Systematic construction of isomorphisms among quantum Grothendieck rings

2021/02/18 Thu 10:30 - 12:00
Hironori Oya
Shibaura Institute of Technology

A quantum Grothendieck ring of the monoidal category of finite-dimensional modules over a quantum loop algebra $U_{q}(L\mathfrak{g})$ is a one parameter deformation of the usual
Grothendieck ring. It is introduced by Nakajima and Varagnolo-Vasserot in the case when $U_{q}(L\mathfrak{g})$ is of simply-laced type through a geometric method, and subsequently by Hernandez when $U_{q}(L\mathfrak{g})$ is of arbitrary untwisted affine type through an algebraic method. In the simply-laced case, quantum Grothendieck rings are known to give an
algorithm for calculating $q$-characters of simple modules, which is an analogue of Kazhdan-Lusztig algorithm.
In this talk, we present a collection of algebra isomorphisms among quantum Grothendieck rings, which respect the $(q, t)$-characters of simple modules. As a corollary, we obtain new positivity results for the simple $(q, t)$-characters of non-simple-laced types. Moreover, comparing our
isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we show that an analogue of Kazhdan-Lusztig algorithm for computing simple $q$-characters is available when $\mathfrak{g}$ is of type $B$.
This result is a vast generalization of our previous work [Hernandez-O, Adv. Math. 347 (2019), 192--272]. Hence, besides the summary of main results, I will explain some details of the proof, focusing on the tools and results obtained after [HO].
This talk is based on a joint work (arxiv:2101.07489) with Ryo Fujita, David Hernandez, and Se-jin Oh.

Remark: This will be a zoom seminar.

Zoom URL:
Passcode: The order of the Weyl group of type $E_6$