Young Flattenings in the Schur module basis

TL;DR

This paper clarifies the theory of Young flattenings in the Schur module basis, provides an efficient software implementation, and demonstrates significant performance improvements over previous methods, especially for powers of linear forms.

Contribution

It presents a complete theory and software implementation of Young flattenings in the Schur module basis, improving computational efficiency and correctness.

Findings

Implementation outperforms previous software by several orders of magnitude.

Efficient computation of Young flattenings for powers of linear forms.

Supports proving border Waring rank lower bounds.

Abstract

There are several isomorphic constructions for the irreducible polynomial representations of the general linear group in characteristic zero. The two most well-known versions are called Schur modules and Weyl modules. Steven Sam used a Weyl module implementation in 2009 for his Macaulay2 package PieriMaps. This implementation can be used to compute so-called Young flattenings of polynomials. Over the Schur module basis Oeding and Farnsworth describe a simple combinatorial procedure that is supposed to give the Young flattening, but their construction is not equivariant. In this paper we clarify this issue, present the full details of the theory of Young flattenings in the Schur module basis, and give a software implementation in this basis. Using Reuven Hodges' recently discovered Young tableau straightening algorithm in the Schur module basis as a subroutine, our implementation outperforms Sam's PieriMaps implementation by several orders of magnitude on many examples, in particular for powers of linear forms, which is the case of highest interest for proving border Waring rank lower bounds.