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Let P be a lattice polytope with the $$h^{*}$$ -vector $$(1, h^*_1, \ldots , h^*_s)$$ . In this note we show that if $$h_s^* \le h_1^*$$ , then the Ehrhart ring $${\mathbb {k}}[P]$$ is generated in degrees at most $$s-1$$ as a $${\mathbb {k}}$$ -algebra. In particular, if $$s=2$$ and $$h_2^* \le h_1^*$$ , then P is IDP. To see this, we show the corresponding statement for semi-standard graded Cohen–Macaulay domains over algebraically closed fields.

We verify the g-conjecture for interval subdivisions of Cohen–Macaulay simplicial complexes, using purely combinatorial methods. More precisely, we show that the g-vector of the interval subdivisio...

This article introduces Gamma-triangles, which are closely related to and more fundamental than F-triangles and H-triangles that have been used in the combinatorics of cluster complexes. It is proved that Gamma-triangles can be expressed as sums of the local gamma-vectors introduced by Athanasiadis, which themselves refine the local h-vectors attached by Stanley to simplicial subdivisions. Gamma triangles of all cluster complexes are then explicitly computed.

We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$, and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where $s=0$. We characterize the minimizers and provide examples of maximizers for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.

International audience; This article introduces Gamma-triangles, which are closely related to and more fundamental than F-triangles and H-triangles that have been used in the combinatorics of cluster complexes. It is proved that Gamma-triangles can be expressed as sums of the local gamma-vectors introduced by Athanasiadis, which themselves refine the local h-vectors attached by Stanley to simplicial subdivisions. Gamma triangles of all cluster complexes are then explicitly computed.; Cet article introduit les gamma-triangles, qui sont liés aux F-triangles et H-triangles utilisés dans l'étude combinatoire des complexes d'amas, et dont ils sont en quelque sorte une version plus fondamentale. On démontre que les gamma-triangles s'expriment comme des sommes de gamma-vecteurs locaux, introduits pas Athanasiadis comme un raffinement des h-vecteurs locaux de subdivisions simpliciales, dus à Stanley. On calcule ensuite explicitement les gamma-triangles des complexes d'amas de type fini.

A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth., Comment: Final version. 13 pages, 2 figures

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture. Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrill\'{o}n-Cruz, Cruz-Estrada and Van Tuyl-Villareal., Comment: To appear in Math. Scand

We prove Davies–Gaffney–Grigor’yan lemma for heat kernels of bounded discrete Hodge Laplacians on simplicial complexes.

Let k be an algebraically closed field of characteristic 0, and A = ⨁ i ∈ N A i a Cohen–Macaulay graded domain with A 0 = k . If A is semi-standard graded (i.e., A is finitely generated as a k [ A 1 ] -module), it has the h-vector ( h 0 , h 1 , … , h s ) , which encodes the Hilbert function of A. From now on, assume that s = 2 . It is known that if A is standard graded (i.e., A = k [ A 1 ] ), then A is level. We will show that, in the semi-standard case, if A is not level, then h 1 + 1 divides h 2 . Conversely, for any positive integers h and n, there is a non-level A with the h-vector ( 1 , h , ( h + 1 ) n ) . Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).

金沢大学理工学域 数物科学系 名誉教授, In this paper, we propose an extension of quantum searches on graphs driven by quantum walks to simplicial complexes. To this end, we define a new quantum walk on simplicial complex which is an alternative of preceding studies by authors. We show that the quantum search on the specific simplicial complex corresponding to the triangulation of n-dimensional unit square driven by this new simplicial quantum walk works well, namely, a marked simplex can be found with probability 1+o(1) 1+o(1) within a time O(N − − √ ) O(N), where N is the number of simplices with the dimension of marked simplex., Embargo Period 12 months

We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its first homology group with coefficients in F2 vanishes and the zero-th homology group is isomorphic to F2. Although this is not intrinsically a monotone property, we show that it has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction.

