40 research outputs found
On multiplicity-free skew characters and the Schubert Calculus
In this paper we classify the multiplicity-free skew characters of the
symmetric group. Furthermore we show that the Schubert calculus is equivalent
to that of skew characters in the following sense: If we decompose the product
of two Schubert classes we get the same as if we decompose a skew character and
replace the irreducible characters by Schubert classes of the `inverse'
partitions (Theorem 4.2).Comment: 14 pages, to appear in Annals. Comb. minor changes from v1 to v2 as
suggested by the referees, Example 3.4 inserted so numeration changed in
section
Equality of multiplicity free skew characters
In this paper we show that two skew diagrams lambda/mu and alpha/beta can
represent the same multiplicity free skew character [lambda/mu]=[alpha/beta]
only in the the trivial cases when lambda/mu and alpha/beta are the same up to
translation or rotation or if lambda=alpha is a staircase partition
lambda=(l,l-1,...,2,1) and lambda/mu and alpha/beta are conjugate of each
other.Comment: 16 pages, changes from v1 to v2: corrected the proof of Theorem 3.5
and some typos, changes from v2 to v3: minor layout change, enumeration
changed, to appear in J. Algebraic Combi
On principal hook length partitions and durfee sizes in skew characters
In this paper we construct for a given arbitrary skew diagram A all
partitions nu with maximal principal hook lengths among all partitions with the
character [nu] appearing in the skew character [A]. Furthermore we show that
these are also partitions with minimal Durfee size. This we use to give the
maximal Durfee size for [nu] appearing in [A] for the cases when A decays into
two partitions and for some special cases of A. Also this gives conditions for
two skew diagrams to represent the same skew character.Comment: 13 pages, minor changes from v1 to v2 as suggested by the referee, to
appear in Annals. Com
The f-vector of the descent polytope
For a positive integer n and a subset S of [n-1], the descent polytope DP_S
is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such
that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express
the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain
factorizations of the associated word over a two-letter alphabet to describe
the f-vector. We show that the f-vector is maximized when the set S is the
alternating set {1,3,5, ...}. We derive a generating function for the
f-polynomial F_S(t) of DP_S, written as a formal power series in two
non-commuting variables with coefficients in Z[t]. We also obtain the
generating function for the Ehrhart polynomials of the descent polytopes.Comment: 14 pages; to appear in Discrete & Computational Geometr
On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions
Given a finite simple graph \cG with vertices, we can construct the
Cayley graph on the symmetric group generated by the edges of \cG,
interpreted as transpositions. We show that, if \cG is complete multipartite,
the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in
terms of the irreducible characters of transpositions, and of the
Littlewood-Richardson coefficients. As a consequence we can prove that the
Laplacians of \cG and of \Cay(\cG) have the same first nontrivial
eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting
that the random walk and the interchange process have the same spectral gap,
holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version
in J. Algebraic Combi
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
A Model for the Development of the Rhizobial and Arbuscular Mycorrhizal Symbioses in Legumes and Its Use to Understand the Roles of Ethylene in the Establishment of these two Symbioses
We propose a model depicting the development of nodulation and arbuscular mycorrhizae. Both processes are dissected into many steps, using Pisum sativum L. nodulation mutants as a guideline. For nodulation, we distinguish two main developmental programs, one epidermal and one cortical. Whereas Nod factors alone affect the cortical program, bacteria are required to trigger the epidermal events. We propose that the two programs of the rhizobial symbiosis evolved separately and that, over time, they came to function together. The distinction between these two programs does not exist for arbuscular mycorrhizae development despite events occurring in both root tissues. Mutations that affect both symbioses are restricted to the epidermal program. We propose here sites of action and potential roles for ethylene during the formation of the two symbioses with a specific hypothesis for nodule organogenesis. Assuming the epidermis does not make ethylene, the microsymbionts probably first encounter a regulatory level of ethylene at the epidermis–outermost cortical cell layer interface. Depending on the hormone concentrations there, infection will either progress or be blocked. In the former case, ethylene affects the cortex cytoskeleton, allowing reorganization that facilitates infection; in the latter case, ethylene acts on several enzymes that interfere with infection thread growth, causing it to abort. Throughout this review, the difficulty of generalizing the roles of ethylene is emphasized and numerous examples are given to demonstrate the diversity that exists in plants