The combinatorial structure of symmetric strongly shifted ideals

TL;DR

This paper explores the algebraic and combinatorial properties of symmetric strongly shifted ideals, revealing their structure, behavior under operations, and connections to combinatorial objects like polymatroids and polytopes.

Contribution

It introduces the concept of partition Borel generators and links symmetric strongly shifted ideals to discrete polymatroids and permutohedral varieties.

Findings

Properties under ideal operations and primary decomposition clarified.

Partition Borel generators connect to polymatroids and convex polytopes.

Structural insights into Rees algebra of these ideals obtained.

Abstract

Symmetric strongly shifted ideals are a class of monomial ideals which come equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper we focus on algebraic and combinatorial properties of symmetric strongly shifted ideals. On the algebraic side, we elucidate properties that pertain to behavior under ideal operations, primary decomposition, and the structure of their Rees algebra. On the combinatorial side, we develop a notion of partition Borel generators which leads to connections to discrete polymatroids, convex polytopes, and permutohedral toric varieties.