Dedekind sums and parsing of Hilbert series

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 $n\geq4$.

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Remarks on Hilbert series of graded modules over polynomial rings

manuscripta mathematica ◽

10.1007/s00229-010-0341-9 ◽

2010 ◽

Vol 132 (1-2) ◽

pp. 159-168 ◽

Cited By ~ 10

Author(s):

Jan Uliczka

Keyword(s):

Hilbert Series ◽

Polynomial Rings ◽

Graded Modules

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The Hilbert Series of a Linear Symplectic Circle Quotient

Experimental Mathematics ◽

10.1080/10586458.2013.863745 ◽

2014 ◽

Vol 23 (1) ◽

pp. 46-65 ◽

Cited By ~ 11

Author(s):

Hans-Christian Herbig ◽

Christopher Seaton

Keyword(s):

Hilbert Series

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

Journal of High Energy Physics ◽

10.1007/jhep01(2021)142 ◽

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, …)

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Identities on poly-Dedekind sums

Advances in Difference Equations ◽

10.1186/s13662-020-03024-x ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Hyunseok Lee ◽

Lee-Chae Jang

Keyword(s):

Modular Group ◽

Reciprocity Relation ◽

Dedekind Sums ◽

Transformation Behavior ◽

Dedekind Eta Function ◽

Bernoulli Function ◽

Bernoulli Functions

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

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

Journal of High Energy Physics ◽

10.1007/jhep02(2021)137 ◽

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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