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Toward Jordan Decompositions of Tensors

Frederic Holweck, Luke Oeding

TL;DR

The work develops a Vinberg-inspired framework to extend Jordan decomposition to tensors by embedding tensor spaces into graded algebras $\mathfrak{a}=\mathfrak{g}\oplus W$ and studying adjoint operators $\operatorname{ad}_T$. It defines the notion of a Jordan decomposition consistent with the $G$-action (GJD) and constructs invariant adjoint data—such as trace powers $f_k(T)=\mathrm{tr}((\operatorname{ad}_T)^k)$ and adjoint rank/root profiles—to distinguish tensor orbits. The authors develop several algebraic variants (e.g., $\mathbb{Z}_2$- and $\mathbb{Z}_3$-graded structures, and an extension of the exterior algebra) and establish rank-subadditivity/semicontinuity results that link adjoint ranks to tensor ranks and orbit dimensions. They demonstrate the utility of adjoint invariants in quantum information by applying the method to 3-, 4-, and 5-qubit systems, showing effective orbit separation and connections to known classifications, hyperdeterminants, and covariants. Overall, the framework provides a computationally viable, algebraically rich approach for tensor orbit classification with potential geometric and quantum-information applications, including quasi-semisimple considerations and extensions to geometric structures.

Abstract

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

Toward Jordan Decompositions of Tensors

TL;DR

The work develops a Vinberg-inspired framework to extend Jordan decomposition to tensors by embedding tensor spaces into graded algebras and studying adjoint operators . It defines the notion of a Jordan decomposition consistent with the -action (GJD) and constructs invariant adjoint data—such as trace powers and adjoint rank/root profiles—to distinguish tensor orbits. The authors develop several algebraic variants (e.g., - and -graded structures, and an extension of the exterior algebra) and establish rank-subadditivity/semicontinuity results that link adjoint ranks to tensor ranks and orbit dimensions. They demonstrate the utility of adjoint invariants in quantum information by applying the method to 3-, 4-, and 5-qubit systems, showing effective orbit separation and connections to known classifications, hyperdeterminants, and covariants. Overall, the framework provides a computationally viable, algebraically rich approach for tensor orbit classification with potential geometric and quantum-information applications, including quasi-semisimple considerations and extensions to geometric structures.

Abstract

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.
Paper Structure (20 sections, 14 theorems, 84 equations, 12 tables)

This paper contains 20 sections, 14 theorems, 84 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Linear algebra review
  3. Historical Remarks on Jordan Decomposition for Tensors
  4. Constructing an algebra extending tensor space
  5. A graded algebra extending a $G$-module
  6. Invariants from adjoint operators
  7. Algebra structures on $\operatorname{End}(V)_0$
  8. A $\mathbb{Z}_2$ graded algebra from a $G$-module.
  9. A $\mathbb{Z}_3$ graded algebra from a $\mathfrak{g}$-module
  10. Extending the exterior algebra
  11. Rank subadditivity and semi-continutity
  12. Dimensions of conical orbits from adjoint rank profiles
  13. Some approachable examples
  14. Connection to geometric constructions
  15. Applications to quantum information
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathbb{F}$ be an algebraically closed field. Every $A\in \operatorname{Mat}_{n\times n }(\mathbb{F})$ is similar to its Jordan canonical form, which is a decomposition: where the $k\times k$ Jordan blocks are $J_k(\lambda) = \left(\right)$. The algebraic multiplicity of the eigenvalue $\lambda$ is the sum $\sum_{\lambda_j = \lambda} k_j$, and the geometric multiplicity of $\lambda$ is the n

Theorems & Definitions (37)

  • Theorem 1.1: Jordan Canonical Form
  • Remark 1.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 27 more