From total positivity to pure free resolutions
Commutative Algebra
2025-03-26 v2 Combinatorics
Representation 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.
Cite
@article{arxiv.2408.10408,
title = {From total positivity to pure free resolutions},
author = {Steven V Sam and Keller VandeBogert},
journal= {arXiv preprint arXiv:2408.10408},
year = {2025}
}
Comments
35 pages; v2: The main example, Theorem 5.3, only works for odd rank quadrics. The proof has been corrected and split off into a separate article