Tensor rank, invariants, inequalities, and applications. (English) Zbl 1295.15016

Allman, Elizabeth S.; Jarvis, Peter D.; Rhodes, John A.; Sumner, Jeremy G.

SIAM J. Matrix Anal. Appl. 34, No. 3, 1014-1045 (2013).

The polynomial \(\Delta\), widely known as tangl (Cayley’s hyperdeterminant) is of particular interest in understanding the case \(2\times2\times2\) of complex tensors of rank 2. The role of \(\Delta\) can be partially understood as a consequence of it being an invariant of the group \(\mathrm{GL}(2, C) \times \mathrm{GL}(2, C) \times \mathrm{GL}(2, C)\), which acts on \(2\times2\times2\) complex tensors in the three indices and preserves their tensor rank. In this work, the authors focus on \(n \times n \times n\) tensors of tensor rank \(n\), over \(\mathbb C\) and over \(\mathbb R\), with the goal of generalizing our detailed understanding of the \(2\times2\times2\) tensors to this particular case. Moreover, they claim that the cases of \(n_1 \times n_2 \times n_3\) tensors of rank \(n\) with \(n \leq n_i\), should follow by applying maps of \(\mathbb C^{n_i} \to\mathbb C^n\).
The main aim of this paper is to generalize \(\Delta\) for arbitrary \(n\). They do it by constructing a polynomial functions whose nonvanishing singles out a dense orbit of the tensors of rank n from their closure. Then the authors study these function in the framework of representation theory for the group \(\mathrm{GL}(nC) \times \mathrm{GL}(nC) \times \mathrm{GL}(nC)\). The research is motivated in part by questions regarding stochastic models with hidden variables.

Reviewer: Rabeya Basu (Pune)

MSC:

15A72	Vector and tensor algebra, theory of invariants
14P10	Semialgebraic sets and related spaces

Keywords:

tensor rank; hyperdeterminant; border rank; latent class model; phylogenetics