191 research outputs found
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1
whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We
present a unifying perspective on ASMs and other combinatorial objects by
studying a certain tetrahedral poset and its subposets. We prove the order
ideals of these subposets are in bijection with a variety of interesting
combinatorial objects, including ASMs, totally symmetric self-complementary
plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux,
Catalan objects, tournaments, and totally symmetric plane partitions. We prove
product formulas counting these order ideals and give the rank generating
function of some of the corresponding lattices of order ideals. We also prove
an expansion of the tournament generating function as a sum over TSSCPPs. This
result is analogous to a result of Robbins and Rumsey expanding the tournament
generating function as a sum over alternating sign matrices.Comment: 24 pages, 23 figures, full published version of arXiv:0905.449
Non-commutative Pieri operators on posets
We consider graded representations of the algebra NC of noncommutative
symmetric functions on the Z-linear span of a graded poset P. The matrix
coefficients of such a representation give a Hopf morphism from a Hopf algebra
HP generated by the intervals of P to the Hopf algebra of quasi-symmetric
functions. This provides a unified construction of quasi-symmetric generating
functions from different branches of algebraic combinatorics, and this
construction is useful for transferring techniques and ideas between these
branches. In particular we show that the (Hopf) algebra of Billera and Liu
related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge
related to enriched P-partitions, and connect this to the combinatorics of the
Schubert calculus for isotropic flag manifolds.Comment: LaTeX 2e, 22 pages Minor corrections, updated references. Complete
and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to
G.-C. Rot
Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
We prove that the noncrossing partition lattices associated with the complex
reflection groups for admit symmetric decompositions
into Boolean subposets. As a result, these lattices have the strong Sperner
property and their rank-generating polynomials are symmetric, unimodal, and
-nonnegative. We use computer computations to complete the proof that
every noncrossing partition lattice associated with a well-generated complex
reflection group is strongly Sperner, thus answering affirmatively a question
raised by D. Armstrong.Comment: 30 pages, 5 figures, 1 table. Final version. The results of the
initial version were extended to symmetric Boolean decompositions of
noncrossing partition lattice
Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties
In the computation of the intersection cohomology of Shimura varieties, or of
the cohomology of equal rank locally symmetric spaces, combinatorial
identities involving averaged discrete series characters of real reductive
groups play a large technical role. These identities can become very
complicated and are not always well-understood (see for example the appendix of
[8]). We propose a geometric approach to these identities in the case of Siegel
modular varieties using the combinatorial properties of the Coxeter complex of
the symmetric group. Apart from some introductory remarks about the origin of
the identities, our paper is entirely combinatorial and does not require any
knowledge of Shimura varieties or of representation theory.Comment: 17 pages, 1 figure; to appear in Algebraic Combinatoric
Discrete Morse theory for totally non-negative flag varieties
In a seminal 1994 paper, Lusztig extended the theory of total positivity by
introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary
(generalized, partial) flag variety G/P. He referred to this space as a
"remarkable polyhedral subspace", and conjectured a decomposition into cells,
which was subsequently proven by the first author. Subsequently the second
author made the concrete conjecture that this cell decomposed space is the next
best thing to a polyhedron, by conjecturing it to be a regular CW complex that
is homeomorphic to a closed ball. In this article we use discrete Morse theory
to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that
the boundaries of the cells are homotopic to spheres, and the closures of cells
are contractible. The latter part generalizes a result of Lusztig's that
(G/P)_{\geq 0} -- the closure of the top-dimensional cell -- is contractible.
Concerning our result on the boundaries of cells, even the special case that
the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere,
is new for all G/P other than projective space.Comment: 30 page
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