Some algebraic properties of ASM varieties

TL;DR

This paper explores algebraic properties of ASM varieties, which are intersections of matrix Schubert varieties linked to alternating sign matrices, focusing on Cohen-Macaulayness, codimension, and pattern avoidance.

Contribution

It investigates the Cohen-Macaulayness, codimension, and pattern avoidance of ASM varieties, providing new insights into their algebraic structure and relationship with ASM pairs.

Findings

Analysis of Cohen-Macaulayness of ASM varieties

Results on codimension calculations for ASM varieties

Insights into ASM pattern avoidance from an algebraic perspective

Abstract

Fulton's matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen-Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.