Face rings of cycles, associahedra, and standard Young tableaux

TL;DR

This paper explores the algebraic and combinatorial properties of the Stanley-Reisner ideal of an n-cycle, connecting its free resolution to associahedra and standard Young tableaux, and providing minimal resolutions and combinatorial bijections.

Contribution

It establishes a connection between the Betti numbers of J_n and standard Young tableaux, and constructs minimal cellular resolutions using discrete Morse theory.

Findings

Betti numbers of J_n correspond to standard Young tableaux of specific shapes

A free resolution supported on the associahedron A_n is constructed

A minimal cellular resolution at the first syzygy is obtained

Abstract

We show that J_n, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron A_n. This resolution is not minimal for n > 5; in this case the Betti numbers of J_n are strictly smaller than the f-vector of A_n. We show that in fact the Betti numbers of J_n are in bijection with the number of standard Young tableaux of shape (d+1, 2, 1^{n-d-3}). This complements the fact that the number of (d-1)-dimensional faces of A_n are given by the number of standard Young tableaux of (super)shape (d+1, d+1, 1^{n-d-3}); a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of J_n that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.