$t$-Young complexes and squarefree powers of $t$-path ideals

TL;DR

This paper introduces $t$-Young complexes derived from Young diagrams, characterizes their topological properties, and connects them to algebraic structures like $t$-path ideals, providing formulas for algebraic invariants.

Contribution

It defines $t$-Young complexes, characterizes their topological and combinatorial properties, and links them to algebraic objects such as $t$-path ideals and their squarefree powers.

Findings

$t$-Young complexes are either contractible or homotopy equivalent to wedges of spheres.

Complete vertex-decomposability characterization of $t$-Young complexes.

Explicit formulas for homotopy types, projective dimension, and Krull dimension of related ideals.

Abstract

We introduce a new class of simplicial complexes, called \emph{$t$-Young complexes}, arising from a Young diagram and a positive integer~$t$. We show that every $t$-Young complex is either contractible or homotopy equivalent to a wedge of spheres. A complete characterization of their vertex-decomposability is provided, and in several cases, we establish explicit formulas for their homotopy types. Interestingly, $t$-Young complexes naturally appear as the Alexander dual complexes of squarefree powers of $t$-path ideals of path graphs, as well as of certain ideals generated by subsets of their minimal generators. As an application, we derive formulas for the projective dimension and Krull dimension of these squarefree powers.