The $\mathbb F_p$-Selberg integral of type $A_n$

TL;DR

This paper establishes a finite field analogue of the Selberg integral for type $A_n$, connecting hypergeometric solutions of KZ equations with polynomial solutions modulo a prime, extending previous results for $A_1$.

Contribution

It introduces a new $F_p$-Selberg integral formula of type $A_n$, generalizing earlier work for $A_1$ and linking hypergeometric and polynomial solutions over finite fields.

Findings

Derived the $F_p$-Selberg integral formula for type $A_n$.

Connected hypergeometric solutions of KZ equations with polynomial solutions mod $p$.

Extended previous $A_1$ results to higher rank $A_n$ cases.

Abstract

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors.