From total positivity to pure free resolutions

TL;DR

The paper develops a bridge between total positivity and Koszul algebras via Jacobi–Trudi structures, proving that equivariant total positivity is equivalent to the existence of a Jacobi–Trudi structure and that PF sequences arise as equivariant Hilbert functions of Koszul algebras. It then constructs quadric Schur functors for quadric hypersurface rings (notably odd-rank) using framed orthogonal spaces, and defines functors $\Phi^X$ that translate GL-representation data into $A$-modules with controlled behavior. Leveraging these tools, the authors extend the Eisenbud–Fløystad–Weyman framework to odd quadric hypersurfaces to produce pure free resolutions (finite or infinite tails) and relate these constructions to the rational normal curve, demonstrating a cohesive interaction among combinatorics, representation theory, and commutative algebra. This work opens pathways to further understandings of equivariant positivity, homotopy Lie algebra connections, and generalized pure resolutions across broader classes of algebras.

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, Fløystad, and Weyman in the case of polynomial rings.

Theorems & Definitions (90)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Remark 2.2
• Theorem 2.3: Zelevinsky
• Remark 2.4
• Proposition 3.1
• proof