Algebraic Modes of Representations
The Canicular Days

July 16-18, 2017 ,
Faculty of Mathematics & Computer Science
The Weizmann Institute of Science


Abstracts

Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups

Shrawan Kumar

We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the flag varieties G/P associated to an arbitrary symmetrizable Kac-Moody group G, where P is any parabolic subgroup. This generalizes the work of Brion and Anderson-Griffeth-Miller from the finite case to the general Kac-Moody case, and affirmatively answers a conjecture of Lam-Schilling-Shimozono regarding the signs of the structure constants in the case of the affine Grassmannian. We also prove the sign-alternation of the structure constants in the basis of dualizing sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the full flag variety G/B associated to any symmetrizable KacMoody group G. Part of this work is done jointly with my student Seth Baldwin.

Representations of affine Lie (super)algebras and (mock)theta functions, false and quasi theta functions

Victor Kac

I will review some old and new results and conjectures on charaters of irreducible highest weight modules of affine Lie (super)algebras.

From spectra of quantum groups to quantum cluster algebras

Milen Yakimov

In the early 90s Anthony Joseph obtained a number of fundamental results on the spectra of quantum groups and their relations to representation theory. We will describe how these results can be used for the construction of quantum cluster algebras. Our first result addresses the BerensteinZelevinsky conjecture on the existence of upper quantum cluster algebra structures on quantum double Bruhat cells which will proved in a stronger form for actual quantum cluster algebras. The second one is a construction of families of clusters on the quantized coordinate rings of open Richardson varieties — a step towards Leclerc’s conjecture for the existence of cluster structures on these rings. This a report on joint work with Ken Goodearl and Tom Lenagan.

Generalized quantum groups and Gelfand-Kirillov conjecture

Jacob Greenstein

In this talk (based on joint work with Arkady Berenstein) I will discuss a uniform generalization of enveloping algebras of Lie algebras and superalgebras and their quantum versions as certain Hopf algebras generated by skew-primitive elements. I will also discuss the main motivation for this setup, namely the appropriate generalization of Gelfand-Kirillov conjecture to the skew fields of fractions of such Hopf algebras and their suitable quotients.

Generalized Joseph's decompositions

Arkady Berenstein

In 1999 Anthony Joseph introduced a remarkable decomposition of (the locally finite part of) quantized enveloping algebras U_q(g). The goal of my talk (based on joint work with J. Greenstein) is to generalize this decomposition to a larger class of Hopf algebras by explicitly using the (generalized) Peter-Weyl theorem for their Hopf duals. In the case when g is semisimple, our approach allows for constructing a natural basis in the center of U_q(g) whose elements behave as characters of finite-dimensional simple g-modules, i.e., as Schur polynomials.

Monomial braidings

Jianrong Li

(This talk is based on a joint work with Arkady Berenstein and Jacob Greenstein)
A braided vector space is a pair (V, f), where V is a vector space and f is an invertible linear operator on V ⊗ V such that f_1 f_2 f_1 = f_2 f_1 f_2. Given a braided vector space (V, f), we constructed a family of braided vector spaces (V, f^ε), where ε is a bitransitive function. Here a bitransitive function is a function ε: [n] x [n] → {1, -1} such that both of {(i,j) : ε (i,j) = 1} and {(i,j) : ε (i,j) = -1} are transitive relations on [n]. The braidings f^ε are monomials in the generators of the braid group Br_n. Therefore we call them monomial braidings. The monomial braidings have some interesting properties. If (V, f) is a Hecke braided vector space (f satisfies (f-q^{-1})(f+q)=0), then f^ε: V^{⊗ 2n} → V^{⊗ 2n} is diagonalizable and all its eigenvalues are of the form q^r,-q^r, r is an integer. Let E_n be the set of all bitransitive functions from [n] x [n] →{1, -1}. The symmetric group S_n acts on E_n. We showed that if ε, ε' in E_n are in the same S_n-orbit, then T^ε and T^ε’ are conjugate in Br_n. We generalized the concept of bitransitive functions to (Γ, C)-transitive functions E → C which satisfy some properties, where C is a finite set and Γ =(V,E) is an oriented graph without multiple oriented edges. Let E_Γ (C) be the set of all (Γ, C)-transitive functions. Then |E_Γ (C)| is a polynomial in c = |C|. We computed the polynomial |E_Γ (C)| explicitly in the case when Γ is a complete oriented graph. It is interesting to study |E_Γ (C)| for any graph Γ. We also introduced the concepts of C-braided categories, C-braided vector spaces, and C-braidings.

