Recent zbMATH articles in MSC 13Fhttps://zbmath.org/atom/cc/13F2024-02-28T19:32:02.718555ZWerkzeugToric rings of perfectly matchable subgraph polytopeshttps://zbmath.org/1527.051452024-02-28T19:32:02.718555Z"Mori, Kenta"https://zbmath.org/authors/?q=ai:mori.kentaSummary: The perfectly matchable subgraph polytope of a graph is a (0,1)-polytope associated with the vertex sets of matchings in the graph. In this paper, we study algebraic properties (compressedness, Gorensteinness) of the toric rings of perfectly matchable subgraph polytopes. In particular, we give a complete characterization of a graph whose perfectly matchable subgraph polytope is compressed.Simplicial complexes in Macaulay2https://zbmath.org/1527.130012024-02-28T19:32:02.718555Z"Hersey, Ben"https://zbmath.org/authors/?q=ai:hersey.ben"Smith, Gregory G."https://zbmath.org/authors/?q=ai:smith.gregory-g"Zotine, Alexandre"https://zbmath.org/authors/?q=ai:zotine.alexandreSummary: We highlight some features of the \textit{SimplicialComplexes} package in \textit{Macaulay}2.Revisiting G-Dedekind domainshttps://zbmath.org/1527.130052024-02-28T19:32:02.718555Z"Zafrullah, M."https://zbmath.org/authors/?q=ai:zafrullah.muhammadSummary: Let \(R\) be an integral domain with \(qf(R)=K\), and let \(F(R)\) be the set of nonzero fractional ideals of \(R\). Call \(R\) a dually compact domain (DCD) if, for each \(I\in F(R)\), the ideal \(I_v=(I^{-1})^{-1}\) is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori, and Krull domains. In addition, we show that a Schreier DCD is a greatest common divisor (GCD) (Greatest Common Divisor) domain with the property that, for each \(A\in F(R)\), the ideal \(A_v\) is principal. We show that a domain \(R\) is G-Dedekind (i.e., has the property that \(A_v\) is invertible for each \(A\in F(R)\)) if and only if \(R\) is a DCD satisfying the property \(\ast\): For all pairs of subsets \(\{a_1,\ldots ,a_m\},\{b_1,\ldots ,b_n\}\subseteq K\backslash \{0\}\), \((\cap_{i=1}^m(a_i)(\cap_{j=1}^n(b_j))=\cap_{i,j=1}^{m,n}(a_ib_j)\). We discuss what the appropriate names for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.On strongly \(\sum\)-\(m\)-clean ringshttps://zbmath.org/1527.130092024-02-28T19:32:02.718555Z"Moutui, Moutu Abdou Salam"https://zbmath.org/authors/?q=ai:moutui.moutu-abdou-salamLet \(R\) be a commutative ring with identity, \(U(R)\) the set of all units of \(R\) and \(J(R)\) the Jacobson radical of \(R\). In this paper, the author defined the notions of strongly \(\sum\)-\(m\)-clean ring and \(m\)-semiboolean ring as natural generalizations of the concept of strongly \(m\)-clean and semiboolean rings respectively. Let \(m\geq 2\) a positive integer. An element \(x\) of \(R\) is said to be strongly \(\sum\)-\(m\)-clean if there exists a positive integer \(n\geq 1\) such that \(x = u_{1} + u_{2} + \dots + u_{n} + f\) with \(u_{1}, u_{2}, \dots, u_{n}\) in \(U(R)\) and \(f^{m} = f\in R\). The ring \(R\) is said to be strongly \(\sum\)-\(m\)-clean if each of its elements is strongly \(\sum\)-\(m\)-clean. An element \(r\) of \(R\) is said to be \(m\)-semiboolean if \(r\) is a sum of \(m\)-potent and an element of \(J(R)\), that is, \(r = a + f\) such that \(f^{m} = f\) and \(a\in J(R)\). Then \(R\) is \(m\)-semiboolean if each of element of \(R\) is \(m\)-semiboolean. The author exhibited connections between these rings and other related classes of rings. He also examined the properties of these notions to various context of commutative ring extensions such us polynomial ring \(R[X]\), power series rings \(R[[X]]\), homomorphic image and direct product, trivial ring extensions and amalgamations.
