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

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.