arXiv subjects

Christos A. Athanasiadis

Publications and source records attributed to Christos A. Athanasiadis.

At least 37 records · Page 2Linked to original sources

arXiv2016

The symmetric group action on rank-selected posets of injective words

The symmetric group $\mathfrak{S}_n$ acts naturally on the poset of injective words over the alphabet $\{1, 2,\dots,n\}$. The induced representation on the homology of this poset has been computed by Reiner and Webb. We generalize their result by computing the representation of $\mathfrak{S}_n$ on the homology of all rank-selected subposets, in the sense of Stanley. A further generalization to the poset of $r$-colored injective words is given.

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arXiv2016

The local $h$-polynomial of the edgewise subdivision of the simplex

The $r$-fold edgewise subdivision is a well studied flag triangulation of the simplex with interesting algebraic, combinatorial and geometric properties. An important enumerative invariant, namely the local $h$-polynomial, of this triangulation is computed and shown to be $γ$-nonnegative by providing explicit combinatorial interpretations to the corresponding coefficients. A construction of a flag triangulation of the seven-dimensional simplex whose local $h$-polynomial is not real-rooted is also described.

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arXiv2015

Edgewise Cohen-Macaulay connectivity of partially ordered sets

The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen--Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen--Macaulay poset of the same rank. A corresponding notion of edgewise Cohen--Macaulay connectivity for partially ordered sets is investigated. Examples and open questions are discussed.

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arXiv2015

A survey of subdivisions and local $h$-vectors

The enumerative theory of simplicial subdivisions (triangulations) of simplicial complexes was developed by Stanley in order to understand the effect of such subdivisions on the $h$-vector of a simplicial complex. A key role there is played by the concept of a local $h$-vector. This paper surveys some of the highlights of this theory and some recent developments, concerning subdivisions of flag homology spheres and their $γ$-vectors. Several interesting examples and open problems are discussed.

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arXiv2015

Power sum expansion of chromatic quasisymmetric functions

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.

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arXiv2014

Edgewise subdivisions, local $h$-polynomials and excedances in the wreath product $\ZZ_r \wr \mathfrak{S}_n$

The coefficients of the local $h$-polynomial of the barycentric subdivision of the simplex with $n$ vertices are known to count derangements in the symmetric group $\mathfrak{S}_n$ by the number of excedances. A generalization of this interpretation is given for the local $h$-polynomial of the $r$th edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be $γ$-nonnegative and a combinatorial interpretation to the corresponding $γ$-coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product $\ZZ_r \wr \mathfrak{S}_n$. A related result on the derangement polynomial for $\ZZ_r \wr \mathfrak{S}_n$, studied by Chow and Mansour, is also derived from results of Linusson, Shareshian and Wachs on the homology of Rees products of posets.

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arXiv2013

A symmetric unimodal decomposition of the derangement polynomial of type $B$

The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n(x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of $d_n (x)$ as the local $h$-polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition and a new formula for the Eulerian polynomial of type $B$. The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness.

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arXiv2013

The absolute order of a permutation representation of a Coxeter group

A permutation representation of a Coxeter group $W$ naturally defines an absolute order. This family of partial orders (which includes the absolute order on $W$) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on $W$ are investigated when $W$ is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of $W$ is defined.

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arXiv2012

Flag subdivisions and $γ$-vectors

The $γ$-vector is an important enumerative invariant of a flag simplicial homology sphere. It has been conjectured by Gal that this vector is nonnegative for every such sphere $Δ$ and by Reiner, Postnikov and Williams that it increases when $Δ$ is replaced by any flag simplicial homology sphere which geometrically subdivides $Δ$. Using the nonnegativity of the $γ$-vector in dimension 3, proved by Davis and Okun, as well as Stanley's theory of simplicial subdivisions and local $h$-vectors, the latter conjecture is confirmed in this paper in dimensions 3 and 4.

