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.
We characterize the Hilbert functions and minimal resolutions of (critical) Cohen-Macaulay graded right modules of Gelfand-Kirillov dimension two over generic quadratic and cubic three dimensional Artin-Schelter regular algebras.
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.
It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.
Let $F$ be an arbitrary field. The Golod-Shafarevich example of a finitely generated nil $F$-algebra which is infinite dimensional -- is revisited. Here we offer a rather elementary treatment of that example, in which induction replaces Hilbert series techniques. This note also contains a detailed exposition of the construction of that example.
In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for …
In this paper, homological methods together with the theory of formal languages of theoretical computer science are proved to be effective tools to determine the growth and the Hilbert series of an associative algebra. Namely, we construct a class of finitely presented associative algebras related to a family of context-free languages. This allows us to connect the Hilbert series of these algebras with the generating functions of such languages. In particular, we obtain a class of finitely presented graded algebras with non-rational algebraic Hilbert series.
In this paper, we give a Gröbner-Shirshov basis of the free dendriform algebra as a quotient algebra of an $L$-algebra. As applications, we obtain a normal form of the free dendriform algebra. Moreover, Hilbert series and Gelfand-Kirillov dimension of finitely generated free dendriform algebras are obtained.
This paper continues the work which attempts to understand the general properties of the graded algebras associated with Hecke symmetries without a restriction on the parameter q of the Hecke relation imposed in earlier results.
There are several remarks on Hilbert series of finitely presented (f. p.) associative algebras over a field and their modules. First, given an integer $D$, the set of Hilbert series of right-sided ideals with generators and relations of degrees at most $D$ in a f. p. algebra is finite. Second, every f. p. module of linear growth over noetherian or coherent algebra has periodic Hilbert function. Third, every ideal in a Koszul family of ideals (defined in math.AG/0412441) in a f. p. algebra has rational Poincare series.
Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.
We give a homological interpretation of the coefficients of the Hilbert series for an algebra associated with a directed graph and its dual algebra. This allows us to obtain necessary conditions for Koszulity of such algebras in terms of homological properties of the graphs. We use our results to construct algebras with a prescribed Hilbert series.
For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$ is rational/algebraic/transcendental, then the same is $H(R,t)$. If the growth of the coefficients of $a(t)$ is polynomial, in the same way we construct a graded algebra $R$ preserving the polynomial growth of the coefficients of its Hilbert series $H(R,t)$. Applying a classical result of Fatou from 1906 we obtain that if a finitely generated graded algebra $…
The denominator of the Hilbert series of a finitely generated R-module M does not always divide the denominator of the Hilbert series of R. For this reason, we define the universal denominator. The universal denominator of a module M is the least common multiple of the denominators of the Hilbert series of all submodules of M. The universal denominator behaves nicely with respect to short exact sequences and tensor products. It also has interesting geometric interpretations. Formulas are given for the universal denominator for rings of invariants. Dixmier gave a conjectural formula for the den…
We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an $\operatorname{SU}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t = 1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary $\operatorname{Sl}_2$- or $\operatorname{SU}_2$-module as well its firs…
This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with exponentially bounded nonnegative integer coefficients. Then we summarize some related facts on algebras with polynomial identity. Further we discuss the problem how to find series $a(t)$ which are rational/algebraic/transcendental over ${\mathbb Q}(t)$. Applying a classical result of Fatou we conclude that if a finitely generated graded algebra has a finite G…
It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tenso…
The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
Mindanao State University-Iligan Institute of Technology（棉兰老州立大学伊利甘理工学院）；Blekinge Institute of Technology（布莱金厄理工学院）
Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More genera…
In this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module $N$ over the free associative algebra $K\langle x_1,\ldots,x_n \rangle$. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing $N$ are annihilated by monomial right ideals whose monomials define regular formal languages. The method is based on the iterative application of the colon right ideal operation to monomial ideals which are given by an eventual infinite basis. By using automata theory, we p…
We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of $Inc(\mathbb{N})$ and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gröbner basis consisting o…
Let $V$ be a finite dimensional representations of the group $\operatorname{SL}_2$ of $2\times 2$ matrices with complex coefficients and determinant one. Let $R=\mathbb{C}[V]^{\operatorname{SL}_2}$ be the algebra of $\operatorname{SL}_2$-invariant polynomials on $V$. We present a calculation of the Hilbert series $\operatorname{Hilb}_R(t)=\sum_{n\ge 0}\dim (R_n)\: t^n$ as well as formulas for the first four coefficients of the Laurent expansion of $\operatorname{Hilb}_R(t)$ at $t=1$.
We begin with proving a formula relating the Hilbert series of a graded algebra $A$ and the Poincaré series for $A$ in two variables. This gives the Fröberg formula in the case where the bigraded $Tor^A(k,k)$ is concentrated on the diagonal, which we take as definition of $A$ being "Koszul". We look at a resolution in the commutative case obtained from the Koszul complex in the "trivially Golod" case. The algebra structure of $Ext_A(k,k)$ is introduced in different ways. Its subalgebra generated by the one-dimensional elements is by definition the "Koszul" dual of $A$. We define the"generalize…
The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra of Q over any field k is Koszul, and has Hilbert series 1/(1-Ct+t^2), where C is the adjacency matrix of the double of Q. (This result, in somewhat less general formulations, was previously obtained by Martinez-Villa and Malkin-Ostrik-Vybornov). We also prove a similar result for the partial preprojective algebra of any connected quiver Q, associated to a subset J of the set I of vertices of Q (by definition, this is the quotient of the path algebra of the double by the preprojective al…
