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

TL;DR

The paper introduces $t$-Young complexes as a topological framework for studying squarefree powers of $t$-path ideals of path graphs, showing each complex is either contractible or a wedge of spheres and characterizing vertex-decomposability. It establishes a precise correspondence between these complexes and the Alexander duals of Stanley–Reisner complexes of $I_{n,t}^{[k]}$, enabling explicit formulas for the projective dimension and Krull dimension of the squarefree powers. The results fuse combinatorial topology (via wedges, suspensions and Helly numbers) with commutative algebra (via squarefree powers and path ideals), delivering sharp dimension bounds and a Leray-number perspective. The work advances understanding of invariants for powers of monomial ideals tied to graph paths and offers computable methods for homological and combinatorial invariants in this setting.

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.

Figures (1)

• Figure 1: Directed graph $G_{9,2}^{[4]}$

Theorems & Definitions (52)

• Theorem 1: Theorem \ref{['Theorem: Projective dimension']}
• Theorem 2: Theorem \ref{['thm:homotopytype']}
• Theorem 3: Theorem \ref{['thm:t=1vd']}, Theorem \ref{['thm:cmdiagram']}
• Lemma 1.1
• Example 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof