Hilbert Series Research Articles

Study of weak-basis invariants in the universal seesaw model using Hilbert series

Universal Seesaw Model is a model which explains the mass hierarchy of the quark sector. This model introduces vector-like quarks. The top quark mass is generated in the electroweak scale and the other quark mass is generated using a seesaw-like mechanism. The invariant theory helps construct a weak-basis invariant. We study the weak-basis invariant (WBI) using Hilbert Series (HS) and apply it to the Universal Seesaw Model, particularly the one-generation case of the quark sector.

Quasisymmetric harmonics of the exterior algebra

AbstractWe study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1)The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$ .(2)There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$ .(3)There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.

Algebraic h-vectors of simplicial complexes through local cohomology, part 1

Given an infinite field k and a simplicial complex Δ, a common theme in studying the f- and h-vectors of Δ has been the consideration of the Hilbert series of the Stanley–Reisner ring k[Δ] modulo a generic linear system of parameters Θ. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of hd−1a(Δ), the dimension over k in degree d−1 of k[Δ]/(Θ), for any complex Δ of dimension d−1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.

Problems of the theory of invariants and Lie algebras applied in the qualitative theory of differential systems

In this work there were formulated 18 problems from the theory of invariant processes, Lie algebras, commutative graded algebras, generating functions and Hilbert series, orbit theory and Lyapunov stability theory that are important to be solved. There was substantiated the necessity of using the solutions of these problems in the qualitative theory of differential systems.

On (σ,τ)-derivations of lie superalgebras

This paper is primarily devoted to studying (?, ?)-derivations of finite-dimensional Lie superalgebras over an algebraically closed field F. We research some properties of (?, ?)-derivations and the relationship between the (?, ?)-derivations and other generalized derivations. Under certain conditions, a left-multiplication structure concerned with (?, ?)-derivations can induces a left-symmetric superalgebra structure. Let L be a Lie superalgebra, we give a subgroup G of Aut(L), exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series when G is a cyclic group.

On the Hilbert series of the tangent cones for some 4-generated pseudosymmetric monomial curves

In this article, we study Hilbert Series of non-Cohen-Maculay tangent cones for some 4-generated pseudo symmetric monomial curves. We show that the Hilbert Function is nondecreasing by explicitly computing it. We also compute standard bases of these toric ideals.

Categorifications of rational Hilbert series and characters of FSop modules

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid.

Higgs branch, Coulomb branch, and Hall-Littlewood index

The Higgs branch of 4D $\mathcal{N}=2$ 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 $\mathcal{S}$ theory corresponding to a genus-zero Riemann surface, they are conjectured to be identical. We present several families of counterexamples. We find that for any class $\mathcal{S}$ theory with four or more ${\mathbb{Z}}_{2}$-twisted punctures, they do not match. We construct 3D mirrors for such theories and analyze their Coulomb branch Hilbert series to compute the Higgs branch Hilbert series of the 4D theory. We further construct $a=c$ theories in class $\mathcal{S}$ using the twisted punctures, and these theories, which includes the ${\stackrel{^}{D}}_{4}(SU(2n+1))$ theories, have Hall-Littlewood index different from the Hilbert series of the Higgs branch. We conjecture that this is the case for all $a=c$ theories with nonempty Higgs branch, including $\mathcal{N}\ensuremath{\ge}3$ superconformal field theories.

The Varchenko-Gel'fand ring of a cone

For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel'fand gave a simple presentation for this ring, along with a filtration and associated graded ring whose Hilbert series is the Poincaré polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul.

The Charney-Davis conjecture for simple thin polyominoes

Let be a simple thin polyomino and a field. Let R be the toric -algebra associated to . Write the Hilbert series of R as . We show that if R is Gorenstein. This shows that the Gorenstein rings associated to simple thin polyominoes satisfy the Charney-Davis conjecture.

Hilbert series and suspensions of graphs

Hilbert series and suspensions of graphs

Cohen-Macaulay local rings with e2 = e1 − e + 1

In this paper we study Cohen-Macaulay local rings of dimension d, multiplicity e and second Hilbert coefficient e2 in the case e2=e1−e+1. Let h=μ(m)−d. If e2≠0 then in our case we can prove that type(A)≥e−h−1. If type(A)=e−h−1 then we show that the associated graded ring G(A) is Cohen-Macaulay. In the next case when type(A)=e−h we determine all possible Hilbert series of A. In this case we show that depthG(A) completely determines the Hilbert Series of A.

