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

2007 ◽  
Vol 311 (2) ◽  
pp. 635-664 ◽  
Author(s):  
Koen De Naeghel
Keyword(s):  
Hilbert Series ◽  
Gk Dimension ◽  
Elliptic Algebras

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 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 Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

2021 ◽  
Vol 9 ◽  
Author(s):  
Alex Chirvasitu ◽  
Ryo Kanda ◽  
S. Paul Smith
Keyword(s):  
Elliptic Curve ◽  
Symmetric Power ◽  
Canonical Homomorphism ◽  
Coordinate Ring ◽  
Characteristic Variety ◽  
Closed Subvariety ◽  
Coordinate Rings ◽  
Cubic Hypersurfaces ◽  
Elliptic Algebras ◽  
Twisted Homogeneous Coordinate Ring

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

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 .

scholarly journals Hilbert Series and Mixed Branches of T [SU(N )] theories

2017 ◽  
Vol 2017 (2) ◽  
Author(s):  
Federico Carta ◽  
Hirotaka Hayashi
Keyword(s):  
Hilbert Series
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