Hilbert functions of certain rings of invariants via representations of the symmetric groups (with an appendix by Dejan Govc)

Ehud Meir * (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K[U]Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U=End(W⊕k) and Γ=GL(W) and on the case where U=End(V⊗W) and Γ=GL(V)×GL(W), though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series, using Mathematica (see the appendix by Dejan Govc).
Original languageEnglish
Pages (from-to)1-35
Number of pages35
JournalJournal of Algebra
Volume572
Early online date16 Dec 2020
DOIs
Publication statusE-pub ahead of print - 16 Dec 2020

Keywords

  • Invariant theory
  • Symmetric groups representations
  • Hilbert series

Fingerprint Dive into the research topics of 'Hilbert functions of certain rings of invariants via representations of the symmetric groups (with an appendix by Dejan Govc)'. Together they form a unique fingerprint.

Cite this