50 research outputs found
Eulerian numbers, tableaux, and the Betti numbers of a toric variety
Let [Sigma] denote the Coxeter complex of Sn, and let X([Sigma]) denote the associated toric variety. Since the Betti numbers of the cohomology of X([Sigma]) are Eulerian numbers, the additional presence of an Sn-module structure permits the definition of an isotypic refinement of these numbers. In some unpublished work, DeConcini and Procesi derived a recurrence for the Sn-character of the cohomology of X([Sigma]), and Stanley later used this to translate the problem of combinatorially describing the isotypic Eulerian numbers into the language of symmetric functions. In this paper, we explicitly solve this problem by developing a new way to use marked sequences to encode permutations. This encoding also provides a transparent explanation of the unimodality of Eulerian numbers and their isotypic refinements.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30110/1/0000482.pd
Equivariant Ehrhart theory
Motivated by representation theory and geometry, we introduce and develop an
equivariant generalization of Ehrhart theory, the study of lattice points in
dilations of lattice polytopes. We prove representation-theoretic analogues of
numerous classical results, and give applications to the Ehrhart theory of
rational polytopes and centrally symmetric polytopes. We also recover a
character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of
a Weyl group on the cohomology of a toric variety associated to a root system.Comment: 40 pages. Final version. To appear in Adv. Mat
Chromatic quasisymmetric functions
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric
function. We derive refinements of both Gasharov's Schur-basis expansion of the
chromatic symmetric function and Chow's expansion in Gessel's basis of
fundamental quasisymmetric functions. We present a conjectural refinement of
Stanley's power sum basis expansion, which we prove in special cases. We
describe connections between the chromatic quasisymmetric function and both the
-Eulerian polynomials introduced in our earlier work and, conjecturally,
representations of symmetric groups on cohomology of regular semisimple
Hessenberg varieties, which have been studied by Tymoczko and others. We
discuss an approach, using the results and conjectures herein, to the
-positivity conjecture of Stanley and Stembridge for incomparability graphs
of -free posets.Comment: 57 pages; final version, to appear in Advances in Mat
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page
The Eulerian Distribution on Involutions is Indeed Unimodal
Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point
free involutions) of {1,...,n} with k descents. Motivated by Brenti's
conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is
log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal
in k, for all n. Furthermore, we conjecture that there are nonnegative integers
a_{n,k} such that This statement is stronger than
the unimodality of I_{n,k} but is also interesting in its own right.Comment: 12 pages, minor changes, to appear in J. Combin. Theory Ser.