Symmetric Semigroup Research Articles

Local complete intersections and Weierstrass points

This work presents a simple proof that the moduli space of complete integral Gorenstein curves with a prescribed symmetric Weierstrass semigroup becomes a weighted projective space, even for fields of positive characteristic, when the associated monomial curve is a local complete intersection.

Almost Gorenstein simplicial semigroup rings

We give a criterion for almost Gorenstein property for semigroup rings associated with simplicial semigroups. We extend Nari's theorem for almost symmetric numerical semigroups to simplicial semigroups with higher rank. By this criterion, we determine 2-dimensional normal semigroup rings which have “Ulrich elements” defined in [8].

Rings of formal power series and symmetric real semigroups

Sections 1–6 of this paper deal with the rings ▪ of (generalized) power series over a formally real field F of coefficients and (non-negative) exponents in an arbitrary totally ordered abelian group G. We determine explicitly the nature of the elements constituting the real spectra of these rings, its structure under the partial order of specialization, and the structure of their associated real semigroups (RS). Theorem A is a corollary of these results. Starting with section 7 we introduce a new class of symmetric real semigroups by requiring that their spaces of characters satisfy analogs to some of the basic laws holding in the real spectra of the rings ▪. These spaces have, amongst others, remarkable symmetry properties that justify their name (see 7.16 and 7.17). In §8 we prove that every finite symmetric RS is isomorphic to the RS associated to some ring of formal power series, but that such representation fails, in general, for symmetric RSs of suitably large infinite cardinality. Our study leads to showing, in §10, that the inclusion map G⁎∪{0}↪G of the reduced special group G⁎ of invertible elements of a symmetric RS G has a retract. As a consequence, every symmetric RS, G, is isomorphic to the extension of G⁎ by a 3-semigroup Δ defined in terms of that retract. Extensions of this form are introduced and studied in §9. Finally, in §11 we present a first order axiomatisation of the class of symmetric RSs in the natural language for real semigroups.

A Bakry–Émery criterion for weighted contractivity and L2-Hardy inequalities

We show a symmetric Markov diffusion semigroup satisfies a weighted contractivity condition if and only if a [Formula: see text]-Hardy inequality holds, and we give a Bakry–Émery-type criterion for the former. We then give some applications.

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) Lp-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d+-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

Complex symmetry in the Fock space of several variables

In this paper we study the complex symmetry in the several variable Fock space by using the techniques of weighted composition operators and semigroups. We characterize unbounded weighted composition operators that are (real) complex symmetric with respect to a concrete conjugation. Using this characterization, we study complex symmetric semigroups and their generators. We realize such generators as first-order differential operators.

Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups

We study vector-valued Littlewood–Paley–Stein theory for semigroups \{T_{t}\}_{t>0} of regular contractions on L_{p}(\Omega) for a fixed 1<p<\infty . We prove that if a Banach space X is of martingale cotype q , then there is a constant C such that \biggl\|\biggl(\int_{0}^{\infty}\biggl\|t\frac{\partial}{\partial t}P_{t} (f)\biggr\|_{X}^{q}\,\frac{dt}{t}\biggr)^{1/q}\biggr\|_{L_p(\Omega)}\le C\|f\|_{L_p(\Omega; X)}, \quad\forall f\in L_{p}(\Omega; X), where \{P_{t}\}_{t>0} is the Poisson semigroup subordinated to \{T_{t}\}_{t>0} . Let \mathsf{L}^{P}_{\mathsf{c}, q, p}(X) be the least constant C , and let \mathsf{M}_{\mathsf{c}, q}(X) be the martingale cotype q constant of X . We show \mathsf{L}^{P}_{\mathsf{c},q, p}(X)\lesssim \max(p^{1/q}, p') \mathsf{M}_{\mathsf{c},q}(X). Moreover, the order \max(p^{1/q}, p') is optimal as p\to1 and p\to\infty . If X is of martingale type q , the reverse inequality holds. If additionally \{T_{t}\}_{t>0} is analytic on L_{p}(\Omega; X) , the semigroup \{P_{t}\}_{t>0} in these results can be replaced by \{T_{t}\}_{t>0} itself. Our new approach is built on holomorphic functional calculus. Compared with the previous approaches, ours is more powerful in several aspects: (a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; (b) it yields the optimal orders of growth on p for most of the relevant constants; (c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood–Paley–Stein inequalities for symmetric submarkovian semigroups are better than those of Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when X is of martingale cotype q and \{P_{t}\}_{t>0} is the classical Poisson or heat semigroup on \mathbb{R}^{d} .

