Hilbert–Poincaré series
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Hilbert series for constructing Lagrangians: Expanding the phenomenologist’s toolbox Open
This note presents the Hilbert series technique to a wider audience in the\ncontext of constructing group-invariant Lagrangians. This technique provides a\nfast way to calculate the number of operators of a specified mass dimension for\na …
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Hilbert series and operator basis for NRQED and NRQCD/HQET Open
We use a Hilbert series to construct an operator basis in the $1/m$ expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET). We present a list of effective operators with m…
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Baryon non-invariant couplings in Higgs effective field theory Open
\n The basis of leading operators which are not invariant under baryon number is constructed within the Higgs effective field theory. This list contains 12 dimension six operators, which preserve the combination $$B-L$$, to be compared to …
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Higgs branch, Coulomb branch, and Hall-Littlewood index Open
The Higgs branch of 4D N = 2 superconformal field theories can be analyzed via the Hilbert series of the Higgs branch or, in special cases, by computing the Hall-Littlewood index. For any class S theory corresponding to a genus-zero Rieman…
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Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture Open
The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant polynomia…
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Hilbert series, machine learning, and applications to physics Open
We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ∼1 mean absolute error, whilst classifiers predict dimension an…
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Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras Open
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 al…
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On the Hilbert series of ideals generated by generic forms Open
There is a longstanding conjecture by Fr\\"oberg about the Hilbert series of\nthe ring $R/I$, where $R$ is a polynomial ring, and $I$ an ideal generated by\ngeneric forms. We prove this conjecture true in the case when $I$ is generated\nby…
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Hilbert series for ALP EFTs Open
A bstract Axions and axion-like particles (ALPs) are ubiquitous in popular attempts to solve supercalifragilisticexpialidocious puzzles of Nature. A widespread and vivid experimental programme spanning a vast range of mass scales and decad…
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Algebraic properties of the monopole formula Open
The monopole formula provides the Hilbert series of the Coulomb branch for a\n3-dimensional N=4 gauge theory. Employing the concept of a fan defined by the\nmatter content, and summing over the corresponding collection of monoids,\nallows …
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Frobenius vectors, Hilbert series and gluings of affine semigroups Open
Let $S_1$ and $S_2$ be two affine semigroups, and let $S$ be the gluing\nof $S_1$ and $S_2$. Several invariants of $S$ are related to those of\n$S_1$ and $S_2$; we review some of the most important properties preserved under gluings.\nThe …
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23, 381, 6242, 103268, 1743183, … : Hilbert series for CP-violating operators in SMEFT Open
A bstract We introduce a systematic method to classify the Standard Model Effective Field Theory (SMEFT) operators based on their CP properties with the Hilbert series techniques. Our method makes it possible to enumerate operators violati…
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Nichols algebras with many cubic relations Open
Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are finite-dimensio…
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Hilbert series and moduli spaces of k U(N ) vortices Open
We study the moduli spaces of k U(N) vortices which are realized by the Higgs branch of a U(k) supersymmetric gauge theory. The theory has 4 supercharges and lives on k D1-branes in a N D3- and NS5-brane background. We realize the vortex m…
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Tight closure of powers of ideals and tight hilbert polynomials Open
Let ( R , ) be an analytically unramified local ring of positive prime characteristic p . For an ideal I , let I * denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \ri…
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Hilbert series for covariants and their applications to minimal flavor violation Open
A bstract We elaborate how to apply the Hilbert series method to enumerating group covariants, which transform under any given representation, including but going beyond group invariants. Mathematically, group covariants form a module over…
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Invariants of GL<sub><i>n</i></sub>(𝔽<sub><i>q</i></sub>) in polynomials modulo Frobenius powers Open
Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups: one for invariants mod Frobenius powers of the irrelevant ideal and one for cofixed spaces of polynomials.
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A WEAK HILBERT SPACE THAT IS A TWISTED HILBERT SPACE Open
We construct a weak Hilbert space that is a twisted Hilbert space.
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Constructing operator basis in supersymmetry: a Hilbert series approach Open
A bstract In this paper we introduce a Hilbert series approach to build the operator basis for a N = 1 supersymmetry theory with chiral superfields. We give explicitly the form of the corrections that remove redundancies due to the equatio…
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Hilbert series of graded Milnor algebras and roots of Bernstein-Sato polynomials Open
We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at…
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Hilbert series of residual intersections Open
We give explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper, we proved that such formulas should exist. We give applications to the numb…
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Smooth Hilbert schemes: Their classification and geometry Open
Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is …
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On compositions with 𝑥²/(1-𝑥) Open
In the past, empirical evidence has been presented that Hilbert series of symplectic quotients of unitary representations obey a certain universal system of infinitely many constraints. Formal series with this property have been called sym…
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Set superpartitions and superspace duality modules Open
The superspace ring $\Omega _n$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $\Omega _n$ , the authors previously defined a family of doubly graded quotients ${\m…
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Type II unprojections, Fano threefolds and codimension four constructions Open
The classification of singular Fano 3-folds remains an open problem in algebraic geometry. The purpose of this thesis is to prove the existence of new families of singular Fano 3-folds, specifically those whose anticanonical embedding is i…
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Classification of quadratic and cubic PBW algebras on three generators Open
We give a complete classification of quadratic algebras A, with Hilbert series $H_A=(1-t)^{-3}$, which is the Hilbert series of commutative polynomials on 3 variables. Koszul algebras as well as algebras with quadratic Gröbner basis among …
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Lefschetz theory for exterior algebras and fermionic diagonal coinvariants Open
Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V…
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Non-commutative algebraic geometry of semi-graded rings Open
In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geo…
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Hilbert Series and Lefschetz Properties of Dimension One Almost Complete Intersections Open
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension …
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Noncommutative algebras, context-free grammars and algebraic Hilbert series Open
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this pu…