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arXiv2012

The local $h$-vector of the cluster subdivision of a simplex

The cluster complex $Δ(Φ)$ is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system $Φ$. The positive part of $Δ(Φ)$ naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of $Φ$. The local $h$-vector of this subdivision, in the sense of Stanley, is computed and the corresponding $γ$-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local $h$-vector and the corresponding $γ$-vector are provided for the classical root systems, in terms of noncrossing partitions of types $A$ and $B$. An analogous result is given for the barycentric subdivision of a simplex.

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arXiv2011

Buchsbaum* complexes

A class of simplicial complexes, which we call Buchsbaum* over a field, is introduced. Buchsbaum* complexes generalize triangulations of orientable homology manifolds as well as doubly Cohen-Macaulay complexes. By definition, the Buchsbaum* property depends only on the geometric realization and the field. Characterizations in terms of simplicial and local cohomology are given. It is proved that Buchsbaum* complexes are doubly Buchsbaum. Enumerative and graph theoretic properties of Buchsbaum* complexes are investigated. It is shown that various constructions, among them one which generalizes convex ear decompositions, yield Buchsbaum* simplicial complexes.

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arXiv2011

On the Mobius function of a lower Eulerian Cohen-Macaulay poset

A certain inequality is shown to hold for the values of the Mobius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen-Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley's nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen-Macaulay meet-semilattice and Adin's nonnegativity conjecture for the cubical h-vector of a Cohen-Macaulay cubical complex.

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arXiv2011

On the flag f-vector of a graded lattice with nontrivial homology

It is proved that the Boolean algebra of rank n minimizes the flag f-vector among all graded lattices of rank n, whose proper part has nontrivial top-dimensional homology. The analogous statement for the flag h-vector is conjectured in the Cohen-Macaulay case.

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arXiv2011

Cubical subdivisions and local $h$-vectors

Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local $h$-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the $h$-vector of a simplicial complex is played by the (short or long) cubical $h$-vector of a cubical complex, defined by Adin, and the role of the local $h$-vector of a triangulation of a simplex is played by the (short or long) cubical local $h$-vector of a cubical subdivision of a cube. The cubical local $h$-vectors are defined in this paper and are shown to share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed.

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arXiv2010

Functions of random walks on hyperplane arrangements

Many seemingly disparate Markov chains are unified when viewed as random walks on the set of chambers of a hyperplane arrangement. These include the Tsetlin library of theoretical computer science and various shuffling schemes. If only selected features of the chains are of interest, then the mixing times may change. We study the behavior of hyperplane walks, viewed on a subarrangement of a hyperplane arrangement. These include many new examples, for instance a random walk on the set of acyclic orientations of a graph. All such walks can be treated in a uniform fashion, yielding diagonalizable matrices with known eigenvalues, stationary distribution and good rates of convergence to stationarity.

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arXiv2009

Some combinatorial properties of flag simplicial pseudomanifolds and spheres

A simplicial complex $Δ$ is called flag if all minimal nonfaces of $Δ$ have at most two elements. The following are proved: First, if $Δ$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $Δ$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $Δ$ of dimension $d-1$ is minimized when $Δ$ is the boundary complex of the $d$-dimensional cross-polytope.

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arXiv2008

On the graph-connectivity of skeleta of convex polytopes

Given a $d$-dimensional convex polytope $P$ and nonnegative integer $k$ not exceeding $d-1$, let $G_k (P)$ denote the simple graph on the node set of $k$-dimensional faces of $P$ in which two such faces are adjacent if there exists a $(k+1)$-dimensional face of $P$ which contains them both. The graph $G_k (P)$ is isomorphic to the dual graph of the $(d-k)$-dimensional skeleton of the normal fan of $P$. For fixed values of $k$ and $d$, the largest integer $m$ such that $G_k (P)$ is $m$-vertex-connected for all $d$-dimensional polytopes $P$ is determined. This result generalizes Balinski's theorem on the one-dimensional skeleton of a $d$-dimensional convex polytope.

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arXiv2008

The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.

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