scholarly journals Minimal Hilbert series for quadratic algebras and the Anick conjecture

2010 ◽  
Vol 59 (4) ◽  
pp. 301 ◽  
Author(s):  
N Iyudu ◽  
S Shkarin
Keyword(s):  
Hilbert Series ◽  
Quadratic Algebras

scholarly journals ALGEBRAS ASSOCIATED TO PSEUDO-ROOTS OF NONCOMMUTATIVE POLYNOMIALS ARE KOSZUL

2005 ◽  
Vol 15 (04) ◽  
pp. 643-648 ◽  
Author(s):  
DMITRI PIONTKOVSKI
Keyword(s):  
Hilbert Series ◽  
Necessary Condition ◽  
Quadratic Algebras ◽  
Noncommutative Polynomials

Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced by I. Gelfand, Retakh, and Wilson in connection with studying the decompositions of noncommutative polynomials. Later they (with S. Gelfand and Serconek) showed that the Hilbert series of these algebras and their quadratic duals satisfy the necessary condition for Koszulity. It is proved in this note that these algebras are Koszul.

Families of quadratic algebras and Hilbert series

2005 ◽  
pp. 119-131
Author(s):  
Alexander Polishchuk ◽  
Leonid Positselski
Keyword(s):  
Hilbert Series ◽  
Quadratic Algebras

scholarly journals Hilbert series of quadratic algebras associated with pseudo-roots of noncommutative polynomials

2002 ◽  
Vol 254 (2) ◽  
pp. 279-299 ◽  
Author(s):  
Israel Gelfand ◽  
Sergei Gelfand ◽  
Vladimir Retakh ◽  
Shirlei Serconek ◽  
Robert Lee Wilson
Keyword(s):  
Hilbert Series ◽  
Quadratic Algebras ◽  
Noncommutative Polynomials

scholarly journals The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

2011 ◽  
Vol 141 (3) ◽  
pp. 609-629 ◽  
Author(s):  
Natalia Iyudu ◽  
Stanislav Shkarin
Keyword(s):  
Commutative Algebra ◽  
Hilbert Series ◽  
Infinite Field ◽  
Asymptotic Results ◽  
Quadratic Algebras ◽  
Princeton University ◽  
Number Of Generators ◽  
Finite Dimensional ◽  
Generic Algebra ◽  
Generic Algebras

We study the question of whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (non-commutative) algebra, is attained. This question was considered by Anick in his 1983 paper, ‘Generic algebras and CW-complexes’ (Princeton University Press), where he proved that the estimate is attained for the number of quadratic relations d ≤ ¼n2 and d ≥ ½n2, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to ½n(n – 1) was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.We prove that over any infinite field, the Anick conjecture holds for d ≥ (n2 + n) and an arbitrary number of generators n, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

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 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 .

RATIONAL DEGENERATION OF ELLIPTIC QUADRATIC ALGEBRAS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 773-779 ◽  
Author(s):  
ALEKSANDR V. ODESSKI
Keyword(s):  
Quadratic Algebras
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