CP violation and flavor invariants in the seesaw effective field theory

In this paper, we systematically study the algebraic structure of the ring of the flavor invariants and the sources of CP violation in the seesaw effective field theory (SEFT), which is obtained by integrating out heavy Majorana neutrinos in the type-I seesaw model at the tree level and thus includes the dimension-five Weinberg operator and one dimension-six operator. For the first time, we calculate the Hilbert series and explicitly construct all the primary flavor invariants in the SEFT. We show that all the physical parameters can be extracted using the primary invariants and any CP-violating observable can be expressed as the linear combination of CP-odd flavor invariants. The calculation of the Hilbert series shows that there is an equal number of primary flavor invariants in the SEFT and in the full seesaw model, which reveals the intimate connection between the flavor space of the SEFT and that of its ultraviolet theory. A proper matching procedure of the flavor invariants is accomplished between the SEFT and the full seesaw model, through which one can establish a direct link between the CP asymmetries in leptogenesis and those in low-energy neutrino oscillations.

Filtering cohomology of ordinary and Lagrangian Grassmannians

This paper studies, for a positive integer $m$, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most $m$. We build in two ways upon a conjecture for the Hilbert series of this subalgebra due to Reiner and Tudose. The first reinterprets it in terms of the operation of $k$-conjugation, suggesting two conjectural bases for the subalgebras that would imply their conjecture. The second introduces an analogous conjecture for the cohomology of Lagrangian Grassmannians.

Cominuscule points and Schubert varieties

We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions.

The equivariant Hilbert series of the canonical ring of Fermat curves

We consider a Fermat curve Fn:xn+yn+zn=0 over an algebraically closed field K of characteristic p≥0 and study the action of the automorphism group G=Z/nZ×Z/nZ⋊S3 on the canonical ring R=⨁H0(Fn,ΩFn⊗m) when p>3, p∤n and n−1 is not a power of p. In particular, we explicitly determine the classes [H0(Fn,ΩFn⊗m)] in the Grothendieck group K0(G,K) of finitely generated K[G]-modules, describe the respective equivariant Hilbert series HR,G(t) as a rational function, and use our results to write a program in Sage that computes HR,G(t) for an arbitrary Fermat curve.

Exploring the orthosymplectic zoo

We study the Higgs branch of the SCFT limit of 5d SO(6) and SO(8) gauge theory with hypermultiplets in the spinor and vector representations. In the case of SO(6) gauge theories, we contrast the magnetic quivers obtained with those of SU(4) gauge theory with hypermultiplets in the fundamental and second rank antisymmetric representations. Since SU(4) gauge theories admit several different values of the Chern-Simons level, we make some observations about how to distinguish those theories from the brane webs of the SO(6) theories. In the case of SO(8) gauge theories, we use SO(8) triality to propose (naively) inequivalent magnetic quivers, which will turn out to have the same moduli spaces of vacua, at least locally around their most singular loci. We encounter several interesting new phenomena occurring in the magnetic quivers, such as hypermultiplets between neighbouring symplectic gauge nodes and matter in two-index representations of unitary nodes. We also give a prescription for computing the local Coulomb branch Hilbert series for quivers involving bad USp(2) gauge nodes.

Exactly Marginal Deformations and Their Supergravity Duals.

We study the space of supersymmetric AdS_{5} solutions of type IIB supergravity corresponding to the conformal manifold of the dual N=1 conformal field theory. We show that the background geometry naturally encodes a generalized holomorphic structure, dual to the superpotential of the field theory, with the existence of the full solution following from a continuity argument. In particular, this work allows us to address the long-standing problem of finding the gravity dual of the generic N=1 deformations of N=4 conformal field theory: even if we are not able to give it in a fully explicit form, we provide a proof-of-existence of the supergravity solution. Using this formalism, we derive a new result for the Hilbert series of the deformed field theories.

Hilbert series of parallelogram polyominoes

We present a conjecture about the reduced Hilbert series of the coordinate ring of a simple polyomino in terms of particular arrangements of non-attacking rooks that can be placed on the polyomino. By using a computational approach, we prove that the above conjecture holds for all simple polyominoes up to rank 11. In addition, we prove that the conjecture holds true for the class of parallelogram polyominoes, by looking at those as simple planar distributive lattices. Finally, we give a combinatorial interpretation of the Gorensteinness of parallelogram polyominoes.

Recursive sequences attached to modular representations of finite groups

The core of a finite-dimensional modular representation M of a finite group G is its largest non-projective summand. We prove that the dimensions of the cores of M⊗n have algebraic Hilbert series when M is Omega-algebraic, in the sense that the non-projective summands of M⊗n fall into finitely many orbits under the action of the syzygy operator Ω. Similarly, we prove that these dimension sequences are eventually linearly recursive when M is what we term Ω+-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences.