Victor Reiner

Resolutions of Stanley-Reisner rings and Alexander duality

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal IΔ in the polynomial ring A = k[x1, …, xn], and its quotient k[Δ] = AIΔ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ∗ which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dimk ToriA(k[Δ],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ∗. As corollaries, we prove that IΔ has a linear resolution as A-module if and only if Δ∗ is Cohen-Macaulay over k, and show how to compute the Betti numbers dimk ToriA (k[Δ],k) in some cases where Δ∗ is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.

Non-crossing partitions for classical reflection groups

Abstract We introduce analogues of the lattice of non-crossing set partitions for the classical reflection groups of types B and D . The type B analogues (first considered by Montenegro in a different guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain decompositions (self-dual for type B ), whose rank-generating functions, zeta polynomials, rank-selected chain numbers have simple closed forms.

The cyclic sieving phenomenon

The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridges q = -1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Polya-Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field q-analogues.

Signed permutation statistics

Abstract We derive multivariate generating functions that count signed permutations by various statistics, using the hyperoactahedral generalization of methods of Garsia and Gessel. We also derive the distributions over inverse descent classes of signed permutations for two of these statistics individually (the major index and inversion number). These results show that, in contrast to the case for (unsigned) permutations, these two statistics are not generally equidistributed. We also discuss applications to statistics on the wreath product Ck ≀ Sn of a cyclic group with the symmetric group.

Noncrossing Partitions for the Group D n

The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W. It is a self-dual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2,...,n} defined by Kreweras in 1972 when W is the symmetric group Sn, and to its type B analogue defined by the second author in 1997 when W is the hyperoctahedral group. We give a combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type Dn, thus answering a question of Bessis. Using this description, we compute a number of fundamental enumerative invariants of this lattice, such as the rank sizes, number of maximal chains, and Mobius function. We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B, and C. This leads to a (case-by-case) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to W-orbits.

On the Charney-Davis and Neggers-Stanley conjectures

For a graded naturally labelled poset P, it is shown that the P-Eulerian polynomial W(P, t): = Σw ∈ L(P) tdes(w) counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the Neggers-Stanley conjecture on real zeroes for W(P, t) in these cases. The result is deduced from McMullens g-Theorem, by exhibiting a simplicial polytopal sphere whose h-polynomial is W(P, t).Whenever this simplicial sphere turns out to be flag, that is, its minimal non-faces all have cardinality two, it is shown that the Neggers-Stanley Conjecture would imply the Charney-Davis Conjecture for this sphere. In particular, it is shown that the sphere is flag whenever the poset P has width at most 2. In this case, the sphere is shown to have a stronger geometric property (local convexity), which then implies the Charney-Davis Conjecture in this case via a result from Leung and Reiner (Duke Math. J. 111 (2002) 253).It is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras, and some evidence is presented.

A Convolution Formula for the Tutte Polynomial

Let M be a finite matroid with rank function r. We will write A M when we mean that A is a subset of the ground set of M, and write M|A and M A for the matroids obtained by restricting M to A and contracting M on A respectively. Let M* denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial TM(x, y) satisfies TM(x, y)= : A M TM|A(0, y) TM A(x, 0). (1)

Free arrangements and rhombic tilings

AbstractLet Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:(1)When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?(2)When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels? Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.

Shifted simplicial complexes are Laplacian integral

We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.

Cohomology of Smooth Schubert Varieties in Partial Flag Manifolds

The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borels presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.