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Abstract: Abstract In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets. PubDate: 2024-02-20

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Abstract: Abstract For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties. PubDate: 2024-02-17

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Abstract: Abstract We give a rank augmentation technique for rank three string C-group representations of the symmetric group \(S_n\) and list the hypotheses under which it yields a valid string C-group representation of rank four thereof. PubDate: 2024-02-16

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Abstract: Abstract A generalized spline on an edge-labeled graph \((G,\alpha )\) is defined as a vertex labeling, such that the difference of labels on adjacent vertices lies in the ideal generated by the edge label. We study generalized splines over greatest common divisor domains and present a determinantal basis condition for generalized spline modules on arbitrary graphs. The main result of the paper answers a conjecture that appeared in several papers. PubDate: 2024-02-15

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Abstract: Abstract We construct a finite Young wall model for a certain irreducible module over \(\imath \) quantum group \({\textbf{U}}^{\jmath }\) . Moreover, we show that this irreducible module is a highest weight module and is determined by a crystal structure on the set of finite Young walls. PubDate: 2024-02-08

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Abstract: Abstract This work is a continuation of [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type A]. Let P be a parabolic subgroup of an irreducible simple algebraic group G. Let \(P'\) be the derived group of P, and let \({\mathfrak {m}}\) be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra \({\mathbb {C}}[{\mathfrak {m}}]^{P'}\) , spanned by the P semi-invariants in \({\mathbb {C}}[{\mathfrak {m}}]\) , is polynomial. A linear subvariety \(e+V\) of \({\mathfrak {m}}\) is called a Weierstrass section for the action of \(P'\) on \({\mathfrak {m}}\) , if the restriction map induces an isomorphism of \({\mathbb {C}}[{\mathfrak {m}}]^{P'}\) onto \({\mathbb {C}}[e+V]\) . Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let \({\mathscr {N}}\) be zero locus of the augmentation \({\mathbb {C}}[{\mathfrak {m}}]^{P'}_+\) . It is called the nilfibre relative to this action. Suppose \(G=\textrm{SL}(n,{\mathbb {C}})\) , and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section \(e+V\) in \({\mathfrak {m}}\) was established by a general combinatorial construction. Notably, \(e \in {\mathscr {N}}\) and is a sum of root vectors with linearly independent roots. The Weierstrass section \(e+V\) looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the “canonical Weierstrass section”. Through [Y. Fittouhi and A. Joseph, loc. cit. Prop. 6.9.2, Cor. 6.9.8], there is always a “canonical” component \({\mathscr {N}}^e\) of \({\mathscr {N}}\) containing e. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment e to an element \(e_\textrm{VS}\) by adjoining root vectors. Then the linear span \(E_\textrm{VS}\) of these root vectors lies in \(\mathscr {N}^e\) and its closure is just \({\mathscr {N}}^e\) . Yet, this same result shows that PubDate: 2024-02-07

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Abstract: Abstract We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension \(n>2\) , we show that the defining ideal has minimal generators of degree at least n. PubDate: 2024-02-06

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Abstract: Abstract We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile–Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile–Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile–Macchia resolutions. PubDate: 2024-02-05

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Abstract: Abstract \((N,\gamma )\) -hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated with totally ramified points of N-fold covers of curves of genus \(\gamma \) . Torres characterized \((2,\gamma )\) -hyperelliptic semigroups of maximal weight whenever their genus is large relative to \(\gamma \) . Here we do the same for \((3,\gamma )\) -hyperelliptic semigroups, and we formulate a conjecture about the general case whenever \(N \ge 3\) is prime. PubDate: 2024-02-05

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Abstract: Abstract A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera’s characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo. PubDate: 2024-01-24

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Abstract: Abstract Denote by \(\rho (G)\) and \(\kappa (G)\) the spectral radius and the signless Laplacian spectral radius of a graph G, respectively. Let \(k\ge 0\) be a fixed integer and G be a graph of size m which is large enough. We show that if \(\rho (G)\ge \sqrt{m-k}\) , then \(C_4\subseteq G\) or \(K_{1,m-k}\subseteq G\) . Moreover, we prove that if \(\kappa (G)\ge m-k+1\) , then \(K_{1,m-k}\subseteq G\) . Both these results extend some known results. PubDate: 2024-01-20

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Abstract: Abstract Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group G if it admits G as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph SC(G, R, L, S) is called quasi-abelian if each of R, L and S is a union of some conjugacy classes of G. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group G has perfect state transfer between distinct vertices g and h, and G has a faithful irreducible character, then \(gh^{-1}\) lies in the center of G and \(gh=hg\) ; in particular, G cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups. PubDate: 2024-01-20

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Abstract: Abstract Cactus groups \(J_n\) are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups and, in particular, twin groups \(Tw_n\) and Mostovoy’s Gauss diagram groups \(D_n\) , which are better understood. Concretely, we construct an injective group 1-cocycle from \(J_n\) to \(D_n\) and show that \(Tw_n\) (and its k-leaf generalizations) inject into \(J_n\) . As a corollary, we solve the word problem for cactus groups, determine their torsion (which is only even) and center (which is trivial), and answer the same questions for pure cactus groups, \(PJ_n\) . In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group \(PJ_4\) . Our tools come mainly from combinatorial group theory. PubDate: 2024-01-10