Quantitative estimates for bounded holomorphic semigroups

We revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood–Paley–Stein theory for symmetric diffusion semigroups.

Inverse polynomials of numerical semigroup rings

We study the defining ideal of a numerical semigroup ring k[H] using the inverse polynomial attached to the Artinian ring k[H]/(th). We give a criterion for H to be symmetric or almost symmetric using the annihilator of the inverse system. Also we give characterizations of symmetric numerical semigroups with small multiplicity and give a new proof of Bresinsky's Theorem for symmetric semigroups generated by 4 elements.

Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra \mathbb{C}[\Gamma] with \Gamma a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten \mathcal{S}_{p} class p \in [2, \infty) , then the n -fold tensor power H^{\otimes n}_{\Gamma} for n \geq \frac{p}{2} is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in \mathcal{S}_{p} for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient- \mathcal{S}_{p} property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p -integrable representation.

The locus of curves with an odd subcanonical point

We present an explicit construction of a compactification of the locus of smooth curves whose symmetric Weierstrass semigroup at a marked point is odd. This construction extends Stöhr’s techniques, which can be seen as a variant of Hauser’s algorithm for computing versal deformation spaces. As an application, we prove the rationality of the locus for genus at most six.

Holomorphic Hörmander-type functional calculus on sectors and strips

In this paper, the abstract multiplier theorems for 0 0 -sectorial and 0 0 -strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405–444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and “holomorphic Schwartz” functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937–974] to obtain an improvement—with respect to the smoothness condition—of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.

Derivations and KMS-Symmetric Quantum Markov Semigroups

We prove that the generator of the L^2 implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule.

A Note on The Some Class of Symmetric Numerical Semigroups

A Note on The Some Class of Symmetric Numerical Semigroups

P-Numerical semigroups with p-symmetric properties

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups

Cipriani and Sauvageot have shown that for any [Formula: see text]-generator [Formula: see text] of a tracially symmetric quantum Markov semigroup on a C*-algebra [Formula: see text] there exists a densely defined derivation [Formula: see text] from [Formula: see text] to a Hilbert bimodule [Formula: see text] such that [Formula: see text]. Here, we show that this construction of a derivation can in general not be generalized to quantum Markov semigroups that are symmetric with respect to a non-tracial state. In particular, we show that all derivations to Hilbert bimodules can be assumed to have a concrete form, and then we use this form to show that in the finite-dimensional case the existence of such a derivation is equivalent to the existence of a positive matrix solution of a system of linear equations. We solve this system of linear equations for concrete examples using Mathematica to complete the proof.

A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis

Given n⩾2 and a1,…,an∈N. Let S=〈a1,…,an〉 be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on a1, which computes the Apéry set and the Frobenius number of S. We also find the Gröbner basis of the toric ideal defined by S, for the weighted degree reverse lexicographical order ≺w to x1,…,xn, without using Buchberger's algorithm. As an application we introduce and study some special classes of semigroups. Namely, when S is generated by generalized arithmetic progressions and generalized almost arithmetic progressions with the ratio a positive or a negative number. We determine symmetric and almost symmetric semigroups generated by a generalized arithmetic progression.

On the Planarity of Graphs Associated with Symmetric and Pseudo Symmetric Numerical Semigroups

Let S(m,e) be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph GS an S(m,e)-graph if there exists a numerical semigroup S∈S(m,e) with V(GS)={x:x∈g(S)} and E(GS)={xy⇔x+y∈S}, where g(S) denotes the gap set of S. The aim of this article is to discuss the planarity of S(m,e)-graphs for some cases where S is an irreducible numerical semigroup.

Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves

In this article, we study minimal graded free resolutions of Cohen-Macaulay tangent cones of some monomial curves associated to 4-generated pseudo symmetric numerical semigroups. We explicitly give the matrices in these minimal free resolutions.