Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hilton-Milner theorem for exterior algebras

Denys Bulavka, Francesca Gandini, Russ Woodroofe

TL;DR

The paper addresses generalizing EKR and Hilton-Milner theorems to exterior algebras by bounding the dimension of self-annihilating subspaces of $\bigwedge^k V$ under a nontriviality constraint. The authors introduce a limit-action framework and a slow-shifting process that preserves annihilation properties while degenerating subspaces to a combinatorially tractable form. They prove a sharp HMext bound $\dim L \le \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$ for nontrivially self-annihilating $L$, and prove a cross-annihilating analogue $\dim K+\dim L \le \binom{n}{k}-\binom{n-k}{k}+1$. The work is characteristic-independent and reframes extremal set theory in a Grassmannian moduli-space framework, expanding the toolkit for EKR-type phenomena in algebraic settings.

Abstract

Recent work of Scott and Wilmer and of Woodroofe extends the Erdős-Ko-Rado theorem from set systems to subspaces of k-forms in an exterior algebra. We prove an extension of the Hilton-Milner theorem to the exterior algebra setting, answering in a strong way a question asked by these authors.

A Hilton-Milner theorem for exterior algebras

TL;DR

The paper addresses generalizing EKR and Hilton-Milner theorems to exterior algebras by bounding the dimension of self-annihilating subspaces of under a nontriviality constraint. The authors introduce a limit-action framework and a slow-shifting process that preserves annihilation properties while degenerating subspaces to a combinatorially tractable form. They prove a sharp HMext bound for nontrivially self-annihilating , and prove a cross-annihilating analogue . The work is characteristic-independent and reframes extremal set theory in a Grassmannian moduli-space framework, expanding the toolkit for EKR-type phenomena in algebraic settings.

Abstract

Recent work of Scott and Wilmer and of Woodroofe extends the Erdős-Ko-Rado theorem from set systems to subspaces of k-forms in an exterior algebra. We prove an extension of the Hilton-Milner theorem to the exterior algebra setting, answering in a strong way a question asked by these authors.
Paper Structure (15 sections, 25 theorems, 17 equations)

This paper contains 15 sections, 25 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorial shifting and shifted families of sets
  4. Exterior algebras
  5. Limit actions on the Grassmannian
  6. Slower shifting
  7. The action of Nij
  8. Comparison with other operations
  9. The action of Mi
  10. Stabilization of slow shifting
  11. Cross-annihilating subspaces
  12. Nontrivially self-annihilating subspaces
  13. Main lemma
  14. Other lemmas
  15. Proof of Theorem \ref{['thm:HMext']}

Key Result

Theorem 1.1

Let $k\leq n/2$. If $\mathcal{F}\subseteq \binom{[n]}{k}$ is a pairwise-intersecting family of sets, then $\left|\mathcal{F}\right|\leq \binom{n-1}{k-1}$. Moreover, if $k<n/2$ and $\left|\mathcal{F}\right|$ achieves the upper bound, then $\mathcal{F}$ consists of all the $k$-subsets containing some

Theorems & Definitions (44)

  • Theorem 1.1: Erdős-Ko-Rado
  • Theorem 1.3: Scott and Wilmer
  • Theorem 1.4: Hilton-Milner
  • Corollary 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 34 more