Tensor invariants for classical groups revisited

TL;DR

The paper resolves a classical invariant-theory problem by providing uniform, combinatorial linear bases for G-invariants in $V^{\otimes p} \otimes V^{*\otimes q}$ across the five classical groups, using arc diagrams and standard Young tableaux. The core method threads standard monomial theory, RSK-type correspondences, and a translation to 1-regular arc diagrams, yielding explicit bases and multidegree control for polynomial and tensor invariants. It further delivers multiple equivalent enumerations of these bases, including stable-range formulas, and ties the constructions to well-known combinatorial objects (tableaux, permutations, involutions), with extensive tables and OEIS connections. The work unifies disparate strands of invariant theory and combinatorics, offering a uniform toolbox for dimensions, bases, and enumerations of tensor invariants.

Abstract

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. The main contribution of this paper is a combinatorial realization of linear bases via standard Young tableaux and arc diagrams, in a uniform manner for all five classical groups. As a secondary contribution, we survey the many equivalent ways -- some old, some new -- to enumerate the elements in these bases.

Figures (2)

• Figure 1: Examples of the arc diagrams specified in Theorem \ref{['thm:poly invariants GL O Sp']}, when $G = \mathop{\mathrm{GL}}\nolimits(V)$, $\operatorname{O}(V)$, or $\mathop{\mathrm{Sp}}\nolimits(V)$. Each arc diagram represents an ordinary monomial in $\mathcal{B}^G_{\mathbf{d}}$, namely the product of the contractions $f_{ij}$ corresponding to its arcs.
• Figure 2: Examples of the arc diagrams in Theorem \ref{['thm:poly invariants SL SO']}, where $G = \mathop{\mathrm{SL}}\nolimits(V)$ or $\mathop{\mathrm{SO}}\nolimits(V)$. Each arc diagram represents an ordinary monomial in $\mathcal{B}^G_{\mathbf{d}}$, where the arcs represent contractions $f_{ij}$ and the hyperedges $A^*$ represent the functions $\det^*(A)$.

Theorems & Definitions (37)

• Proposition 2.1
• Remark 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 2.5
• proof
• Proposition 3.1
• proof