Highest weight vectors of tensors

TL;DR

The paper develops a unified framework for constructing and relating highest weight vectors in both the symmetric and exterior tensor algebras under $G=\mathrm{GL}(n)^{\times d}$, with dimensions governed by generalized Kronecker coefficients. It introduces spanning sets $\{\Delta_T\}$ (polynomials) and $\{\nabla_S\}$ (forms) indexed by $d$-dimensional $\mathbb{N}$-hypermatrices and establishes a complete description of linear relations and a duality isomorphism between dual highest weight spaces, governed by parity of $d$. A central duality formula expresses $\Delta_T$ in terms of inner products with $\nabla_S$, linking combinatorics of double cosets of Young subgroups to Kronecker coefficients and to invariant theory. The authors connect these algebraic constructions to invariant theory by showing power expansions of Cayley’s form $\omega$ (odd $d$) and Cayley’s first hyperdeterminant $\delta$ (even $d$), interpreting the coefficients via Latin hypercubes and Alon–Tarsi numbers and highlighting implications for positivity of Kronecker coefficients. Methodologically, the paper develops tools such as boundary operators, slice symmetries, and horizontal concatenation, enabling a semigroup perspective on Kronecker coefficients and deep connections between representation theory, combinatorics, and algebraic geometry with potential impact on geometric complexity theory.

Abstract

We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We present a unified explicit construction for corresponding spanning sets of highest weight vectors and completely describe the linear relations among them. We prove that these highest weight vectors satisfy a natural yet nontrivial duality. As applications, we also give conceptual interpretations to power expansions of Cayley's first hyperdeterminant and its dual exterior Cayley form.

Figures (1)

• Figure 1: A 3-d hypermatrix $T$ with marginals $(64,532,442)$ and its $3$-line notation. Empty cells regarded as equal to $0$.

Theorems & Definitions (71)

• Theorem 1.1: Characterization of the linear relations for $\Delta$ and $\nabla$, cf. Theorem \ref{['th:relations1']}
• Theorem 1.2: Duality isomorphism of highest weight spaces, cf. Theorem \ref{['th:isomorphism']}
• Theorem 1.3: Duality for $\Delta, \nabla$, cf. Theorem \ref{['th:duality2']}
• Corollary 1.4: Power expansions of the Cayley form and first hyperdeterminant
• Remark 2.1
• Definition 3.1: Signature function
• Remark 3.2
• Definition 3.3: HW $p$-vectors
• Remark 3.4
• Proposition 3.5