scholarly journals A graded algebra with a non-rational Hilbert series

1980 ◽  
Vol 62 (1) ◽  
pp. 228-231 ◽  
Author(s):  
James B Shearer
Keyword(s):  
Hilbert Series ◽  
Graded Algebra

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals Hilbert series associated to symplectic quotients by SU2

2020 ◽  
Vol 30 (07) ◽  
pp. 1323-1357
Author(s):  
Hans-Christian Herbig ◽  
Daniel Herden ◽  
Christopher Seaton
Keyword(s):  
Explicit Expression ◽  
Hilbert Series ◽  
Laurent Expansion ◽  
Graded Algebra ◽  
Regular Functions ◽  
Nonzero Coefficient ◽  
Symplectic Quotients ◽  
The Way

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an [Formula: see text]-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at [Formula: see text]. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most [Formula: see text]. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary [Formula: see text]- or [Formula: see text]-module as well as its first three Laurent coefficients.

Extremal rays of Betti cones

2019 ◽  
Vol 19 (02) ◽  
pp. 2050027
Author(s):  
H. Ananthnarayan ◽  
Rajiv Kumar
Keyword(s):  
Hilbert Series ◽  
Graded Algebra ◽  
One Dimensional ◽  
The Given ◽  
Algebra Over A Field

We study the Betti cone of modules of a standard graded algebra over a field, and find a class of modules whose Betti diagrams span extremal rays of the Betti cone. We also identify a class of one-dimensional Cohen–Macaulay rings with certain Hilbert series, for which the Betti cone is spanned by these extremal rays. These results lead to an algorithm for the decomposition of Betti table of modules, into the extremal rays, over such rings, and also help to obtain bounds for the multiplicity of the given module.

scholarly journals The Hilbert Series of a Linear Symplectic Circle Quotient

2014 ◽  
Vol 23 (1) ◽  
pp. 46-65 ◽  
Author(s):  
Hans-Christian Herbig ◽  
Christopher Seaton
Keyword(s):  
Hilbert Series

scholarly journals 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lukáš Gráf ◽  
Brian Henning ◽  
Xiaochuan Lu ◽  
Tom Melia ◽  
Hitoshi Murayama
Keyword(s):  
Hilbert Series ◽  
Charge Conjugation ◽  
External Fields ◽  
Dynkin Diagrams ◽  
Non Linear ◽  
Very High ◽  
Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

scholarly journals A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

scholarly journals GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

2021 ◽  
pp. 1-54
Author(s):  
MANUEL L. REYES ◽  
DANIEL ROGALSKI
Keyword(s):  
Regularity Property ◽  
Graded Algebra ◽  
General Study ◽  
Locally Finite ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Dimension 2 ◽  
Separable Algebras ◽  
Gk Dimension

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

scholarly journals The Coulomb and Higgs branches of $$ \mathcal{N} $$ = 1 theories of Class $$ {\mathcal{S}}_k $$

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Thomas Bourton ◽  
Alessandro Pini ◽  
Elli Pomoni
Keyword(s):  
String Theory ◽  
Special Class ◽  
Hilbert Series ◽  
Superconformal Index

Abstract Even though for generic $$ \mathcal{N} $$ N = 1 theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $$ \mathcal{N} $$ N = 1 SCFTs, these of Class $$ {\mathcal{S}}_k $$ S k for which it is possible to define Coulomb and Higgs branches precisely as for the $$ \mathcal{N} $$ N = 2 theories of Class $$ \mathcal{S} $$ S from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional (1, 1) Little String theory can be deconstructed from a toroidal quiver in Class $$ {\mathcal{S}}_k $$ S k .

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