The Capelli problem for basic Lie superalgebras

Vera Serganova

We study the Capelli problem about the spectrum of invariant differential operators on symmetric superspaces G/K related to Jordan superalgebras via TKK construction. Our results show that in all cases the spectra are described by certain supersymmetric polynomials which are some specifications of Macdonald polynomials. (Joint work with S. Sahi and H. Salmasian).

A Kazdhan-Lusztig theory for Rep(OSp(m|2n))

Catharina Stroppel

The representation theory of Lie supergroups is even in the finite dimensional case full of interesting structures and non-trivial combinatorics which show several analogies to the representation theory of algebraic groups in positive characteristics, but also to parabolic category O for semisimple Lie algebras. Due to the lack of a localisation theorem, there is no geometric description and therefore one likes to find replacements for standard methods from geometric representation theory or wants to connect these categories to known categories with geometric interpretations or Kazdhan-Lusztig theories. In this talk I will do a few steps in this direction. I plan to describe the combinatorics of the category of finite dimensional representations of the supergroup OSp(m|2n) and compare it with the one of category O in the non-supercase. Using Deligne's universal tensor category we can give a diagrammatic and algebraic description of the category and connect it with the combinatorics of canonical bases etc of Hecke algebras. The goal of the talk is to explain some of the basic ideas and results hereby and in particular outline a version of Kazdhan-Lusztig theory in this setup.

Super-harmonics

Yuri Bazlov

By Laplace's separation of variables theorem, functions on R3 expand in terms of powers of r multiplied by spherical harmonics. That is, the ring of functions is a free module over the invariants of G = SO(3). This result holds for all semisimple Lie groups G and finite reflection groups, and remains true if "functions" are replaced by the universal enveloping algebra of g = Lie(G), classical or quantised, where harmonics are suitably defined. It fails, however, for the superalgebra Λg. Almost two decades ago, Joseph conjectured a multiplicity formula for the adjoint component of Λg. My proof of the formula led to a conjecture that this component is a free module over "almost all" invariants. This conjecture was recently proved by De Concini, Papi and Procesi, with the use of Givental's convolution of invariants. In my talk, I will report on mine and my student Ibukun Ademehin's work which uses Cherednik's double affine Hecke algebra to extend the separation of variables theorem to other representations of G.

Mishchenko-Fomenko algebras and nilpotent bicone

Anne Moreau

The Mishchenko-Fomenko subalgebra of a simple Lie algebra at a regular element x, constructed by the so-called argument shift method, is known to be a maximal Poisson-commutative subalgebra of the symmetric algebra. Moreover, it is a polynomial algebra. In this talk I will explain how to prove that the free generators of this algebra form a regular sequence using geometrical properties of the nilpotent bicone. This was proven by Ovsienko in the particular case of sl(n) and x semisimple regular. I will also discuss some open problems in the case of Mishchenko-Fomenko subalgebras associated with centralizers of nilpotent elements.

Computing the annihilator of a highest weight sl(∞)-module

Ivan Penkov

The goal of this talk is to present an analogue for sl(∞) of A. Joseph's descrip- tion of U(sl(n;C)). In recent work we have parametrized the primitive ideals of U(sl(∞;C)) via quadruples consisting of two Young diagrams and two non-negative integers. In the talk we present an algorithm which computes the primitve ideal of any simple highest weight sl(∞;C)-module. This algorithm is a modifcation of the Robinson-Schensted algorithm. Joint work with Alexey Petukhov.

Joseph ideals and Higgs branches

Tomoyuki Arakawa

The Joseph ideal is a special primitive ideal in the universal enveloping algebra of a simple Lie algebra. In my talk I will upgrade the Joseph ideals to the setting of affine Kac-Moody algebras for Lie algebras in the Deligne exceptional series, and discuss its connection with four-dimensional N=2 superconformal field theories.
This is a joint work with Anne Moreau.