Reviewer: A. Mimouni (Dhahran)Commutative rings whose proper ideals are pure-semisimplehttps://zbmath.org/1527.130112024-02-28T19:32:02.718555Z"Baghdari, S."https://zbmath.org/authors/?q=ai:baghdari.samaneh"Behboodi, M."https://zbmath.org/authors/?q=ai:behboodi.mahmood"Moradzadeh-Dehkordi, A."https://zbmath.org/authors/?q=ai:moradzadeh-dehkordi.aliRecall that for a left \(R\)-module \(M\), \(\sigma(M)\) denotes the smallest Grothendieck subcategory of \(R\)-Mod containing \(M\), i.e., \(\sigma[M]\) consists of all left \(R\)-module \(N\) which is isomorphic to a factor module of \(M^{(I)}\) for some index set \(I\). Recall that an \(R\)-module \(M\) is called pure-semisimple if every module in the category \(\sigma[M]\) is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that a commutative ring \(R\) is pure-semisimple if and only if every \(R\)-module is a direct sum of cyclic modules, and the latter holds true if and only if, \(R\) is an Artinian principal ideal ring. Consequently, every (or finitely generated, cyclic) ideal of \(R\) is pure-semisimple if and only if \(R\) is an Artinian principal ideal ring. Therefore, a natural question is the following: Whether the same is true if one only assumes that every proper ideal of \(R\) is pure-semisimple? The goal of this paper is to answer this question. The structure of such rings is completely described as either an Artinian principal ideal ring or a local rings \(R\) with a maximal ideal \(M = Rx \oplus T\) in which \(Rx\) is Artinian uniserial while \(T\) is semisimple. Also, they give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple.
Reviewer: Tongsuo Wu (Shanghai)A Taylor resolution over complete intersectionshttps://zbmath.org/1527.130152024-02-28T19:32:02.718555Z"Sobieska, Aleksandra"https://zbmath.org/authors/?q=ai:sobieska.aleksandraLet \(Q=\mathbb{K}[x_1,\dots,x_n]\) be a polynomial ring with coefficients in a field, \(I\subseteq Q\) a monomial ideal and \(\mathfrak{a}\subseteq Q\) an ideal generated by a \(Q\)-regular sequence, such that \(\mathfrak{a}\subseteq I\). Let \(R\) denote the complete intersection ring \(Q/\mathfrak{a}\). Using the Shamash construction --- [\textit{J. Shamash}, J. Algebra 12, 453--470 (1969; Zbl 0189.04004)] and [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)] --- the author describes a resolution of \(R/I\) over \(R\), obtained from the Taylor resolution of \(Q/I\) (Theorem 3.2). As an application, Corollary 3.3 contains upper bounds for the total Betti numbers of \(R/I\).
Reviewer: Jorge Sentiero Neves (Coimbra)On virtually Cohen-Macaulay simplicial complexeshttps://zbmath.org/1527.130192024-02-28T19:32:02.718555Z"Kenshur, Nathan"https://zbmath.org/authors/?q=ai:kenshur.nathan"Lin, Feiyang"https://zbmath.org/authors/?q=ai:lin.feiyang"McNally, Sean"https://zbmath.org/authors/?q=ai:mcnally.sean"Xu, Zixuan"https://zbmath.org/authors/?q=ai:xu.zixuan"Yu, Teresa"https://zbmath.org/authors/?q=ai:yu.teresaSummary: We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen-Macaulay.Recursive formulas for the Kronecker quantum cluster algebra with principal coefficientshttps://zbmath.org/1527.130202024-02-28T19:32:02.718555Z"Ding, Ming"https://zbmath.org/authors/?q=ai:ding.ming.3|ding.ming.1|ding.ming.4|ding.ming|ding.ming.2"Xu, Fan"https://zbmath.org/authors/?q=ai:xu.fan|xu.fan.1"Chen, Xueqing"https://zbmath.org/authors/?q=ai:chen.xueqing.1|chen.xueqingCluster algebras were first defined by \textit{S. Fomin} and \textit{A. Zelevinsky} in [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)]. They are subalgebras of an ambient field, with a combinatorially defined family of generators, the cluster variables, grouped into overlapping sets of the same size, the clusters. Examples of cluster algebras arise as rings of regular functions on double Bruhat cells in reductive groups, in connection with canonical bases of quantum groups and the study of total positivity in algebraic groups.
