Séminaires : Séminaire d'Algèbre

The talk, based on joint work in progress with V. Schechtman, will first
recall our description of perverse sheaves on $Sym^n(\mathbb{C})$, the symmetric
product of the complex line with its natural stratification by multiplicities.
This description proceeds in terms of contingency matrices, which are certain
integer matrices appearing (besides their origin in statistics) in three
different contexts:
-- A natural cell decomposition of $Sym^n(\mathbb{C})$.
-- Compatibility of multiplication and comultiplication in $\mathbb{Z}_+$-graded Hopf algebras.
-- Parabolic Bruhat decomposition for $GL_n$.
Perverse sheaves on $Sym^n(\mathbb{C})$ are described in terms of certain data
of mixed functoriality on contingency matrices which we call Janus sheaves.
I will then explain our approach to categorifying the concept of Janus sheaves,
in which sums are replaced by filtrations with respect to the Bruhat order.
Such data can be called Janus schobers. Examples can be obtained from
$\mathbb{Z}_+$-graded Hopf categories, a concept going back to Crane-Frenkel,
of which we consider two examples related to representations of groups $GL_n$
over finite fields (Joyal-Street) and $p$-adic fields (Bernstein-Zelevinsky).
(Zoom talk shared with https://nc-shapes.info )