This workshop is supported by the stat-up fund of L-station at Hokkaido University, Japan Society for the Promotion of Science Grand-in-Aid for Young Scientists (B) 23740014, Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (C) 22540002, Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (S) 50022687, and Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (B) 22340005.
There are notes taken by Steven Sam: [Link]
| 9:30--10:30 | 11:00--12:00 | 13:30--14:30 | 15:00--16:00 | 16:30--17:30 | |
|---|---|---|---|---|---|
| 19 | Enomoto I | Sam I | Juteau I | ||
| 20 | Tsuchioka | Solleveld I | Enomoto II | Juteau II | Sam II |
| 21 | Kodera | Sam III | Hikita | Muthiah I | Solleveld II |
| 22 | Kato | Kimura | Muthaih II | Juteau III | Solleveld III |
The registration will be started at 13:00 in February 19.
Lascoux-Leclerc-Thibon-Ariki (LLTA) type theory connects a modular representation theory of various Hecke algebras to a representation theory of quantum enveloping algebras.
In the first talk, we will illustrate a framework of this theory. Moreover we will review more detailed structure for affine Hecke algebras of type A and of type B. The LLTA type theory for affine Hecke algebras of type B is conjectured by the speaker and Masaki Kashiwara. We introduced the notion of “symmetric crystals”. Varagnolo and Vasserot solved this conjecture using recent developments of a graded representation theory and my quiver construction of symmetric crystals.
In the second talk, first of all, we give an overview of Lusztig's geometric construction of a lower global basis of the half quantum enveloping algebra $U_v^-(\mathfrak{g})$. In the latter half of this talk, we will explain a quiver construction of a lower global basis associated to symmetric crystals. This is analogous to Lusztig's construction. We will use a theory of perverse sheaves on the moduli space of representations of quivers with an involution.
Let $\hat{\mathcal{B}_{v}}$ be an affine Springer fiber of type $A$ associated to a regular semisimple nil elliptic element $v\in\mathfrak{sl}_{n}[[\epsilon]]$. The symmetric group $\mathfrak{S}_{n}$ acts on the homology $H_{\ast}(\hat{\mathcal{B}_{v}},\mathbb{C})$. By the work of Sommers combined with Gordon's result, this $\mathfrak{S}_{n}$-module for a particular choice of $v$ is isomorphic (up to the sign representation) to the ring of diagonal coinvariants $DR_{n}$.
The ring $DR_{n}$ has a natural bigrading compatible with the $\mathfrak{S}_{n}$-action and its bigraded character has been studied in algebraic combinatorics. In particular, Haglund, Haiman, Loehr, Remmel, and Ulyanov gave a conjectural combinatorial formula for the bigraded Frobenius series for $DR_{n}$.
On the other hand, there is no a priori bigrading on $\hat{\mathcal{B}_{v}}$, while there is a natural grading coming from the homological degree. We construct a filtration on $\hat{\mathcal{B}_{v}}$ which is compatible with the $\mathfrak{S}_{n}$-action and the homological grading. Then, we show that the bigraded Frobenius series of the associated graded is given by a formula generalizing the formula of Haglund, Haiman, Loehr, Remmel, and Ulyanov.
In geometric representation theory a central role is played by the decomposition theorem, which asserts the semi-simplicity of the direct image of an intersection cohomology complex under any proper map. The decomposition theorem is only valid when the coefficients of the sheaves are of characteristic zero, and indeed it is easy to give examples with positive characteristic coefficients where the analogous statement is no longer valid.
However, for applications in modular representation theory it is desirable to understand this “failure” of the decomposition theorem, and to have some replacement when this “failure” occurs. This project is probably too ambitious in general. However the varieties which one meets in representation theory (Schubert varieties, the nilpotent cone etc.) often have special features which one can hope to exploit to develop a theory.
This course will give an introduction to the theory of parity sheaves, which provide one way of understanding the failure of the decomposition theorem in geometric representation theory. Where possible I will explain links to representation theory, so that one gets a sense of the rich interplay between the topology of complex algebraic maps and (modular) representation theory.
Kostka polynomials are certain family of polynomials indexed by two copies of simple modules of a Weyl group $W$. They are intimately connected with unipotent characters in the sense of Deligne-Lusztig, and they admits a characterization by the Lusztig-Shoji algorithm. In this talk, we reinterpret the Lusztig-Shoji algorithm (in more general setting where $W$ is a complex reflection groups) in terms of homological algebra. This naturally upgrades Kostka polynomials to a family of indecomposable modules that we call Kostka systems. They give a new characterization of Kostka polynomials in the case $W$ is a Weyl group.