Subsequently, \textit{A. Berenstein} and \textit{A. Zelevinsky} [Adv. Math. 195, No. 2, 405--455 (2005; Zbl 1124.20028)] defined quantum cluster algebras, which are quantum version of cluster algebras. Cluster multiplication formulas and atomic bases for cluster algebras were first obtained by \textit{P. Sherman} and \textit{A. Zelevinsky} [Mosc. Math. J. 4, No. 4, 947--974 (2004; Zbl 1103.16018)], who initiated the use of Chebyshev polynomials in problems regarding cluster algebras.
In the paper under review, the authors use a quantum version of Chebyshev polynomials to obtain recursive cluster multiplication formulas and an atomic basis for the quantum cluster algebra with principal coefficients associated with the Kronecker quiver.
Reviewer: Francesco Esposito (Padova)Exchange graphs of cluster algebras have the non-leaving-face propertyhttps://zbmath.org/1527.130212024-02-28T19:32:02.718555Z"Fu, Changjian"https://zbmath.org/authors/?q=ai:fu.changjian"Geng, Shengfei"https://zbmath.org/authors/?q=ai:geng.shengfei"Liu, Pin"https://zbmath.org/authors/?q=ai:liu.pinGeneralized associahedra were introduced by Fomin and Zelevinsky in connection to cluster algebras of finite type. According to [\textit{D. D. Sleator} et al., J. Am. Math. Soc. 1, No. 3, 647--681 (1988; Zbl 0653.51017); \textit{C. Ceballos} and \textit{V. Pilaud}, Eur. J. Comb. 51, 109--124 (2016; Zbl 1337.52010)], all associahedra of type \(A\), \(B\), \(C\), \(D\) have the non-leaving-face property, namely, any geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both.
For a finite Coxeter system \((W,S)\), \textit{N. Williams} [Eur. J. Comb. 62, 272--285 (2017; Zbl 1358.05061)] established the non-leaving-face property for \(W\)-permutahedra and \(W\)-associahedra. However, it is known that there are examples related to the associahedron that do not satisfy the non-leaving-face property (cf. [Ceballos and Pilaud, loc. cit.]).
The exchange graph is an important combinatorial invariant of a cluster algebra. The non-leaving-face property for the exchange graphs arising from unpunctured marked surfaces was established in [\textit{T. Brüstle} and \textit{J. Zhang}, Front. Math. China 14, No. 3, 521--534 (2019; Zbl 1436.13044)]. The reachable-in-face property for (abstract) exchange graphs was introduced in[\textit{C. Fu} et al., J. Algebra 628, 189--211 (2023; Zbl 1521.16010)] and it was proved that the exchange graph of a finite-dimensional gentle algebra is connected and has the reachable-in-face property. The reachable-in-face property for the exchange graphs of cluster algebras was conjectured by Fomin and Zelevinsky and has been proved recently by \textit{P. Cao} and \textit{F. Li} [Math. Ann. 377, No. 3--4, 1547--1572 (2020; Zbl 1454.13031)]. The non-leaving-face property implies the reachable-in-face property, but the converse is not true.
In the paper under review, the authors show that the exchange graph for any cluster algebra has the non-leaving-face property.
Reviewer: Haicheng Zhang (Nanjing)On 2nd-stage quantization of quantum cluster algebrashttps://zbmath.org/1527.130222024-02-28T19:32:02.718555Z"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang"Pan, Jie"https://zbmath.org/authors/?q=ai:pan.jieSummary: Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same.
