124 research outputs found
Powers of the Vandermonde determinant, Schur Functions, and recursive formulas
Since every even power of the Vandermonde determinant is a symmetric
polynomial, we want to understand its decomposition in terms of the basis of
Schur functions. We investigate several combinatorial properties of the
coefficients in the decomposition. In particular, we give recursive formulas
for the coefficient of the Schur function s_{\m} in the decomposition of an
even power of the Vandermonde determinant in variables in terms of the
coefficient of the Schur function s_{\l} in the decomposition of the same
even power of the Vandermonde determinant in variables if the Young diagram
of \m is obtained from the Young diagram of \l by adding a tetris type
shape to the top or to the left. An extended abstract containing the statement
of the results presented here appeared in the Proceedings of FPSAC11Comment: 23 pages; extended abstract appeared in the Proceedings of FPSAC1
Special Isogenies and Tensor Product Multiplicities
We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann’s Path Model, and geometrically, using isogenies between algebraic groups defined over an algebraically closed field of positive characteristic
Generalized Involution Models for Wreath Products
We prove that if a finite group has a generalized involution model, as
defined by Bump and Ginzburg, then the wreath product also has a
generalized involution model. This extends the work of Baddeley concerning
involution models for wreath products. As an application, we construct a
Gelfand model for wreath products of the form with abelian, and
give an alternate proof of a recent result due to Adin, Postnikov, and Roichman
describing a particularly elegant Gelfand model for the wreath product \ZZ_r
\wr S_n. We conclude by discussing some notable properties of this
representation and its decomposition into irreducible constituents, proving a
conjecture of Adin, Roichman, and Postnikov's.Comment: 29 page
Coloured peak algebras and Hopf algebras
For a finite abelian group, we study the properties of general
equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of with
the symmetric group \SG_n, also known as the -coloured symmetric group. We
show that under certain conditions, some equivalence relations give rise to
subalgebras of \k G_n as well as graded connected Hopf subalgebras of
\bigoplus_{n\ge o} \k G_n. In particular we construct a -coloured peak
subalgebra of the Mantaci-Reutenauer algebra (or -coloured descent algebra).
We show that the direct sum of the -coloured peak algebras is a Hopf
algebra. We also have similar results for a -colouring of the Loday-Ronco
Hopf algebras of planar binary trees. For many of the equivalence relations
under study, we obtain a functor from the category of finite abelian groups to
the category of graded connected Hopf algebras. We end our investigation by
describing a Hopf endomorphism of the -coloured descent Hopf algebra whose
image is the -coloured peak Hopf algebra. We outline a theory of
combinatorial -coloured Hopf algebra for which the -coloured
quasi-symmetric Hopf algebra and the graded dual to the -coloured peak Hopf
algebra are central objects.Comment: 26 pages latex2
The Tchebyshev transforms of the first and second kind
We give an in-depth study of the Tchebyshev transforms of the first and
second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform
(of the first kind) preserves desirable combinatorial properties, including
Eulerianess (due to Hetyei) and EL-shellability. It is also a linear
transformation on flag vectors. When restricted to Eulerian posets, it
corresponds to the Billera, Ehrenborg and Readdy omega map of oriented
matroids. One consequence is that nonnegativity of the cd-index is maintained.
The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on
the space of quasisymmetric functions QSym. It coincides with Stembridge's peak
enumerator for Eulerian posets, but differs for general posets. The complete
spectrum is determined, generalizing work of Billera, Hsiao and van
Willigenburg.
The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's
classical quasisymmetric function of a poset, this map is a comodule morphism
with respect to the quasisymmetric functions QSym.
Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform,
and the Tchebyshev transforms motivate a general study of chain maps. One such
occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on
the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and
Sottile's result on the terminal object in the category of combinatorial Hopf
algebras. In contrast, the chain map of the first kind is both an algebra map
and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page
Crystal Graphs and -Analogues of Weight Multiplicities for the Root System
We give an expression of the -analogues of the multiplicities of weights
in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal
graph attached to the corresponding U_q(\sl_{n+1})-modules. As an
application, we describe multivariate polynomial analogues of the
multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to
appear in Lett. Math. Phy
The impact of chronic endurance and resistance training upon the right ventricular phenotype in male athletes
Objectives
The traditional view of differential left ventricular adaptation to training type has been questioned. Right ventricular (RV) data in athletes are emerging but whether training type mediates this is not clear. The primary aim of this study was to evaluate the RV phenotype in endurance- vs. resistance-trained male athletes. Secondary aims included comparison of RV function in all groups using myocardial speckle tracking, and the impact of allometric scaling on RV data interpretation.
Methods
A prospective cross-sectional design assessed RV structure and function in 19 endurance-trained (ET), 21 resistance-trained (RT) and 21 sedentary control subjects (CT). Standard 2D tissue Doppler imaging and speckle tracking echocardiography assessed RV structure and function. Indexing of RV structural parameters to body surface area (BSA) was undertaken using allometric scaling.
Results
A higher absolute RV diastolic area was observed in ET (mean ± SD: 27 ± 4 cm2) compared to CT (22 ± 4 cm2; P < 0.05) that was maintained after scaling. Whilst absolute RV longitudinal dimension was greater in ET (88 ± 9 mm) than CT (81 ± 10 mm; P < 0.05), this difference was removed after scaling. Wall thickness was not different between ET and RT and there were no between group differences in global or regional RV function.
Conclusion
We present some evidence of RV adaptation to chronic ET in male athletes but limited structural characteristics of an athletic heart were observed in RT. Global and regional RV functions were comparable between groups. Allometric scaling altered data interpretation in some variables
Vicious walkers, friendly walkers and Young tableaux II: With a wall
We derive new results for the number of star and watermelon configurations of
vicious walkers in the presence of an impenetrable wall by showing that these
follow from standard results in the theory of Young tableaux, and combinatorial
descriptions of symmetric functions. For the problem of -friendly walkers,
we derive exact asymptotics for the number of stars and watermelons both in the
absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the
statement of Theorem 4 and its proof were correcte
Scaling limit of vicious walks and two-matrix model
We consider the diffusion scaling limit of the one-dimensional vicious walker
model of Fisher and derive a system of nonintersecting Brownian motions. The
spatial distribution of particles is studied and it is described by use of
the probability density function of eigenvalues of Gaussian random
matrices. The particle distribution depends on the ratio of the observation
time and the time interval in which the nonintersecting condition is
imposed. As is going on from 0 to 1, there occurs a transition of
distribution, which is identified with the transition observed in the
two-matrix model of Pandey and Mehta. Despite of the absence of matrix
structure in the original vicious walker model, in the diffusion scaling limit,
accumulation of contact repulsive interactions realizes the correlated
distribution of eigenvalues in the multimatrix model as the particle
distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio
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