Department of

# Mathematics

Seminar Calendar
for Integrability and Representation Theory events the year of Thursday, September 14, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, September 14, 2017

3:00 pm in 345 Altgeld Hall,Thursday, September 14, 2017

#### Symmetrization vs Constant Term Identities for generalized Macdonald operators: a (directed) walk through Double Affine Hecke Algebras, Toroidal Algebras, and Shuffle algebras. Part II

###### Philippe Di Francesco (Illinois)

Abstract: We introduce a generalization of the A-type Macdonald difference operators via a symmetrization identity (S) that maps symmetric functions to difference operators. Recall that Macdonald operators have Macdonald polynomials as common eigenfunctions, and play a crucial role in deep combinatorial theorems (factorial n theorem, shuffle theorem, general questions about Schur positivity). We will show how our generalized Macdonald operators can be obtained by implementing the modular group symmetry of the A-type Double Affine Hecke Algebra in its functional representation. The corresponding difference operators obey commutation relations that can be viewed as t-deformations of some particular quantum cluster algebra relations pertaining to the A-type Q-system. We’ll show how these can be canned into a representation of the affine gl1 quantum toroidal algebra with zero central charge. We finally present an alternative definition of our generalized Macdonald difference operators via a constant term identity (CT) that maps symmetric functions to difference operators. This leads to a natural definition of Shuffle product, for which (S) and (CT) are algebra morphisms. We then show how commutation relations for our operators reduce to simple Shuffle identities. This is illustrated in the quantum cluster algebra limit t->infinity.

Thursday, September 28, 2017

3:00 pm in 345 Altgeld Hall,Thursday, September 28, 2017

#### W-Operators and Combinatorial Hurwitz Numbers

###### Hao Sun (University of Illinois)

Abstract: We will introduce the W-operator and show some interesting combinatorial properties about the W-operator. Then, we will show how the W-operator relates to the study of combinatorial Hurwitz numbers.

Thursday, October 12, 2017

3:00 pm in 345 Altgeld Hall,Thursday, October 12, 2017

#### Generalized Weyl modules and nonsymmetric q-Whittaker functions

###### Daniel Orr   [email] (Virginia Tech)

Abstract: For the current algebra of a finite-dimensional simple Lie algebra, there are universal highest weight modules known as Weyl modules. By works of Cherednik and Braverman-Finkelberg, it is known that the graded characters of these modules form a joint eigenfunction of the q-Toda difference operators (and such functions are known as q-Whittaker functions). These works reveal connections to semi-infinite flag manifolds, double affine Hecke algebras (DAHAs), and Macdonald polynomials. E. Feigin and Makedonskyi introduced "generalized" Weyl modules for the Iwahori subalgebra of the current algebra, in order to give a representation-theoretic interpretation of positivity exhibited by certain specialized nonsymmetric Macdonald polynomials. I will explain some results from our joint work which show that the graded characters of generalized Weyl modules form an eigenfunction of a nonsymmetric variant of the q-Toda system arising from a representation of the nil-DAHA.

Thursday, October 19, 2017

3:00 pm in 345 Altgeld Hall,Thursday, October 19, 2017

#### Hall algebras and the Fukaya category

###### Peter Samuleson (University of Edinburgh)

Abstract: The Hall algebra is an invariant of an abelian (or triangulated) category C whose multiplication comes from "counting extensions in C." Recently, Burban and Schiffmann defined the "elliptic Hall algebra" using coherent sheaves over an elliptic curve, and this algebra has found applications in knot theory, mathematical physics, combinatorics, and more. In this talk we discuss some background and then give a conjectural description of the Hall algebra of the Fukaya category of a topological surface. This is partially motivated by an isomorphism between the elliptic Hall algebra and the skein algebra of the torus, which we also discuss. (Joint works with H. Morton and with B. Cooper.)

Thursday, October 26, 2017

3:00 pm in 345 Altgeld Hall,Thursday, October 26, 2017

#### Toroidal actions on fermionic q-Fock space

###### Joshua Wen (Illinois)

Abstract: One origin story for quantum toroidal algebras comes from the observation that some integrable representations of quantum affine algebras admit an alternate action of that same algebra: the toroidal algebras are what glue these two structures together. One of the first worked-out examples of this phenomenon is the case of fermionic q-Fock space (by Varagnolo-Vasserot and Takemura-Uglov-Saito), where level-1 representations of quantum affine $\mathfrak{sl_n}$ also admit a level-0 action. One can understand this level-0 action by diagonalizing a commutative subalgebra, which was done by Takemura-Uglov using nonsymmetric Macdonald polynomials. I’ll explain this story and work of Nagao relating it to equivariant K-theory of cyclic quiver varieties, wherein the above diagonalization comes for free via fixed-point classes.

Thursday, November 2, 2017

3:00 pm in 345 Altgeld Hall,Thursday, November 2, 2017

#### Integrability of generalized Toda lattice systems

###### Matej Penciak (Illinois)

Abstract: In this talk I will define a collection of integrable systems depending on a Lie algebra g called generalized Toda lattice systems. In the case when = sl_n, this system recovers the definition of the periodic Toda lattice. I will then describe Kostant's proof of the integrability of the generalized Toda lattice, working through the sl_n example along the way.

Thursday, November 9, 2017

3:00 pm in 345 Altgeld Hall,Thursday, November 9, 2017

#### Toroidal actions on fermionic q-Fock space II

###### Joshua Wen (Illinois)

Abstract: Continuing from before, I’ll describe how our two actions on finite q-wedges can each be stabilized in the semi-infinite limit and how they can glued into a level (0,1) action of the quantum toroidal algebra by ‘rotating the Dynkin diagram’. The description of the level 0 action as quantum affine Schur-Weyl duality applied to the polynomial representation of the affine Hecke algebra acting via Macdonald operators allows one to use nonsymmetric Macdonald polynomials to construct interesting bases of q-Fock space. Switching gears, I’ll present Nagao’s work relating the toroidal action on q-Fock space with that on K-theory of cyclic quiver varieties, where such a basis is mapped to the fixed point classes. This assignment of some kind of Macdonald polynomial to a fixed point class in a Nakajima quiver variety should ring a bell for Procesi bundle enthusiasts, and if time permits, I’ll present possible directions along that thread.

Thursday, November 30, 2017

3:00 pm in 345 Altgeld Hall,Thursday, November 30, 2017

#### Integrability of generalized Toda lattice systems: Part 2

###### Matej Penciak (University of Illinois)

Abstract: In this continuation of my first talk I will recall the description of the Toda lattice phase space, and Hamiltonians. After this I will describe the quantization of the system. The quantized Hamiltonians will turn out to be restrictions of differential operators to so called Whittaker functions on the simply connected Lie group associated to $\mathfrak{g}$.