Speakers:  David Hernandez ( Université Paris Cité)

Title:  Monoidal Jantzen Filtrations

Abstract:  We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal categories with generic braidings.  It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials.

As a first main example, for finite-dimensional representations of simply-laced quantum affine Kac-Moody algebras, we prove the associativity and that the resulting quantization coincides with the quantum Grothendieck ring constructed in a geometric manner.  Hence, it yields a unified representation-theoretic interpretation of the quantum Grothendieck ring.

As a second main example, we establish an analogous result for representations of quiver Hecke algebras associated with adapted reduced words.

This is a joint work with Ryo Fujita.