As an example, we give the 2nd-stage quantized cluster algebra \(A_{p,q}(SL(2))\) of \(Fun_{\mathbb{C}}(SL_q(2))\) in {\S}7.1 and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization. Finally, we prove that the compatible Poisson structure of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.Multiple generalized cluster structures on \(D(\text{GL}_n)\)https://zbmath.org/1527.130232024-02-28T19:32:02.718555Z"Voloshyn, Dmitriy"https://zbmath.org/authors/?q=ai:voloshyn.dmitriySummary: We produce a large class of generalized cluster structures on the Drinfeld double of \(\operatorname{\text{GL}}_n\) that are compatible with Poisson brackets given by Belavin-Drinfeld classification. The resulting construction is compatible with the previous results on cluster structures on \(\operatorname{\text{GL}}_n\).Gorenstein homogeneous subrings of graphshttps://zbmath.org/1527.130272024-02-28T19:32:02.718555Z"Cruz, Lourdes"https://zbmath.org/authors/?q=ai:cruz.lourdes"Reyes, Enrique"https://zbmath.org/authors/?q=ai:reyes.enrique"Toledo, Jonathan"https://zbmath.org/authors/?q=ai:toledo.jonathanLet \(G\) denote a graph with \(n\) vertices and consider \[ S = K[x_1t,\dots,x_nt,x^{v_1}t,\dots,x^{v_q}t,t], \] where \(v_1,\dots,v_q\) are the characteristic vectors of the edges of \(G\). In the main result of this article, the authors show that if \(S\) is normal and Gorenstein, then \(G\) is unmixed, the cover number of \(G\), \(\tau\), is equal to \(\lceil \frac{n}{2} \rceil\) and there exist induced subgraphs \(G_1,\dots,G_{s-1}\in E_G\) and \(G_s\subseteq G\), such that \(V(G_1),\dots,V(G_s)\) form a partition of \(V(G)\) and \[ \tau = \lceil \tfrac{n}{2} \rceil = s-1 + \tau(G_s), \] where \(G_s\) is either an edge or an odd cycle of length \(\leq 7\) (Theorem 3.18). In Theorem 3.21, they show that if \(S\) is normal, \(G\) is unmixed, \(\tau=\lceil \frac{n}{2}\rceil\), the existence of \(G_1,\dots,G_s\) as above is guaranteed and an additional assumption is verified, then \(S\) is Gorenstein. The authors conjecture that the additional assumption may be removed (Conjecture 3.23).
Reviewer: Jorge Sentiero Neves (Coimbra)The tensor product and quasiorder of an algebra related to Cohen-Macaulay ringshttps://zbmath.org/1527.130292024-02-28T19:32:02.718555Z"Molkhasi, Ali"https://zbmath.org/authors/?q=ai:molkhasi.aliSummary: This paper shows how the tensor products of the distributive lattices and the finite solvable groups can used to WB-height-unmixed of the method of Stanley and Reisner.Formal Bott-Thurston cocycle and part of a formal Riemann-Roch theoremhttps://zbmath.org/1527.140152024-02-28T19:32:02.718555Z"Osipov, D. V."https://zbmath.org/authors/?q=ai:osipov.denis-vLet \(A\) be a commutative ring and \(A((t))\) the algebra of Laurent series over \(A\). Using the concept of Contou-Carrère symbol (see [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 73, 147--181 (1991; Zbl 0749.14011)]), the author defines the formal Bott-Thurston cocycle as a certain 2-cocycle on the group of continuous \(A\)-automorphisms of \(A((t))\) taking values in the group of invertible elements of \(A\). The main result of the paper under review is a formal version of the Riemann-Roch theorem,
applicable to separated schemes over \(\mathbb Q\).
The proof is partly based on ideas and constructions similar to those discussed in [\textit{M. Kapranov} and \textit{É. Vasserot}, Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113--133 (2007; Zbl 1129.14022)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)Computing with jetshttps://zbmath.org/1527.140252024-02-28T19:32:02.718555Z"Galetto, Federico"https://zbmath.org/authors/?q=ai:galetto.federico"Iammarino, Nicholas"https://zbmath.org/authors/?q=ai:iammarino.nicholasThis paper describes a package for the commutative algebra software system Macaulay2 for computation of the jet schemes of an affine algebraic variety. The explicit description of jet schemes is useful for the computation of invariants of singularities in algebraic geometry and commutative algebra, but such computations are often difficult to do by hand. This paper describes the implementation of the construction of jet schemes, as well as several more specific methods. These methods include the computation of the reduced jet schemes of varieties defined by monomial ideals, as well as their minimal primes and multiplities. This also includes algorithms for the jet schemes of a squarefree quadratric ideal associated to a (hyper)graph, and the corresponding graph of this jet scheme. Finally, the paper describes the computation of the defining equations for the principal component of the \(m\)-jet scheme (i.e., the irreducible component of the \(m\)-jet scheme of an irreducible variety obtained as the closure of the \(m\)-jets lying over smooth points). Example computations of the jet schemes of determinal varieties are given.
