Welcome to the homepage of the Online Representation Theory Seminar. The Seminar is an initiative to bring together faculty working in Algebra and Geometry, and mostly based in Italian Universities, with interests in Representation Theory. The talks cover a wide range of topics, including Lie Theory, Combinatorics, and Geometric Representation Theory.

The activity is currently being coordinated by Andrea Appel (Parma), Fabio Gavarini (Roma – Tor Vergata), Martina Lanini (Roma – Tor Vergata), and Francesco Sala (Pisa).

We usually meet on Fridays from 3:00pm to 4:00pm (Central European Time) on a Microsoft Teams Channel. If you want to stay informed about the seminars and receive the link to attend the seminar, please subscribe to the following mailing list: [rainmaker_form id=”1291″]

Title: Shifted Yangians and quantum affine algebras revisited

Abstract: In the first part of the talk, I will recall some basic results about shifted Yangians (and their trigonometric versions-the shifted quantum affine algebras), which first appeared in the work of Brundan-Kleshchev relating type A Yangians and finite W-algebras and have become a subject of renewed interest over the last 5 years due to their close relation to quantized Coulomb branches introduced by Braverman-Finkelberg-Nakajima.

In the second part of the talk, I will try to convince that the case of antidominant shifts (opposite to what was originally studied in the work of Brundan-Kleshchev in type A and of Kamnitzer-Webster-Weekes-Yacobi in general type) is of particular importance as the corresponding algebras admit the RTT realization (at least in the classical types). In particular, this provides a conceptual explanation of the coproduct homomorphisms, gives rise to the integral forms of shifted quantum affine algebras, and also yields a family of (conjecturally) integrable systems on the corresponding Coulomb branches. As another application, the GKLO-type homomorphisms used to define truncated version of the above algebras provide a wide class of rational/trigonometric Lax matrices in classical types.

This talk is based on the joint works with Michael Finkelberg as well as Rouven Frassek and Vasily Pestun.

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: Newton-Okounkov bodies for cluster varieties

Abstract: Cluster varieties are schemes glued from algebraic tori. Just as tori themselves, they come in dual pairs and it is good to think of them as generalizing tori. Just as compactifications of tori give rise to interesting varieties, (partial) compactifications of cluster varieties include examples such as Grassmannians, partial flag varieties or configurations spaces. A few years ago Gross–Hacking–Keel–Kontsevich developed a mirror symmetry inspired program for cluster varieties. I will explain how their tools can be used to obtain valuations and Newton–Okounkov bodies for their (partial) compactifications. The rich structure of cluster varieties however can be exploited even further in this context which leads us to an intrinsic definition of a Newton–Okounkov body. The theory of cluster varieties interacts beautifully with representation theory and algebraic groups. I will exhibit this connection by comparing GHKK’s technology with known mirror symmetry constructions such as those by Givental, Baytev–Ciocan-Fontanini–Kim–van Straten, Rietsch and Marsh–Rietsch. (joint work in progress with M.Cheung, T. Magee and A. Nájera Chávez.)

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: Perverse sheaves with nilpotent singular support for curves and quivers

Abstract: Perverse sheaves on the representation stacks of quivers are fundamental in the categorification of quantum groups. I will explain how to prove that semisimple perverse sheaves with nilpotent singular support on the stack of representations of an affine quiver form Lusztig category and how to extend this question to quivers with loops. The analogous question for curves is to determine perverse sheaves on the stack of coherent sheaves whose singular support is a union of irreducible components of the global nilpotent cone. We solve this problem for elliptic curves, for which we also show that the characteristic cycle map induces a bijection between simple Eisenstein spherical perverse sheaves and irreducible components of the global nilpotent cone. This constitutes a step towards the understanding of the degree zero part of the cohomological Hall algebra of a curve.

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: An asymptotic cellular category for G(e, e, n)

Abstract: Given a Coxeter group W, one may consider its Hecke algebra, which is a deformation of the group algebra of W. Kazhdan and Lusztig have constructed the celebrated Kazhdan–Lusztig basis, which has many interesting properties. This basis can be used to construct a partition of W into Kazhdan–Lusztig cells, a partition of the irreducible complex representations of W into families and also a partition of the “unipotent characters” of W into families. There exist categorical counterparts of these objects, and the goal of this talk is to explain a tentative towards a partial generalization for the complex reflection group G(e, e, n).

First, I will describe the situation of a Coxeter group and then explain briefly what can be extended to (some) complex reflection groups. Finally, I will turn to an description of the asymptotic category, which is constructed from representations of quantum sl(n) at a 2e-th root of unity, and try to justify the term “asymptotic cellular category”.

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: Perverse Sheaves, Finite Dimensional Algebras and Quivers

Abstract: In this talk I will introduce the category of perverse sheaves on a topologically stratified space X and give some examples. Then, I will show that when X has finitely many strata, each with finite fundamental group, such category is equivalent to a category of modules over a finite dimensional algebra A. Finally, I will discuss some algebraic approaches one can use in order to describe the algebra A. This talk is based on joint work with Jon Woolf.

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: R-matrices and Yangians

Abstract: An R-matrix is a solution to the Yang-Baxter equation (YBE), a central object in Statistical Mechanics, discovered in 1970’s. The R-matrix also features prominently in the theory of quantum groups formulated in the eighties. In recent years, many areas of mathematics and physics have found methods to construct R-matrices and solve the associated integrable system. In this talk, I will present one such method, which produces meromorphic solutions to (YBE) starting from the representation theory of a family of quantum groups called Yangians. Our techniques give (i) a constructive proof of the existence of the universal R-matrix of Yangians, which was obtained via cohomological methods by Drinfeld in 1983, and (ii) prove that Drinfeld’s universal R-matrix is analytically well behaved. This talk is based on joint works with Valerio Toledano Laredo and Curtis Wendlandt.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Cellularity of the p-Kazhdan-Lusztig Basis for Symmetric Groups

Abstract: After recalling the most important results about Kazhdan-Lusztig cells for symmetric groups, I will introduce the p-Kazhdan-Lusztig basis and give a complete description of p-cells for symmetric groups. After that I will mention important consequences of the Perron-Frobenius theorem for p-cells which provide one of the last missing ingredients for the proof of the cellularity of the p-Kazhdan-Lusztig basis in finite type A.

