Non-degenerate 2 × k × (k + 1) hypermatrices

ABSTRACT

We show that if $\mathbb{F}$ is a topological field, then there is a transitive, free and continuous action of a natural quotient of $G{L}_k(\mathbb{F})\times G{L}_{k+1}(\mathbb{F})$ on the set $M_k(\mathbb{F})$ of $2\times k\times (k+1)$ hypermatrices over $\mathbb{F}$ with non-zero hyperdeterminant. We use this action to study the homotopy type of $M_k(\mathbb{C})$ and $M_k(\mathbb{R})$ and count elements of $M_k({\mathbb{F}}_q)$ (generalizing an unpublished result of Lewis and Sam).

KEYWORDS:

• Hyperdeterminants
• hypermatrices
• boundary format

AMS SUBJECT CLASSIFICATIONS:

• 15A69
• 15A15

Acknowledgements

This research was part of the 2015 summer REU program at the University of Minnesota, Twin Cities. I am very grateful to Joel Lewis and Elise DelMas for their mentorship and valuable advice and comments. I would also like to thank the anonymous referee for many helpful comments and references.

Notes

No potential conflict of interest was reported by the authors.

1 By ‘relatively GL-invariant’, we mean that for any $1\le i\le r$, there is an integer $l_i$ such that for any element $g\in G{L}_{k_i+1}(\mathbb{F})$ and $(k_1+1)\times \cdots \times (k_r+1)$ hypermatrix M, we have $\text{Det}(g·M)=det{(g)}^{l_i}\text{Det}(M)$.

2 Recall that we are including multiples of $M_0$ in the matrix pencil!.

Additional information

Funding

This work was supported by the NSF RTG [grant number NSF/DMS-1148634].