Decompositions of Bernstein-Sato polynomials and slices

TL;DR

This paper explores how the Bernstein-Sato polynomial of a semi-invariant polynomial can be decomposed using representation theory and a slice method, enabling elementary computations for classical invariants and semi-invariants of quivers.

Contribution

It introduces a new slice method for decomposing Bernstein-Sato polynomials based on representation-theoretic properties, simplifying calculations for various invariants.

Findings

Decomposition of Bernstein-Sato polynomials via a slice method.

Elementary computation of classical invariants' Bernstein-Sato polynomials.

Application of the method to semi-invariants of quivers.

Abstract

Let $G$ be a linearly reductive group acting on a vector space $V$, and $f$ a (semi-)invariant polynomial on $V$. In this paper we study systematically decompositions of the Bernstein-Sato polynomial of $f$ in parallel with some representation-theoretic properties of the action of $G$ on $V$. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein-Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a "slice method" which shows that the decomposition of $V$ as a representation of $G$ can induce a decomposition of the Bernstein-Sato polynomial of $f$ into a product of two Bernstein-Sato polynomials - that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein-Sato polynomials for a large class of semi-invariants of quivers.