Department of


Seminar Calendar
for Integrability and Representation Theory events the year of Thursday, October 12, 2017.

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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.)