Reviewer: Devlin Mallory (Salt Lake City)Exotic Lagrangian tori in Grassmannianshttps://zbmath.org/1527.140982024-02-28T19:32:02.718555Z"Castronovo, Marco"https://zbmath.org/authors/?q=ai:castronovo.marcoLagrangian tori of symplectic manifolds have relevance in various areas, e.g. Hamiltonian mechanics, mirror symmetry, low-dimensional topology, Seiberg-Witten invariants. In the paper under review, the author gives a construction of a Lagrangian torus \(L_{\mathfrak{s}}\) of the Grassmannian \(Gr(k, n)\) of \(k\)-dimensional subspaces of \(\mathbb{C}^n\). The torus \(L_{\mathfrak{s}}\) is attached to a Plucker sequence \(\mathfrak{s}\) of type \((k, n)\) and it is obtained by a technique of degeneration to a toric variety. In this way, the author gives examples of Lagrangian tori that are not displaceable and are not Hamiltonian isotopic one to the other. Moreover, the derived Fukaya category of \(Gr(2, n)\), in various cases is shown to be split generated by objects supported on a finite number of these tori.
Reviewer: Francesco Esposito (Padova)Quantum cluster characters of Hall algebras revisitedhttps://zbmath.org/1527.170082024-02-28T19:32:02.718555Z"Fu, Changjian"https://zbmath.org/authors/?q=ai:fu.changjian"Peng, Liangang"https://zbmath.org/authors/?q=ai:peng.liangang"Zhang, Haicheng"https://zbmath.org/authors/?q=ai:zhang.haichengSummary: Let \(Q\) be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of \(Q\). As an application, we recover the surjective homomorphism defined in [\textit{Ming Ding}, \textit{Fan Xu} and the third author, Math. Z. 296, No. 3-4, 945--968 (2020; Zbl 1509.17010)], which realizes the principal coefficient quantum cluster algebra \(\mathcal{A}_q(Q)\) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero-Chapoton formula, as well as some multiplication formulas between quantum Caldero-Chapoton characters.Discrete dynamics in cluster integrable systems from geometric \(R\)-matrix transformationshttps://zbmath.org/1527.820102024-02-28T19:32:02.718555Z"George, Terrence"https://zbmath.org/authors/?q=ai:george.terrence"Ramassamy, Sanjay"https://zbmath.org/authors/?q=ai:ramassamy.sanjaySummary: Cluster integrable systems are a broad class of integrable systems modelled on bipartite dimer models on the torus. Many discrete integrable dynamics arise by applying sequences of local transformations, which form the cluster modular group of the cluster integrable system. This cluster modular group was recently characterized by the first author and Inchiostro. There exist some discrete integrable dynamics that make use of non-local transformations associated with geometric \(R\)-matrices. In this article we characterize the generalized cluster modular group -- which includes both local and non-local transformations -- in terms of extended affine symmetric groups. We also describe the action of the generalized cluster modular group on the spectral data associated with cluster integrable systems.A generic instability in clustering dark energy?https://zbmath.org/1527.830432024-02-28T19:32:02.718555Z"Hassani, Farbod"https://zbmath.org/authors/?q=ai:hassani.farbod"Adamek, Julian"https://zbmath.org/authors/?q=ai:adamek.julian"Kunz, Martin"https://zbmath.org/authors/?q=ai:kunz.martin"Shi, Pan"https://zbmath.org/authors/?q=ai:shi.pan"Wittwer, Peter"https://zbmath.org/authors/?q=ai:wittwer.peterSummary: In this paper, we study the effective field theory (EFT) of dark energy (DE) for the \(k\)-essence model beyond linear order. Using particle-mesh \(N\)-body simulations that consistently solve the DE evolution on a grid, we find that the next-to-leading order in the EFT expansion, which comprises the terms of the equations of motion that are quadratic in the field variables, gives rise to a generic instability in the regime of low speed of sound (high Mach number). We rule out the possibility of a numerical artefact by considering simplified cases in spherically and plane symmetric situations analytically. If the speed of sound vanishes exactly, the non-linear instability makes the evolution singular in finite time, signalling a breakdown of the EFT framework. The case of finite (but small) speed of sound is subtle, and the local singularity could be replaced by some other type of behaviour with strong non-linearities. While an ultraviolet completion may cure the problem in principle, there is no reason why this should be the case in general. As a result, for a large range of the effective speed of sound \(c_s\), a linear treatment is not adequate.