Cyclic sieving of noncrossing partitions for complex reflection groups
We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for well-generated complex reflection groups.
math.CO↗
arXiv subjects
Victor Reiner
Publications and source records attributed to Victor Reiner.
Coincidences among skew Schur functions
New sufficient conditions and necessary conditions are developed for two skew diagrams to give rise to the same skew Schur function. The sufficient conditions come from a variety of new operations related to ribbons (also known as border strips or rim hooks). The necessary conditions relate to the extent of overlap among the rows or among the columns of the skew diagram.
math.CO↗
Differential posets and Smith normal forms
We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.
math.CO↗
Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer
We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.
math.AC↗
Shifted set families, degree sequences, and plethysm
We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. The first part collects for the first time in one place, various implications such as: Threshold implies Uniquely Realizable implies Degree-Maximal implies Shifted, which are equivalent concepts for 2-families (=simple graphs), but strict implications for k-families with k > 2. The implication that uniquely realizable implies degree-maximal seems to be new. The second part recalls Merris and Roby's reformulation of the characterization due to Ruch and Gutman for graphical degree sequences and shifted 2-families. It then introduces two generalizations which are characterizations of shifted k-families. The third part recalls the connection between degree sequences of k-families of size m and the plethysm of elementary symmetric functions e_m[e_k]. It then uses highest weight theory to explain how shifted k-families provide the ``top part'' of these plethysm expansions, along with offering a conjecture about a further relation.
math.CO↗
Bimahonian distributions
Motivated by permutation statistics, we define for any complex reflection group W a family of bivariate generating functions. They are defined either in terms of Hilbert series for W-invariant polynomials when W acts diagonally on two sets of variables, or equivalently, as sums involving the fake degrees of irreducible representations for W. It is also shown that they satisfy a ``bicyclic sieving phenomenon'', which combinatorially interprets their values when the two variables are set equal to certain roots of unity.
math.CO↗
Betti numbers of monomial ideals and shifted skew shapes
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.
math.AC↗
Faces of Generalized Permutohedra
The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and gamma-vectors involving descent statistics. This includes a combinatorial interpretation for gamma-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal's conjecture on nonnegativity of gamma-vectors. We calculate explicit generating functions and formulae for h-polynomials of various families of graph-associahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb's problem. We give (and conjecture) upper and lower bounds for f-, h-, and gamma-vectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.
math.CO↗
Alternating subgroups of Coxeter groups
We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.
math.CO↗
Rigidity theory for matroids
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M. The ideas of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this winder setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial) and have little to do with the special nature of graphic matroids or the field R. Our main result is a "nesting theorem" relating the various kinds of independence. Immediate corollaries include generalizations of Laman's Theorem, as well as the equality of 2-rigidity and 2-parallel independence. A key tool in our study is the space of photos of M, a natural algebraic variety whose irreducibility is closely related to the notions of rigidity independence and parallel independence. The number of points on this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation.
math.CO↗
Bergman Complexes, Coxeter Arrangements, and Graph Associahedra
Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi, then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi) equals the nested set complex of A. In addition, we prove that for any orientable matroid M, one can find |mu(M)| different reorientations of M such that the corresponding positive Bergman complexes cover B(M), where mu(M) denotes the Mobius function of the lattice of flats of M.
math.CO↗
Classification of Ding's Schubert varieties: finer rook equivalence
K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups H^*(X_λ; Zz), H^*(X_μ; Zz) are additively isomorphic exactly when the Ferrers boards B_λ, B_μsatisfy the combinatorial condition of rook-equivalence. We classify the varieties X_λup to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.
math.AG↗
Reciprocal domains and Cohen-Macaulay $d$-complexes in $R^d$
We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the issue of whether a Cohen-Macaulay complex of dimension d embedded piecewise-linearly in d-space is necessarily a d-ball. This is observed to be true for d at most 3, but false for d=4.
math.CO↗
Cyclotomic and simplicial matroids
Two naturally occurring matroids representable over Q are shown to be dual: the {\it cyclotomic matroid} $μ_n$ represented by the $n^{th}$ roots of unity $1,ζ,ζ^2,...,ζ^{n-1}$ inside the cyclotomic extension $Q(ζ)$, and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of $Q$-bases for $Q(ζ)$ among the $n^{th}$ roots of unity, which is tight if and only if $n$ has at most two odd prime factors. In addition, we study the Tutte polynomial of $μ_n$ in the case that $n$ has two prime factors.
math.CO↗
Conjectures on the cohomology of the Grassmannian
We give a series of successively weaker conjectures on the cohomology ring of the Grassmannian, starting with the Hilbert series of a certain natural filtration.
math.CO↗
Factorizations of some weighted spanning tree enumerators
We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors.
math.CO↗
The signature of a toric variety
We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by Charney and Davis in their work, which in particular showed that its non-negativity is closely related to a conjecture of Hopf on the Euler characteristic of a non-positively curved manifold. We prove positive (or non-negative) lower bounds for this quantity under geometric hypotheses on the polytope. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch Signature Formula. Moreoever, we show that under these hypotheses on the polytope, the i-th L-class of the corresponding toric variety is (-1)^i times an effective class for any i.
math.AG↗
The Sign Representation for Shephard Groups
Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representation turns out to have many equivalent guises making it analogous to the sign representation of a finite Coxeter group. One of these guises is (up to a twist) the cohomology of the Milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invariant of minimal degree.
math.GR↗