Alg/Top-seminar: Serge Bouc, Simple correspondence functors

Title : Simple correspondence functors

Speaker: Serge Bouc (Amiens)

Time: Friday, 8 August 2014, 13:15

Place: Aud 8

Abstract:
In this joint work with Jacques Thévenaz, we consider the category kC of finite
sets, in which morphisms are k-linear combinations of correspondences (where
k is a commutative ring), and linear representations of this category. Our first
step in a previous work was the description of the essential algebra of a finite set,
and in particular its simple modules, leading to a parametrization of the simple
functors on kC by triples (E,R,V) consisting of a finite set E, an order relation R on E,
and a simple k-linear representation V of the automorphism group of R.

In this talk, I would first like to introduce other examples of functors on kC: to each finite
lattice T, we associate the functor F_T of "functions from a set to T". We show
in particular that this functor F_T is projective if and only if the lattice T is distributive.

The case where T is a total order is of particular interest: the endomorphism algebra
of F_T turns out to be naturally isomorphic to a direct product of matrix algebras over k.
As a consequence, when k is a field and R is a total order on E, the simple functor
parameterized by (E,R,k) is also projective and injective.

In general, we obtain an explicit description of the simple functor S indexed by the triple
(E,R,V) : we first choose a lattice T such that the poset of irreducible elements of T
is isomorphic to (E,R). We then introduce a specific subset G of T, containing E, and
 invariant by the group of automorphisms of (E,R), and the simple functor S appears as
a quotient of F_T associated to G and V. As a consequence, the dimension of each evaluation of S can be explicitly computed.