Graphs of relations and Hilbert series

TL;DR

This paper explores combinatorial structures and Hilbert series of quadratic algebras, confirming conjectures for small cases and analyzing graph structures related to algebraic relations.

Contribution

It provides new examples confirming the Anick conjecture for small n and characterizes the graph structures of RIT algebras with maximal Hilbert series.

Findings

Confirmed Anick conjecture for n ≤ 7

Described graph structures for maximal Hilbert series

RIT algebra's maximal Hilbert series depends on graph isomorphism

Abstract

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by n generators and n(n-1)/2 relations for n less or equal then 7. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.