof on such actions on deformation quantizations (arXiv:1602.00532) and on algebraic quantizations (arXiv:1605:00560).

### Valerio Toledano Laredo (Northeastern University)

Quasi-Coxeter categories, the Casimir connection and quantum Weyl groups

A quasi-Coxeter category is a braided tensor category which
carries an action of a generalised braid group \(B_W\) on the
tensor powers of its objects.

The data which defines the action of \(B_W\) is similar in flavour
to the associativity constraints in a monoidal category, but
is related to the coherence of a family of fiber functors on \(C\).

I will outline how to construct such a structure on integrable,
category \(\mathcal{O}\) representations of a symmetrisable Kac-Moody algebra
\(\mathfrak{g}\), in a way that incorporates the monodromy of the KZ and Casimir
connections of \(\mathfrak{g}\).

The rigidity of this structure implies in particular that the
monodromy of the latter connection is given by the quantum Weyl
group operators of the quantum group \(U_h(\mathfrak{g})\).

This is joint work with Andrea Appel.

### Monica Vazirani (UC Davis)

A Schur-Weyl-like construction of \(L(k^N)\) for the DAHA

Building on the work of Calaque-Enriquez-Etingof, Lyubashenko-Majid, and Arakawa-Suzuki, Jordan constructed a functor from quantum D-modules on special linear groups to representations of the double affine Hecke algebra (DAHA) in type A. When we input quantum functions on SL(N) the output is \(L(k^N)\), the irreducible DAHA representation indexed by an N by k rectangle. For the specified parameters, \(L(k^N)\) is Y-semisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Y-weight basis. This is joint work with David Jordan.

### Raphael Rouquier (UC Los Angeles)

Tensor products and 2-representations

I will explain how Hopf-type categorical structures arise in Lie theory and I will discuss conjectural relations with low-dimensional topology and counting invariants.

### Eric Friedlander (University of Southern California)

Support varieties and cohomology for unipotent algebraic groups

Cohomology and support varieties have been used to study the representations of finite groups and more general finite group schemes. Using 1-parameter subgroups, I have developed some aspects of a theory of support varieties for certain linear algebraic groups over an algebraically closed field of positive characteristic. This theory satisfies most of of the standard properties, but lacks a close connection to cohomology in this generality. Because unipotent groups have interesting cohomology algebras, I have initiated a computation of the (rational, continuous) cohomology of certain unipotent algebraic groups and a comparison of the resulting cohomological support varieties to support varieters defined in terms of 1-parameter subgroups.

### Gustavo Jasso (Universitat Bonn)

Higher Nakayama Algebras

This is a report of join work with Julian Kulshammer. We introduce a class of combinatorially described finite dimensional algebras which we call higher Nakayama algebras. These algebras exhibit properties analogous to those of classical Nakayama algebras from the point of view of Iyama's higher Auslander--Reiten theory.

## Friday June 3rd

### José Pablo Pelaez (IM-UNAM)

A triangulated approach to the Bloch-Beilinson filtration

We will construct a finite filtration on the Chow groups which satisfies several of the properties of the still conjectural Bloch-Beilinson filtration. The construction is carried out in Voevodsky’s triangulated category of motives DM.

### Vladislav Karchenko (FES-C UNAM)

Quantum Lie Theory

The numerous attempts over the previous 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary ''quantum'' Lie bracket have not been widely accepted. In the talk we discuss an alternative approach that includes multivariable operations. There are many fields in which multivariable operations replace the Lie bracket, such as investigations of skew derivations in ring theory, local analytic loop theory, and theoretical research on generalizations of Nambu mechanics. Among the problems discussed in the talk are the following: multilinear quantum Lie operations, the principle generic quantum Lie operation, the basis of symmetric generic operations, Shestakov-Umirbaev operations for the Lie theory of nonassociative products. We also discuss extent to which a natural binary bracket may define somerthing that looks like a ''quantum Lie algebra'' inside of quantizations of Kac-Moody type in line with the PBW theorem for character Hopf algebras.

### Birge Huisgen-Zimmerman (UC Santa Barbara)

Generic representation theory of quivers with relations

Instead of shooting for a complete classification of the indecomposable modules with any fixed dimension vector over a finite dimensional algebra (a hopeless objective), one adopts the following strategy to attain a more modest goal: Namely, (1) one tries to determine the irreducible components of the algebraic varieties parametrizing these modules, and (2), one targets the ''generic structure'' of the modules encoded by each of the individual components (a module property is ''generic'' for a component if it is shared by all modules in some dense open subset). We will outline the origin and present status of the theory. Then we will focus on a specific class of algebras and illustrate recent results with examples.

### Peter McNamara (University of Queensland)

Geometric Extension Algebras

Geometric extension algebras are a family of algebras which includes the KLR algebras, quiver Schur algebras and algebras governing category \(\mathcal{O}\). We study the question of when these algebras are affine quasihereditary, or equivalently when their module categories are affine highest weight.