Coherent systems of probability measures on graphs for representations of free Frobenius towers

TL;DR

This paper extends the concept of coherent probability measures on graphs to non-semisimple free Frobenius towers, linking representation theory with probabilistic structures and dualities.

Contribution

It generalizes the framework of coherent systems to non-semisimple algebras and connects these systems to central elements and module dualities.

Findings

Defined two coherent systems for free Frobenius towers

Connected one system to central elements of the algebras

Captured duality between simple modules and projective modules

Abstract

First formally defined by Borodin and Olshanski, a coherent system on a graded graph is a sequence of probability measures which respect the action of certain down/up transition functions between graded components. In one common example of such a construction, each measure is the Plancherel measure for the symmetric group $S_{n}$ and the down transition function is induced from the inclusions $S_{n} \hookrightarrow S_{n+1}$. In this paper we generalize the above framework to the case where $\{A_n\}_{n \geq 0}$ is any free Frobenius tower and $A_n$ is no longer assumed to be semisimple. In particular, we describe two coherent systems on graded graphs defined by the representation theory of $\{A_n\}_{n \geq 0}$ and connect one of these systems to a family of central elements of $\{A_n\}_{n \geq 0}$. When the algebras $\{A_n\}_{n \geq 0}$ are not semisimple, the resulting coherent systems reflect the duality between simple $A_n$-modules and indecomposable projective $A_n$-modules.