Graphs of relations and Hilbert series

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.Comment: 14 page

Optimal 5-step nilpotent quadratic algebras

By the Golod--Shafarevich Theorem, an associative algebra R given by n generators and d<n^2/3 homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer n, there is an algebra R given by n generators and n^2/3 homogeneous quadratic relations such that R is 5-step nilpotent

Asymptotically optimal kk-step nilpotency of quadratic algebras and the Fibonacci numbers

It follows from the Golod--Shafarevich theorem that if R is an associative algebra given by n generators and $d<\frac{n^2}{4}\cos^{-2}(\frac{\pi}{k+1})$ quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal, and establish number of related results. For example, we show that for any k this estimate is attained for ifinitely many n.Comment: to appear in Combinatoric

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CW-complexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations $d \leq \frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the case for any number of quadratic relations. The particular point where the number of relations is equal to $\frac{n(n-1)}{2}$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We prove that over any infinite field, the Anick conjecture holds for $d \geq \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.Comment: 17 pages, to appear in the Proceedings of the Royal Society Edinburgh