Frobenius splitting of certain rings of invariants

TL;DR

This paper proves that specific rings of invariants associated with special linear and orthogonal groups are Frobenius split, enhancing understanding of their algebraic and geometric properties.

Contribution

It establishes Frobenius splitting for rings of invariants under certain classical groups, a result not previously known for these cases.

Findings

Rings of invariants for SL acting on multiple copies are Frobenius split.

Rings of invariants for SO acting on multiple copies are Frobenius split.

Frobenius splitting implies desirable geometric and algebraic properties.

Abstract

Two classical rings of invariants are shown to be Frobenius split: for the special linear group acting on the direct sum of several copies of the defining representation and several copies of the dual of the defining representation; and for the special orthogonal group acting on several copies of the defining representation.