Algebraic and Enumerative Combinatorics Seminar - Maciej Dolega

Thursday, February 10, 2022 1:00 pm - 1:00 pm EST (GMT -05:00)

Title:  A curious identity between the orthogonal Brezin--Gross--Witten integral and Schur symmetric functions via b-deformed monotone Hurwitz numbers

Speaker: Maciej Dolega
Affiliation: IMPAN
Zoom: Please email Olya Mandelshtam


This talk is intended for an algebraic combinatorial community and no prior knowledge is required. All the difficult words (Hurwitz numbers, KP hierarchy, HCIZ and BGW integrals, Jack symmetric functions, the b-conjecture) will be explained and gently introduced.

The monotone Hurwitz numbers can be understood combinatorially as cardinalities of monotone transitive walks in the Cayley graph of the symmetric group. They share many beautiful properties with ordinary Hurwitz numbers, but one of their most interesting properties is that their generating function coincides with the topological expansion of the celebrated Harish-Chandra--Itzykson--Zuber integral (in the case of double numbers) and the Brezin--Gross--Witten integral (in the case of single numbers). Using standard tools from algebraic combinatorics, one can express this generating function in terms of Schur symmetric functions and prove that it is a solution of the infinite system of partial differential equations called the KP hierarchy. Inspired by a mysterious conjecture of Goulden and Jackson which connects generating function of Jack symmetric functions with enumeration of combinatorial maps, we define (following joint work with Chapuy) the generating function of b-deformed monotone Hurwitz numbers by replacing Schur symmetric functions by their one-parameter deformation -- Jack symmetric functions. We show that it has an explicit combinatorial interpretation, which gives a topological expansion of the \beta-HCIZ integral. Finally, we show that for b=1 this generating function has a very interesting structure -- it is a solution of the infinite system of Partial Differential Equations called the BKP hierarchy. We prove it by finding an explicit expansion in Schur symmetric functions, which surprisingly involves dimensions of the irreducible representations of the orthogonal group. As an application, we deduce an explicit Pfaffian formula for the Brezin--Gross--Witten integral over the orthogonal group. This is joint work with Valentin Bonzom and Guillaume Chapuy.