Schur apolarity

TL;DR

This paper introduces Schur apolarity theory, extending classical apolarity to irreducible representations of SL(V), enabling structured tensor decompositions and providing algorithms for low border rank tensor analysis.

Contribution

It develops the Schur apolarity framework, generalizing classical apolarity to new tensor structures and deriving a key lemma for tensor decomposition.

Findings

Schur apolarity lemma analogous to classical apolarity lemma

Descriptions of tensor decompositions involving flags of V

Algorithms for low border rank tensor analysis

Abstract

Inspired by the classic apolarity theory of symmetric tensors, the aim of this paper is to introduce the Schur apolarity theory, i.e. an apolarity for any irreducible representation of the special linear group $SL(V)$. This allows to describe decompositions of structured tensors whose elementary elements are tensors that represent flags of the vector space $V$. The main result is the Schur apolarity lemma which is the analogous of the apolarity lemma of symmetric apolarity theory. Eventually we study the rank tensors of low border rank related to specific varieties giving rise also to simple algorithms.