arXiv math/05021492005-02-08math.RAmath.AC

Sets of Hilbert series and their applications

Dmitri Piontkovski

备注：: 10 pages

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Extremal Hilbert 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.

2017-11-03math.ACmath.RA

Universal denominators of Hilbert series

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…

2003-05-07math.ACmath.AGmath.RA

Monomial right ideals and the Hilbert series of noncommutative modules

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…

2016-05-27math.ACmath.RA

Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$

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…

2018-09-20math.ACmath.RAmath.SG

Equivariant Hilbert Series in non-Noetherian Polynomial Rings

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…

2015-10-19math.ACmath.RA

Rationality of Hilbert series in noncommutative invariant theory

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…

2017-08-21math.ACmath.RAmath.RT

更多相关论文收起更多论文

Hilbert series of modules of GK-dimension two over elliptic algebras

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.

2006-06-06math.RA

Noncommutative algebras, context-free grammars and algebraic Hilbert series

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.

2019-06-03math.RA

Rationality of the Hilbert series of Hopf-invariants of free algebras

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.

2011-05-26math.RA

The Golod Shafarevich counter-example without Hilbert series

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.

2009-01-11math.RA

Multigraded Hilbert Series of noncommutative modules

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 …

2018-09-04math.RA

Context-free languages and associative algebras with algebraic Hilbert series

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.

2020-01-24math.RA

Gröbner-Shirshov Bases and Hilbert Series of Free Dendriform Algebras

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.

2010-09-01math.RA

On the graded algebras associated with Hecke symmetries, II. The Hilbert series

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.

2019-03-21math.RA

秩 $1$ 李群的某些不变式、协变式和辛商的多级 Hilbert 级数

Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups

本文计算圆群和正交群 O₂ 表示的不变式与协变式的单变量及多级 Hilbert 级数，涵盖最大分级和双分级，并给出算法及示例，推广至半直积情形。

2022-06-08math.ACmath.RAmath.SG

Hilbert series of algebras associated to direct graphs and order homology

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.

2011-11-14math.GTmath.RA

Graded algebras with 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 $…

2020-01-04math.NTmath.RA

预投射代数的 Hilbert 级数

The Hilbert series of the preprojective algebras

本文利用带 Adams 分次的 dg 代数 Hilbert 级数公式，证明任意特征下预投射代数的 Hilbert 级数，并进一步计算其特殊中心扩张——箭图 Heisenberg 代数的 Hilbert 级数。

2024-05-09math.RAmath.RT

Color Lie超代数的Hilbert级数表示

Representations of Color Lie Superalgebras by Hilbert Series

本文研究Color Lie超代数的Hilbert级数表示，提出了一个类似于自由Color Lie超代数和特定Color Lie p-超代数的Witt公式的维数公式。

2023-06-30math.RA

The Hilbert series of $\operatorname{SL}_2$-invariants

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 $t=1$.

2018-06-19math.ACmath.AGmath.RA

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 component…

2008-01-19math.COmath.RA

非结合系数环上的非结合斜幂级数环与非结合斜洛朗级数环的希尔伯特基定理

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）

本文为非结合斜幂级数环和斜洛朗级数环建立希尔伯特基定理，通过引入拟结合性质给出Noetherian条件，并指出一般非结合情形下定理不成立。

2026-09-28math.RA

Algebraic sets, ideals of points and the Hilbert's Nullstellensatz theorem for skew PBW extensions

In this paper we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended, in particular, a Zariski topology will be constructed. Assuming additionally that the extension is quasi-commutative with polynomial center and the ring of coefficients is an algebraically closed field, we will prove an adapted version of the Hilbert's Nullstellensatz theorem that covers the classical one. The Gröbner bases of skew PBW extensions will be use…

2021-06-24math.AGmath.RA