IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Structure of Linear Hyperpath Nullvarieties

  • Nov. 5, 2021
  • 2:30 p.m.

Abstract

Call a hypergraph $k$-uniform if every edge is a $k$-subset of the vertices. Moreover, a $k$-uniform linear hyperpath with $n$ edges is a hypergraph $G$ with edges $e _ 1, ... , e _ n$ so that $e _ i \cap e _ j = \emptyset$ if $|i-j| \neq 1$, and $|e _ i \cap e _ {i+1}| = 1$ for all $1 \leq i\leq n-1$. Since the adjacency matrix of a graph is real symmetric, sets of eigenvectors form vector spaces. The extension of adjacency matrices to adjacency tensors in the hypergraph setting implies eigenvectors are no longer closed under arbitrary linear combinations. Instead, eigenvectors form varieties, and there is interest in finding decompositions of eigenvarieties into irreducible components. In this talk we find such a decomposition of the nullvariety for 3-uniform hyperpaths of arbitrary length. Moreover, we explore the repercussions of this decomposition on a conjecture of Hu and Ye.

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