scholarly journals Hilbert series of PI relatively free G -graded algebras are rational functions

2011 ◽  
Vol 44 (3) ◽  
pp. 520-532 ◽  
Author(s):  
Eli Aljadeff ◽  
Alexei Kanel-Belov
Keyword(s):  
Hilbert Series ◽  
Rational Functions ◽  
Graded Algebras

Hilbert Series of Positive Braids

2011 ◽  
Vol 18 (spec01) ◽  
pp. 1017-1028 ◽  
Author(s):  
Zaffar Iqbal
Keyword(s):  
Hilbert Series ◽  
Rational Functions

Deligne proved that the Hilbert series of all Artin monoids are rational functions. We give an algorithm to compute the Hilbert series of the braid monoids [Formula: see text]. We also show that the Hilbert series of the positive words in [Formula: see text] with a given prefix are rational functions.

scholarly journals Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 881 ◽  
Author(s):  
Oswaldo Lezama ◽  
Jaime Gomez
Keyword(s):  
Mirror Symmetry ◽  
Hilbert Series ◽  
Poincaré Series ◽  
Graded Rings ◽  
Yoneda Algebra ◽  
Graded Algebras ◽  
Poincare Series ◽  
Rings And Algebras ◽  
Lattice Of Ideals ◽  
Point Modules

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series.

scholarly journals A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants (condensed version)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
J. Haglund
Keyword(s):  
Hilbert Series ◽  
Quotient Ring ◽  
Rational Functions ◽  
Constant Term ◽  
Diagonal Action ◽  
Integer Coefficients ◽  
Polynomial Expression ◽  
International Audience ◽  
Special Case ◽  
Invent Math

International audience A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series. Un cas spécial de l'identité de Haiman [Invent. Math. \textbf149 (2002), pp. 371–407] pour le caractère de l'anneau quotient des coinvariants diagonaux sous l'action du groupe symétrique fournit une formule pour la série de Hilbert bigraduée comme somme de fonctions rationnelles en q,t. Dans cet article nous montrons comment une identité de sommation de Garsia et Zabrocki pour les coefficients de Pieri des polynômes de Macdonald peut être utilisée pour transformer la formule de Haiman pour la série de Hilbert en un polynôme explicite en q,t à coefficients entiers. Nous présentons également une formule équivalente pour la série de Hilbert comme terme constant d'une série de Laurent multivariée.

scholarly journals Hilbert series for leptonic flavor invariants in the minimal seesaw model

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Bingrong Yu ◽  
Shun Zhou
Keyword(s):  
Hilbert Series ◽  
Rational Functions ◽  
High Energy ◽  
Energy Scale ◽  
Effective Theory ◽  
Practical Applications ◽  
Phenomenological Studies ◽  
Sufficient And Necessary Conditions ◽  
High Energy Scale ◽  
First Time

Abstract In this paper, we examine the leptonic flavor invariants in the minimal see-saw model (MSM), in which only two right-handed neutrino singlets are added into the Standard Model in order to accommodate tiny neutrino masses and explain cosmological matter-antimatter asymmetry via leptogenesis mechanism. For the first time, we calculate the Hilbert series (HS) for the leptonic flavor invariants in the MSM. With the HS we demonstrate that there are totally 38 basic flavor invariants, among which 18 invariants are CP-odd and the others are CP-even. Moreover, we explicitly construct these basic invariants, and any other flavor invariants in the MSM can be decomposed into the polynomials of them. Interestingly, we find that any flavor invariants in the effective theory at the low-energy scale can be expressed as rational functions of those in the full MSM at the high-energy scale. Practical applications to the phenomenological studies of the MSM, such as the sufficient and necessary conditions for CP conservation and CP asymmetries in leptogenesis, are also briefly discussed.

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