Sunday, October 22, 2023

Seamus Albion (Vienna, Austria) - Thursday Oct 26, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Seamus Albion of the University of Vienna. The announcement is as follows.

Talk Announcement: 
Title:   An elliptic $A_n$ Selberg integral
Speaker: Seamus Albion (Vienna, Austria)
When: Thursday, Thursday Oct 26, 2023 - 4:00 PM (IST) (12:30 PM CEST)
Where: Zoom: Write to the organisers to get the link

Selberg's multivariate extension of the beta integral appears 
all over mathematics: in random matrix theory, analytic number theory,  multivariate orthogonal polynomials and conformal field theory. The goal of my talk will be to explain a recent unification of two important generalisations of the Selberg integral, namely the Selberg integral associated with the root system of type A_n due to Warnaar and the elliptic Selberg integral conjectured by van Diejen and Spiridonov and proved by Rains. The key tool in our approach is the ellipticinterpolation kernel, also due to Rains. This is based on joint work with Eric Rains and Ole Warnaar.

Saturday, October 7, 2023

David Wahiche (Univeriste de Tours, France) - Thursday, October 12, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by David Wahiche of the University of Tours, France. The title and abstract is below. 

Talk Announcement: 
Title:   From Macdonald identities to Nekrasov--Okounkov type formulas
Speaker: David Wahiche (Universite' de Tours, France)
When: Thursday, Oct 12, 2023, 4:00 PM- 5:00 PM IST 
Where: Zoom: Ask the organisers for a link


Between 2006 and 2008, using various methods coming from representation theory (Westbury), gauge theory (Nekrasov--Okounkov) and combinatorics (Han), several authors proved the so-called Nekrasov–Okounkov formula which involves hook lengths of integer partitions.

This formula does not only cover the generating series for P, but more generally gives a connection between powers of the Dedekind η function and integer partitions. Among the generalizations of the Nekrasov--Okounkov formula, a (q, t)-extension was proved by Rains and Warnaar, by using refined skew Cauchy-type identities for Macdonald polynomials. The same result was also obtained independently by Carlsson–Rodriguez-Villegas by means of vertex operators and the plethystic exponential. As mentioned in both of these papers, the special case q=t of their formula correspond to a q version of the Nekrasov--Okounkov formula, which was already obtained by Dehaye and Han (2011) and Iqbal et al. (2012).

Motivated by the work of Han et al. around the generalizations of the Nekrasov--Okounkov formula, one way of deriving Nekrasov--Okounkov formula is by using the Macdonald identities for infinite affine root systems (Macdonald 1972), which can be thought as extension of the classical Weyl denominator formula.

In this talk, I will try to explain how some reformulations of the Macdonald identities (Macdonald 1972, Stanton 1989, Rosengren and Schlosser 2006) can be decomposed in the basis of characters for each infinite of the 7 infinite affine root systems by the Littlewood decomposition. This echoes a representation theoretic interpretation of the Macdonald identities (see the book of Carter for instance) and an ongoing project with Cédric Lecouvey, I will mention some partial results we get.

At last, I will briefly explain how to go from these reformulations of Macdonald identities to q Nekrasov--Okounkov type formulas.