Symmetric permutation invariants in some tensor products

TL;DR

This paper develops methods for constructing fundamental invariants and calculating the Hilbert series of invariant subalgebras in tensor products of polynomial rings under symmetric group actions, utilizing Schur functions and combinatorial identities.

Contribution

It introduces a new approach combining Schur functions and combinatorial identities to analyze invariants in tensor products under symmetric groups.

Findings

Explicit formulas for Hilbert series of invariant subalgebras

New combinatorial identities involving Schur functions

Framework for constructing fundamental invariants

Abstract

This is a note for constructing fundamental invariants and computing the Hilbert series of the invariant subalgebras of tensor products of polynomial rings under the action by a direct product of symmetric groups. Our computation relies on Schur functions bringing together several identities of combinatorial generating functions including that of plane partitions.