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: Birational sheets in linear algebraic groups

Abstract: The sheets of a variety X under the action of an algebraic group G are the irreducible components of subsets of elements of X with equidimen- sional G-orbits. For G complex connected reductive, the sheets for the adjoint action of G on its Lie algebra g were studied by Borho and Kraft in 1979. In 2016, Losev introduced finitely-many subvarieties of g consisting of equidimensional orbits, called birational sheets: their definition is less immediate than the one of a sheet, but they enjoy better geometric and representation–theoretic properties and are central in Losev’s suggestion of an Orbit method for semisimple Lie algebras.

In the opening part of the seminar we give a brief overview of sheets and recall some basics about Lusztig–Spaltenstein induction of conjugacy classes in terms of the so-called Springer generalized map and analyse its interplay with birationality. This will give the instruments to introduce Losev’s birational sheets in g.

The main part is aimed at investigating analogues of birational sheets of conjugacy classes in G. To conclude, assuming that the derived subgroup of G is simply connected, we illustrate the main features of these varieties, comparing them with the objects defined by Losev.

Part of the talk is based on joint works with G. Carnovale and F. Esposito, and M. Costantini.

The seminar will be broadcasted online. More information will be posted before the date of the seminar.

Title: Elliptic root systems of non-reduced type

Abstract: After explaining some known basic facts about elliptic root systems (ERS) of reduced type, I will show the classification and automorphism groups of ERS of non-reduced type. Some future problems will be discussed. These results are obtained in collaboration with A. Fialowski and Y. Saito.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Old and new identities for the nabla operator and counting affine permutations

Abstract: Amazing nabla operator discovered by Bergeron and Garsia is a cornerstone of the theory of Macdonald polynomials. Applying it to various symmetric functions produces interesting generating functions of Dyck paths and parking functions. These kinds of results are sometimes known as “shuffle theorems”. I will try to give an overview of these results and explain how working with affine permutations and certain generalized P-tableaux allows to view them from a uniform point of view. The “new” in the title refers to the formula conjectured by Loehr and Warrington giving an explicit expansion of nabla of a Schur function in terms of nested Dyck paths.

This is a joint work with Erik Carlsson.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Approximations of a Nichols algebra from a geometric point of view

Abstract: The talk is based on work in progress with Francesco Esposito and Lleonard Rubio y Degrassi.

Recently Kapranov and Schechtman have settled an equivalence between the category of graded connected co-connected bialgebras in a braided monoidal category V and the category of factorizable systems of perverse sheaves on all symmetric products Sym ⁿ(C) with values in V. The Nichols (shuffle) algebra associated with an object V corresponds to the system of intersection cohomology extensions of a precise local system on the open strata. Motivated by the study of Fomin-Kirillov algebras and their relation with Nichols algebras, we describe the factorizable perverse sheaves counterpart of some algebraic constructions, including the n-th approximation of a graded bialgebra, and we translate into geometric statements when a Nichols algebra is quadratic.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Stokes phenomena, Poisson-Lie groups and quantum groups.

Abstract: Let G be a complex reductive group, G* its dual Poisson-Lie group, and g the Lie algebra of G. G-valued Stokes phenomena were exploited by P. Boalch to linearise the Poisson structure on G*. I will explain how U( g)-valued Stokes phenomena can be used to give a purely transcendental construction of the quantum group U _h(g), and show that the semiclassical limit of this construction recovers Boalch’s. The latter result is joint work with Xiaomeng Xu.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Lax type operator for W-algebras

Abstract: In 1985 Zamolodchikov constructed a “non-linear” extension of the Virasoro algebra known as W_3 algebra. This is one of the first appearances of a rich class of algebraic structures, known as W-algebras, which are intimately related to physical theories with symmetries and revealed many applications in mathematics. In the talk I will review some basic facts about the general theory of W-algebras and provide a description using Lax operators. This approach shows the deep connection of the theory of W-algebras with Yangians and integrable Hamiltonian hierarchies of Lax type equations. This is a joint work with A. De Sole and V.G. Kac.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Ideals of subspace arrangements and representation theory

Abstract: I’ll discuss results linking the study of certain highly symmetric arrangements of linear subspaces in affine space to representation theory, describing how to obtain qualitative information (e.g., is the arrangement Cohen-Macaulay?) and quantitative information (e.g., what is the Hilbert series of the ideal of the arrangement?) using techniques from the representation theory of Cherednik algebras.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.

Title: Cohomological Hall algebras associated to ADE surface singularities

Abstract: To a (reasonable) CY category C of global dimension two one can attach an associative algebra — its cohomological Hall algebra (COHA) — which is an algebra structure on the Borel-Moore homology of the stack of objects in C. In examples related to quivers (i.e. when C is the category of representations of the preprojective algebra of a quiver Q) this yields (positive halves) of Kac-Moody Yangians. In ongoing joint work with Diaconescu, Sala and Vasserot, we consider the case of the category of coherent sheaves supported on the exceptional locus of a Kleinian surface singularity. This is related to the above quiver case by a ‘2d-COHA’ version of Cramer’s theorem relating the (usual) Hall algebras of two hereditary categories which are derived equivalent.

The seminar will be broadcasted online. The access link will be sent to the mailing list a day in advance.