Cayley's hyperdeterminant: A combinatorial approach via representation theory
In 1845, Cayley introduced the concept of hyperdeterminant of a multidimensional matrix. In the case of \(2\times2\times2\) matrices, the hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of the matrix, called Cayley hyperdeterminant.NEWLINENEWLINEIt is well-known that the Cayley hyperdeterminant is invariant under the natural action of \(\mathrm{SL}_2(\mathbb{C})\times \mathrm{SL}_2(\mathbb{C})\times \mathrm{SL}_2(\mathbb{C})\) on \(\mathbb{C}^2\otimes \mathbb{C}^2\otimes \mathbb{C}^2\) and moreover, it generates all the invariant polynomials. In this paper, the authors gives a new proof of this fact, based on a combinatorical approach.
scientific article; zbMATH DE number 6034517
scientific article