Skew-symmetric tensor decomposition

Abstract

We introduce the "skew apolarity lemma" and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in d V C with d ≤ 3 and dim V C ≤ 8. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.2010 Mathematics Subject Classification. Primary:15A75; secondary: 14M15, 16W22, 16Z05.

Schur apolarity

“…Before going along with the description, we have to recall the definition of skew-symmetric apolarity action given in [2]. Definition 6.…”

“…Hence the intersection I(p) ∩ ∧ • V is the usual ideal of the point p contained in ∧ • V * used in the skew-symmetric apolarity theory. See [2] for more details. and it represents the subspace spanned by v 1 and v 2 .…”

“…Coming back to (skew-)symmetric tensors, clearly one would like to compute the (skew-)symmetric rank of any (skew-)symmetric tensor. In both cases but chronologically first in the symmetric, see [11], and after for the other one, see [2], apolarity theories have been developed to compute the ranks of such tensors. Even though they may seem different, these two theories share a greatest common idea which is the evaluation.…”

Decomposability of Tensors

“…Notice that the above definition of skew-symmetric apolarity works well for computing the dimension of secant varieties to Grassmannians since it defines the apolar of a subspace that is exactly what is needed for Terracini's lemma, but if one would like to have an analogous definition of apolarity for skew symmetric tensors, then there are a few things that have to be done. Firstly, one needs to extend by linearity the above definition to all the elements of k V. Secondly, in order to get the equivalent notion of the apolar ideal in the skew symmetric setting, one has to define the skew-symmetric apolarity in any degree ≤ d. This is done in [116], where also the skew-symmetric version of the apolarity lemma is given. Moreover, in [116], one can find the classification of all the skew-symmetric-ranks of any skew-symmetric tensor in 3 C n for n ≤ n (the same classification can actually be found also in [117,118]), together with algorithms to get the skew-symmetric-rank and the skew-symmetric decompositions for any for those tensors (as far as we know, this is new).…”

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

“…Firstly, one needs to extend by linearity the above definition to all the elements of k V . Secondly, in order to get the equivalent notion of the apolar ideal in the skew symmetric setting, one has to define the skew-symmetric apolarity in any degree d. This is done in [116], where also the skew-symmetric version of the apolarity lemma is given. Moreover, in [116], one can find the classification of all the skew-symmetric-ranks of any skew-symmetric tensor in 3 C n for n n (the same classification can actually be found also in [118,117]), together with algorithms to get the skew-symmetric-rank and the skew-symmetric decompositions for any for those tensors (as far as we know, this is new).…”