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family $G(d,1,n)$. We highlight consequences for the $q$-Narayana and $q$-Kirkman polynomials, giving simple product formulas for both, and give a $q$-analogue of the identity transforming the $h$-vector to the $f$-vector for the coincidental finite type cluster/Cambrian complexes of Fomin--Zelevinsky and Reading., Version 2 fixed some typos, added results of K. Koike, and completed Hilbert series calculations for all irreducible reflection groups

We get a classification of 1-truncated simplicial algebras under certain conditions by using GAP.

We classify all possible $h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and codimension $\leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $h$-vector is $(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a long-standing open question in this area. All of our results are characteristic free.

This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using Cech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.

We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley-Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified., Comment: 25 pages to be appeared on JPAA

In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.

We give a combinatorial formula for the Ehrhart h ⁎ -vector of the hypersimplex. In particular, we show that h d ⁎ ( Δ k , n ) is the number of hypersimplicial decorated ordered set partitions of type ( k , n ) with winding number d, thereby proving a conjecture of N. Early. We do this by proving a more general conjecture of N. Early on the Ehrhart h ⁎ -vector of a generic cross-section of a hypercube.

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1 &le, s &le, 5 , an O-sequence ( h 0 , h 1 , &hellip, h 2 s - 1 ) &isin, Z &ge, 0 2 s satisfying the properties that (i) h 0 &le, h 1 &le, ⋯ &le, h s - 1 , (ii) h 2 s - 1 = h 0 , h 2 s - 2 = h 1 and (iii) h 2 s - 1 - i = h i + ( - 1 ) i , 2 &le, i &le, s - 1 , can be the h-vector of a Cohen-Macaulay standard G-domain.

We show that for every simplicial polytope an inscribed simplicial polytope exists that has the same dimension, number of vertices, number of edges, and number of 2-faces. This proves that the g-theorem for simplicial polytopes also holds for the class of inscribed simplicial polytopes (up to dimension 7). The proof includes an incremental construction scheme for Delaunay triangulations.

The generalized Dehn–Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn–Sommerville relations for the combinatorial Hopf algebra of hypergraphs. We characterize a wide class of eulerian hypergraphs in terms of the combinatorics of underlying clutters. The analogous results hold for simplicial complexes by the isomorphism which is induced by the correspondence between clutters and simplicial complexes.

To a simplicial complex $\Delta$, we associate a square-free monomial ideal $\mathcal{F}(\Delta)$ in the polynomial ring generated by its facet over a field. Furthermore, we could consider $\mathcal{F}(\Delta)$ as the Stanley--Reisner ideal of another simplicial complex $\delta_{N}(\mathcal{F}(\Delta))$ from facet ideal theory and Stanley--Reisner theory. In this paper, we determine what families of simplicial complexes $\Delta$ have the property that their Stanley--Reisner complexes $\delta_{N}(\mathcal{F}(\Delta))$ are shellable. Furthermore, we show that the simplicial complex with the free vertex property is sequentially Cohen--Macaulay. This result gives a new proof for a result of Faridi on the sequentially Cohen--Macaulayness of simplicial forests.

A comparison of the etale homotopy type of a geometrically pointed simplicial sheaf and that of its standard and twisted fibred sites is demonstrated. This result is applied to yield an etale van Kampen theorem for representable geometrically pointed connected simplicial sheaves under suitable hypotheses on the ambient site.

R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes., Comment: 17 pages, corrected typos

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of $X$ by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds., 16 pages, 5 figures

For a square-free monomial ideal I ⊂ S = k[x 1, x 2,…, x n ], we introduce the notion of quasi-linear quotients. By using the quasi-linear quotients, we give a new algebraic criterion for the shellability of a pure simplicial complex Δ over [n]. Also, we provide a new criterion for the Cohen–Macaulayness of the face ring of a pure simplicial complex Δ. Moreover, we show that the face ring of the spanning simplicial complex (defined in [2]) of an r-cyclic graph is Cohen–Macaulay.

