Generic initial ideals and fibre products

TL;DR

This paper investigates the properties of generic initial ideals under fibre products, providing explicit descriptions and applications to algebraic shifted complexes of simplicial complexes, advancing understanding in combinatorial algebraic geometry.

Contribution

It determines the generic initial ideal of fibre products in reverse lex order and applies this to compute symmetric algebraic shifted complexes, confirming a conjecture by Kalai.

Findings

Explicit formula for the generic initial ideal of fibre products

Computed symmetric algebraic shifted complex of disjoint simplicial complexes

Extended results to symmetric analogues of known exterior algebraic theorems

Abstract

We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of Nevo who determined the exterior algebraic shifted complex of two disjoint simplicial complexes as predicted by Kalai.