If time allows, we discuss its abstract version and applications to the KLR-algebras.
Inspired by Nakajima's work on the monoidal categorification of bipartite (quantum) cluster algebra, we construct acyclic quantum cluster algebra via graded quiver varieties “adapted to” acyclic quiver and perverse sheaves on them. (This is a joint work with Fan Qin.)
In a homological study of representation theory, it is important to understand semisimple filtrations of smallest length for a module, which are called Loewy series. This notion includes the radical and the socle series as typical examples.
We study representations of the Lie algebras of one-variable polynomial currents. Among finite-dimensional modules, graded Weyl modules are quite interesting objects. They are defined as the universal finite-dimensional highest weight modules, and for current Lie algebras of type ADE, also known to be isomorphic to level-one Demazure modules for affine Lie algebras, and standard modules defined as the homology groups of quiver varieties. One of the aims of this talk is to explain that the grading gives a unique Loewy series of every Weyl module when the current Lie algebra is of type ADE.
Further I discuss applications of this result to the corresponding quiver varieties. In particular, a formula for the Poincare polynomials of quiver varieties in terms of certain crystal bases is given.
This talk is based on a joint work with Katsuyuki Naoi.
The theory of MV cycles associated to a complex reductive group has proven to be a rich source of structures related to representation theory. I will discuss double MV cycles, which are the analogues of MV cycles in the case of an affine Kac-Moody group.
During the first hour, I will review some aspects of the theory of MV cycles for finite-dimensional groups. This story gives rise to MV polytopes and a surprising connection with Lusztig's canonical basis.
In the second half, I will introduce double MV cycles. Here the finite-dimensional story does not naively generalize. Nonetheless, in type A, I will present a method to parameterize double MV cycles using the action on Fermionic Fock space. This method gives rise to exactly the combinatorics of the Naito-Sagaki-Saito crystal.
If time permits, I would like to discuss some conjectural connections between double MV cycles and the Baumann-Kamnitzer-Tingley notion of affine MV polytopes. I will focus on the case of $\widehat{\mathfrak{sl}_2}$, which already exhibits many of the essential features of the theory.
These talks will be about two topics: the saturation problem for tensor product multiplicities and the homological properties of determinantal varieties.
Given a complex reductive algebraic group G and a dominant weight $\lambda$, let $V_\lambda$ denote the corresponding highest weight irreducible representation. The saturation problem involves finding a positive integer $k$ such that the existence of a nonzero $G$-invariant in \[V_{N\lambda} \otimes V_{N\mu} \otimes V_{N\nu}\] implies the existence of a nonzero G-invariant in \[V_{k\lambda} \otimes V_{k\mu} \otimes V_{k\nu}.\]
I will survey some of the results for this problem. I will also discuss my own work for the classical groups using the invariant theory of symmetric quivers and the properties of certain complete intersections.
The complete intersections that arise give natural analogues of “determinantal varieties”. Motivated by the previous examples, I will describe a class of varieties which might be called determinantal varieties. I will explain some of the homological properties of these determinantal varieties and explain some connections with branching rules for complex reductive algebraic groups, generalizations of Cauchy's identity for symmetric functions, and representations of classical Lie superalgebras.
Let $H$ be a graded affine Hecke algebra with parameters $k$. We will show that the representation theory of $H$ is essentially independent of $k$, where “essentially” will be made precise in several ways. A crucial role is by periodic cyclic homology, which serves as a bridge between spectral and ring-theoretic aspects of $H$.
The first talk will be a general introduction to periodic cyclic homology. This functor was designed as a noncommutative counterpart of De Rham cohomology, but the analogy is precise only for nice commutative algebras. We will show that the periodic cyclic homology of noncommutative algebras with a large centre can still be considered as the cohomology of the dual space of the algebra, even though the dual space need not be separated.
In the second talk we intend to apply this machinery to graded affine Hecke algebras. To do so we need a reasonable understanding of the dual space of $H$, which we will obtain from an extended form of the Langlands classification.
In the final talk we prove that the homology of $H$ does not depend on $k$. For $k=0$ everything can be made explicit, so this yields a description of the homology of $H$. We discuss some consequences about relations between representations of $H$ for different values of $k$.
We introduce two families of superalgebras $R_n$ and $RC_n$ which are “Morita superequivalent” each other. $R_n$ is a generalization of the Khovanov-Lauda-Rouquier algebras. We show that, after suitable specialization and completion, $RC_n$ is isomorphic to the affine Hecke-Clifford superalgebras and its rational degeneration. This is a joint work with Seok-Jin Kang and Masaki Kashiwara.