Tensor invariants for classical groups revisited

We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been we...

Full description

Saved in:
Bibliographic Details
Main Authors Erickson, William Q, Hunziker, Markus
Format Journal Article
LanguageEnglish
Published 30.01.2024
Subjects
Mathematics - Combinatorics
Mathematics - Representation Theory
Online AccessGet full text

Cover

More Information
Summary:We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. Beginning in the broader setting of polynomial invariants, we write down multigraded linear bases which we realize as certain arc diagrams (which may include hyperedges); then the bases for tensor invariants are obtained by restricting our attention to the diagrams that are 1-regular. We survey the many equivalent ways -- some old, some new -- to enumerate these diagrams.
DOI:10.48550/arxiv.2401.17496