Shuffle of trees : the state of the art, Cumulants of bimonoids of species and characteristic operations

03.11.2023, 10:00  –  Room 2.22, Building 9
Arbeitsgruppenseminar Analysis

Pierre Clavier (Mulhouse), Yannic Vargas (Graz)

Pierre Clavier (Mulhouse):

This talk will be introductory and non-technical, with the aim to introduce the participants to
current research questions about shuffle of rooted forests. I will start by recalling the definition and
properties of the usual shuffles of words. Then I will present a family of shuffles of rooted forests
generalising these shuffle of words and state an important result that relates them to Rota-Baxter
operators. Finally I will present some recent results on their (not quite) universal properties and
state open questions regarding their coalgebraic.

Yannic Vargas (Graz):

In this introductory talk, we present the definition of cumulants associated with bimonoids in the category of species, as defined by Aguiar and Mahajan. Bimonoids in species is a categorical analog of bimonoids in the category of graded vector spaces.  When the bimonoid is connected, cocommutative, and locally finite, each of its n-th cumulants is a non-negative integer. Moreover, this number is related to the dimension of its space of primitives. We will illustrate this fact using two approaches: first by using species versions of the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems, and then by using the characteristic operations of a bimonoid and the action of the Tits algebra of compositions on the space of primitives of the bimonoid. Finally, we give an overview of the notion of free cumulants of a bimonoid, related to the free independence of Voiculescu's theory of Free Probability.