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Abstract: There are two established gradings on Leavitt path algebras associated with ultragraphs, namely the grading by the integers group and the grading by the free group on the edges. In this paper, we characterize properties of these gradings in terms of the underlying combinatorial properties of the ultragraphs. More precisely, we characterize when the gradings are strong or epsilon-strong. The results regarding the free group on the edges are new also in the context of Leavitt path algebras of graphs. Finally, we also describe the relationship between the strongness of the integer grading on an ultragraph Leavitt path algebra and the saturation of the gauge action associated with the corresponding ultragraph C*-algebra. PubDate: 2023-01-23
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Abstract: Given a reductive group G and a parabolic subgroup \(P\subset G\) , with maximal torus T, we consider (following Dabrowski’s work) the closure X of a generic T-orbit in G/P and determine in combinatorial terms when the toric variety X is \(\mathbb {Q}\) -Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenskiĭ and Klyachko. As an application, we apply the well-known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures—this list includes the reflexive root polytopes. PubDate: 2023-01-23
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Abstract: We will study evolution algebras A that are free modules of dimension two over domains. We start by making some general considerations about algebras over domains: They are sandwiched between a certain essential D-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and we characterize the perfect and quasiperfect evolution algebras in terms of the determinant of its structure matrix. We classify the two-dimensional perfect evolution algebras over domains parametrizing the isomorphism classes by a convenient moduli set. PubDate: 2023-01-23
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Abstract: This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them. PubDate: 2023-01-21
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Abstract: Manin and Schechtman introduced a family of arrangements of hyperplanes generalizing classical braid arrangements, which they called the discriminantal arrangements. Athanasiadis proved a conjecture by Bayer and Brandt providing a full description of the combinatorics of discriminantal arrangements in the case of very generic arrangements. Libgober and Settepanella described a sufficient geometric condition for given arrangements to be non-very generic in terms of the notion of dependency for a certain arrangement. Settepanella and the author generalized the notion of dependency introducing r-sets and \(K_{\mathbb {T}}\) -vector sets, and provided a sufficient condition for non-very genericity but still not convenient to verify by hand. In this paper, we give a classification of the r-sets, and a more explicit and tractable condition for non-very genericity. PubDate: 2023-01-18
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Abstract: We introduce categories \(\mathcal {M}\) and \(\mathcal {S}\) internal in the tricategory \(\textrm{Bicat}_3\) of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory V. Their horizontal tricategories are the tricategories of matrices and spans in V. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following Fiore et al. (J Pure Appl Algebra 215(6):1174–1197, 2011), we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices \(\mathcal {M}\) and spans \(\mathcal {S}\) in a 1-strict tricategory V are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in V, respectively. As a by-product of our tricategorical constructions, we recover some results from Femić (Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories. arXiv:2101.01460v2). Truncating to 1-categories, we recover results from Cottrell et al. (Tbilisi Math J 10(3):239–254, 2017) and Ehresmann and Ehresmann (Cah Topol Géom Differ Catég 19/4:387–443, 1978) on the equivalence of enriched and discretely internal 1-categories. PubDate: 2023-01-04
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Abstract: In this paper, we give three infinite families of chiral 4-polytopes, with the automorphism group of each member of a family being symmetric. This construction is based on the construction depicted in Conder et al. (J Algebr Combin 42:225–244, 2015), but with various preassigned facets for polytopes in distinct families, and with the degree of symmetric automorphism groups of members in each family growing linearly with the last entry of its type (Schläfli symbol). PubDate: 2023-01-02
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Abstract: We consider a class of Nichols algebras \(\mathscr {B}(\mathfrak L_q( 1, \mathscr {G}))\) introduced in Andruskiewitsch et al. which are domains and have many favourable properties like AS-regular and strongly noetherian. We classify their finite-dimensional simple modules and their point modules. PubDate: 2023-01-02
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Abstract: The Gelfand representation of \({\mathcal {S}}_n\) is the multiplicity-free direct sum of the irreducible representations of \({\mathcal {S}}_n\) . In this paper, we use a result of Adin, Postnikov, and Roichman to find a generating function for the Gelfand character. In order to find this generating function, we investigate descents of so-called \(\lambda \) -unimodal involutions. PubDate: 2022-12-29
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Abstract: The characterization of distance-regular Cayley graphs originates from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a partial classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter 3 satisfying some additional conditions. PubDate: 2022-12-29
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Abstract: In this note, we report the classification of all symmetric 2-(36, 15, 6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code. PubDate: 2022-12-29
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Abstract: A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges and even otherwise. Pontus von Brömssen (né Andersson) showed that the existence of such an automorphism is independent of the orientation and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic even graphs on n vertices is equal to the number of pairwise non-isomorphic tournaments on n vertices. We prove this conjecture using a counting argument with several applications of the Cauchy–Frobenius theorem. PubDate: 2022-12-29
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Abstract: Association schemes whose thin residues are finite abelian groups have been studied in many papers. The existence, constructions, and schurity problems are among the major topics in the research in this area. In this paper, we use table algebras as a tool to construct association schemes. We will first study the properties of table algebras with thin thin residues. Then, using these properties we will develop a method to construct association schemes in which the thin residues are finite abelian groups. The schurity of these association schemes will also be discussed. PubDate: 2022-12-29
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Abstract: Let \(J_{n}^{B}\) denote the set of all fixed-point free involutions of the hyperoctahedral group \(B_{n}\) , and let \(\hbox {des}_{B}(\pi )\) denote the number of descents of the permutation \(\pi \in B_{n}\) . We show that \(J_{n}^{B}(t):=\sum _{\pi \in J_{n}^{B}}t^{\text {des}_{B}(\pi )}\) is symmetric, unimodal and \(\gamma \) -positive for \(n\ge 2\) . PubDate: 2022-12-29
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Abstract: An (association) scheme is said to be Frobenius if it is the (orbital) scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group G with abelian kernel to be determined up to isomorphism only by the character table of G. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with n vertices and Frobenius automorphism group is equal to 2 unless \(n\in \{p,p^2,p^3,pq,p^2q\}\) , where p and q are distinct primes. PubDate: 2022-12-29
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Abstract: We construct the first infinite families of locally 2-arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger’s program for the classification of locally 2-arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs. PubDate: 2022-12-29
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Abstract: We calculate all irreducible representations over a subfamily of pointed Hopf algebras with group-likes the dihedral group analyzing the possible decompositions of the restriction to the dihedral group and calculating the Jacobson radical of the Hopf algebra. PubDate: 2022-12-29
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Abstract: We show the reduced \(C^*\) -algebra of a graded ample groupoid is a strongly graded \(C^*\) -algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We use this to show that the graph \(C^*\) -algebra of a countable directed graph is a strongly \({\mathbb {Z}}\) -graded \(C^*\) -algebra if and only if the Leavitt path algebra is a strongly \({\mathbb {Z}}\) -graded ring. The latter of these was shown previously to be equivalent to the graph satisfying property (Y). PubDate: 2022-12-29
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Abstract: We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type \(A_2\) : this deforms to the Lyness family of integrable symplectic maps in the plane. For types \(A_3\) and \(A_4\) , we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation. PubDate: 2022-12-29
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