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δχ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δχ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δχ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.

In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This suggests that a pure $O$-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank $d$ matroids on at most $2d$ elements, then it holds for all matroids.

We study the combinatorial structure of the poset H consisting of h-vectors of length s of codimension c standard determinantal schemes, defined by the maximal minors of a t × (t + c − 1) homogeneous, polynomial matrix. We show that H obtains a natural stratification, where each strata contains a maximum h-vector. Moreover, we prove that any h-vector in H is bounded from above by a h-vector of the same length and which corresponds to a codimension c level standard determinantal scheme. Furthermore, we show that the only strata in which there exists also a minimum h-vector is the one consisting of h-vectors of level standard determinantal schemes.

By means of a countermodel we show that the homotopy equivalence of the fibers of a Kan fibration over a connected base cannot be proved constructively.

The $$l{\mathrm{th}}$$ partial barycentric subdivision is defined for a $$(d-1)$$ -dimensional simplicial complex $$\Delta $$ and studied along with its combinatorial and geometric aspects. We analyze the behavior of the f- and h-vector under the lth partial barycentric subdivision extending previous work of Brenti and Welker on the standard barycentric subdivision – the case $$l = d$$ . We discuss and provide properties of the transformation matrices sending the f- and h-vector of $$\Delta $$ to the f- and h-vector of its lth partial barycentric subdivision. We conclude with open problems.

In this paper, we consider the path semigroup l 1 -algebra for a quiver and the inverse semigroup l 1 -algebra of a quiver, the latter of which can be used in the construction of Cuntz–Krieger algebras. The main objectives of the paper are to determine the simplicial and cyclic cohomology groups of these algebras. First, we determine the simplicial and cyclic cohomology of the path algebra of the quiver, showing the simplicial cohomology groups of dimension n vanish for n > 1 . We then determine the simplicial and cyclic cohomology of the inverse semigroup algebra. The work uses the Connes–Tzygan long exact sequence.

For \(X\) a pointed simplicial set with a simplicial monoid action, we construct a simplicial monoid and give a sufficient condition for its classifying space to be the homotopy cofiber of the inclusion of \(X\) into its reduced Borel construction. The known formulae for the respective classifying spaces of the four classical constructions of James, Milnor, Carlsson and Wu are special cases of this result. Thus, we unify categorially the four above-mentioned constructions. We also give the classical constructions and an example as the applications of the main result.

We show that the vertex coloring problem for finite simplicial complexes can be translated into the algebraic homotopy problem of ellipticity for rational spaces. We follow the ideas of Lechuga---Murillo for the classical vertex coloring of graphs.

Let G be a Cohen–Macaulay very well-covered graph. We prove that the h-vector of the independence complex of G is precisely the f-vector of a flag complex. Moreover, we show that the h-vector of clique-whiskered graphs is exactly the h-vector of Cohen–Macaulay very well-covered graphs. In particular, we show that Kalai’s conjecture holds if G is a very well covered Cohen–Macaulay graph.

We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm (which finds a minimum spanning subtree of a weighted graph) to abstract simplicial complexes.

The survey is devoted to the theory of a combinatorial invariant of simple convex polytopes and simplicial complexes that was introduced by V.M. Buchstaber on the basis of constructions of toric topology. We describe methods for calculating this invariant and its relation to other classical and modern combinatorial invariants and constructions, calculate the invariant for special classes of polytopes and simplicial complexes, and find a criterion for this invariant to be equal to a given small number. We also describe a relation to matroid theory, which allows one to apply the results of this theory to the description of the real Buchstaber number in terms of subcomplexes of the Alexander dual simplicial complex.