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Abstract: Abstract Let \(\mathcal{M}\) be an orientably regular (resp. regular) map with the number n vertices. By \(G^+\) (resp. G) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of \(\mathcal{M}\) . Let \(\pi \) be the set of prime divisors of n. A Hall \(\pi \) -subgroup of \(G^+\) (resp. G) is meant a subgroup such that the prime divisors of its order all lie in \(\pi \) and the primes of its index all lie outside \(\pi \) . It is mainly proved in this paper that (1) suppose that \(\mathcal{M}\) is an orientably regular map where n is odd. Then \(G^+\) is solvable and contains a normal Hall \(\pi \) -subgroup; (2) suppose that \(\mathcal{M}\) is a regular map where n is odd. Then G is solvable if it has no composition factors isomorphic to \(\hbox {PSL}(2,q)\) for any odd prime power \(q\ne 3\) , and G contains a normal Hall \(\pi \) -subgroup if and only if it has a normal Hall subgroup of odd order. PubDate: 2024-01-01

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Abstract: Abstract Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, \({\text {depth}}R/I \ge 1\) and \({\text {depth}}R/I^t = 0\) for \(t \gg 1\) . This paper studies the problem when \({\text {depth}}R/I^t = 1\) for some \(t \ge 1\) and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for \({\text {depth}}R/I^{(t)} = 1\) for \(t \gg 1\) and show that the condition \({\text {depth}}R/I^{(t)} = 1\) is persistent, where \(I^{(t)}\) denotes the t-th symbolic powers of I. PubDate: 2024-01-01

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Abstract: Abstract Given a finite group G acting on a set X let \(\delta _k(G,X)\) denote the proportion of elements in G that have exactly k fixed points in X. Let \(\mathcal {S}_n\) denote the symmetric group acting on \([n]=\{1,2,\dots ,n\}\) . For \(A\leqslant \mathcal {S}_m\) and \(B\leqslant \mathcal {S}_n\) , the permutational wreath product \(A\wr B\) has two natural actions and we give formulas for both, \(\delta _k(A\wr B,[m]{\times }[n])\) and \(\delta _k(A\wr B,[m]^{[n]})\) . We prove that for \(k=0\) the values of these proportions are dense in the intervals \([\delta _0(B,[n]),1]\) and \([\delta _0(A,[m]),1]\) . Among further results, we provide estimates for \(\delta _0(G,[m]^{[n]})\) for subgroups \(G\leqslant \mathcal {S}_m\wr \mathcal {S}_n\) containing \(\mathcal {A}_m^{[n]}\) . PubDate: 2024-01-01

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Abstract: Abstract Path algebras from quivers are a fundamental class of algebras with wide applications. Yet it is challenging to describe their universal properties since their underlying path semigroups are only partially defined. A new notion, called locality structures, was recently introduced to deal with partially defined operation, with motivation from locality in convex geometry and quantum field theory. We show that there is a natural correspondence between locality sets and quivers which leads to a concrete class of locality semigroups, called Brandt locality semigroups, which can be obtained by the paths of quivers. Further these path Brandt locality semigroups are precisely the free objects in the category of Brandt locality semigroups with a rigidity condition. This characterization gives a universal property of path algebras and at the same time a combinatorial realization of free rigid Brandt locality semigroups. PubDate: 2023-12-28

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Abstract: Abstract Let p be a prime, and let \(\Lambda _{2p}\) be a connected cubic arc-transitive graph of order 2p. In the literature, a lot of works have been done on the classification of edge-transitive normal covers of \(\Lambda _{2p}\) for specific \(p\le 7\) . An interesting problem is to generalize these results to an arbitrary prime p. In 2014, Zhou and Feng classified edge-transitive cyclic or dihedral normal covers of \(\Lambda _{2p}\) for each prime p. In our previous work, we classified all edge-transitive N-normal covers of \(\Lambda _{2p}\) , where p is a prime and N is a metacyclic 2-group. In this paper, we give a classification of edge-transitive N-normal covers of \(\Lambda _{2p}\) , where \(p\ge 5\) is a prime and N is a metacyclic group of odd prime power order. PubDate: 2023-12-26

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Abstract: Abstract In this paper, we try to answer some questions raised by Cangelmi (Eur J Comb 33(7):1444–1448, 2012). We reinterpret the Riemann–Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann–Roch theorems for orientable hypermaps by defining the divisor of a function f on darts. In addition, we extend Riemann–Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann–Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory. PubDate: 2023-12-26

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Abstract: Abstract We study chains of nonzero edge ideals that are invariant under the action of the monoid \({{\,\textrm{Inc}\,}}\) of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of \({{\,\textrm{Inc}\,}}\) -invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of \({{\,\textrm{Inc}\,}}\) -invariant chains of edge ideals. PubDate: 2023-12-21