Geometric Aspects of Iterated Matrix Multiplication

TL;DR

This paper explores the geometric properties of the Iterated Matrix Multiplication polynomial, including symmetry and dual varieties, using representation theory, to understand potential implications for computational complexity.

Contribution

It introduces a geometric perspective on the polynomial, analyzing symmetry groups and dual varieties with representation theory, which is a novel approach in this context.

Findings

Identified the symmetry group of the polynomial.

Computed the dual variety and Jacobian loci.

Applied representation theory of quivers to these geometric properties.

Abstract

This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial, the dual variety and the Jacobian loci of the hypersurface, that are computed with the aid of representation theory of quivers.