From total positivity to pure free resolutions

TL;DR

This paper draws parallels between totally positive sequences and Koszul algebras, constructing new analogues of Schur modules and pure free resolutions for quadric hypersurface rings using combinatorial and Lie-theoretic methods.

Contribution

It introduces a novel connection between total positivity and algebraic resolutions, extending Schur module constructions to quadric hypersurfaces.

Findings

Constructed new analogues of Schur modules for quadric hypersurfaces.

Developed pure free resolutions analogous to polynomial rings.

Established parallels between total positivity and Koszul algebra theory.

Abstract

Using the Jacobi-Trudi identity as a base, we establish parallels between the theory of totally positive integer sequences and Koszul algebras. We then focus on the case of quadric hypersurface rings and use this parallel to construct new analogues of Schur modules. We investigate some of their Lie-theoretic properties (and in more detail in a followup article) and use them to construct pure free resolutions for quadric hypersurface rings which are completely analogous to the construction given by Eisenbud, Fl{\o}ystad, and Weyman in the case of polynomial rings.