Some Non-Unimodal Level Algebras

TL;DR

This paper proves the existence of non-unimodal level algebras by extending previous results and introducing new combinatorial methods involving L-Matrices, advancing understanding of Hilbert functions in algebra.

Contribution

It extends Iarrobino's 1984 results and introduces L-Matrices to construct and analyze non-unimodal level algebras with combinatorial techniques.

Findings

Proved non-unimodality for certain level algebras.

Introduced L-Matrices with useful properties.

Established conditions for nonsingularity of specific L-Matrices.

Abstract

In 2005, building on his own recent work and that of F. Zanello, A. Iarrobino discovered some constructions that, he conjectured, would yield level algebras with non-unimodal Hilbert functions. This thesis provides proofs of non-unimodality for Iarrobino's level algebras, as well as for other level algebras that the author has constructed along similar lines. The key technical contribution is to extend some results published by Iarrobino in 1984. Iarrobino's results provide insight into some naturally arising vector subspaces of the vector space R_d of forms of fixed degree in a polynomial ring in several variables. In this thesis, the problem is approached by combinatorial methods and results similar to Iarrobino's are proved for a different class of vector subspaces of R_d. The combinatorial methods involve the definition of a new class of matrices called L-Matrices, which have useful properties that are inherited by their submatrices. A particular class of square L-Matrices, associated with some specialized partially ordered sets having interesting combinatorial properties, is identified. For this class of L-Matrices, necessary and sufficient conditions are given that they be nonsingular. Several larger questions are discussed whose answers are incrementally improved by the knowledge that the new non-unimodal level algebras exist.