In this paper the problem of projection onto a simplicial cone is studied. By using Moreau’s decomposition theorem for projecting onto closed convex cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone is always well defined, and the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented. It is also shown that under a somewhat restrictive assumption, the semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone has finite convergence. Besides, under a mild assumption on the simplicial cone, the generated sequence converges linearly to the solution of the associated system of equations.

In this paper we deal with the simplicial nerve (Street’s geometric nerve) of bicategories and tricategories. We prove that if 𝔸 is a trigroupoid then Ner(𝔸) is a Kan simplicial set which moreover turns out to be a (Duskin-Glenn) 3-hypergroupoid, generalizing in this way, the analogous result for the simplicial nerve of bigroupoids, due to Duskin. We associate to any Kan simplicial set X, its homotopy bigroupoid Π2X, and its homotopy trigroupoid Π3X. We prove that there exists a simplicial map v:X→Ner(Π2X) which is a surjective weak 2-equivalence and an isomorphism if X is a Kan 2-hypergroupoid, giving another proof of Duskin’s characterization for the simplicial nerve of bigroupoids. In the 3-dimensional case, there also exists a simplicial map v : X → Ner(Π3X) that is a weak 3-equivalence but, in this case, not necessarily surjective, not even under the hypergroupoid hypothesis. As a corollary, we conclude that any Kan simplicial set with trivial homotopy groups at dimensions ≥ 4 is homotopy equivalent to the nerve of a trigroupoid.

For a complete intersection set of points in a projective space ℙ n over an algebraically closed field, we prove that the h-vector of the graded Artinian ring R/(I 𝕏 + I 𝕐) is an SI-sequence provided 𝕏 is also a complete intersection subset of ℤ and 𝕐 = ℤ \ 𝕏. Furthermore, we also show that if {𝕏, 𝕐} is a regular division of a general basic configuration ℤ in ℙ n , then R/(I 𝕏 + I 𝕐) has the Weak Lefschetz property. These two results give positive answers to the interesting question of [39].

We show that there are exactly 126 combinatorially distinct simplicial, neighbourly 5-polytopes with nine vertices, and give their constructions.

Simplicial Dirac structures as finite analogues of the canonical Stokes Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds. (C) 2013 Elsevier Ltd. All rights reserved.

For any flag nestohedron, we define a flag simplicial complex whose f-vector is the ź-vector of the nestohedron. This proves that the ź-vector of any flag nestohedron satisfies the Frankl---Furedi---Kalai inequalities, partially solving a conjecture by Nevo and Petersen (Discrete Comput. Geom. 45:503---521, 2010). We also compare these complexes to those defined by Nevo and Petersen (Discrete Comput. Geom. 45:503---521, 2010) for particular flag nestohedra.

In our previous work, we investigate a simplicial group model for Ω S 3 obtained from link groups of naive cablings on framed link. In the present paper, we explore some different properties between these simplicial groups and Kanʼs free simplicial group construction, which is a classical simplicial group model for Ω S 3 .

We expose some basic concepts of combinatorial topology (simplicial complex, polyhedron, simplicial map, simplicial approximation of a continuous map) together with a brief description of the popular examples of Voronoi cells and of Delaunay triangulation. Mathematics Subject Classication: 57Q05

In connection with the publication of the catalogue B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Mathematica Contemporanea 2 (2009), 1-25 of known simplicial arrangements of lines in the real projective plane, and the note B. Grünbaum, Small unstretchable simplicial arrangements of pseudolines, Geombinatorics 18 (2009), 153-160 about small simplicial arrangements of pseudolines, several developments of these topics deserve to be mentioned. The present paper puts these results in perspective, and provides appropriate illustrations. Članek podaja nadaljni razvoj in ilustracije rezultatov iz kataloga znanih trikotnih ureditev premic v realni projektivni ravnini B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Mathematica Contemporanea 2 (2009), 1-25 in majhnih trikotnih ureditev psevdo premic B. Grünbaum, Small unstretchable simplicial arrangements of pseudolines, Geombinatorics 18 (2009), 153-160.