1,421 research outputs found
Multiplication Rules for Schur and Quasisymmetric Schur Functions
An important problem in algebraic combinatorics is finding expansions of products of symmetric functions as sums of symmetric functions. Schur functions form a well-known basis for the ring of symmetric functions. The Littlewood-Richardson rule was introduced to expand the product of two Schur functions as a positive sum of Schur functions. Remmel and Whitney introduced an algorithmic way to find the coefficients of Schur functions appearing in the expansion. Haglund et al. introduced quasisymmetric Schur functions as a refinement of Schur functions. For quasisymmetric Schur functions, the Littlewood-Richardson rule was introduced to expand the product of a Schur and quasisymmetric Schur function as the positive sum of quasisymmetric Schur functions. We determine an algorithm similar to the Remmel-Whitney rule to find the coefficients of quasisymmetric Schur functions appearing in the expansion
Non-Abelian Chern-Simons Particles in an External Magnetic Field
The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons
particles (non-Abelian anyons) in an external magnetic field are addressed. We
derive the N-body Hamiltonian in the (anti-)holomorphic gauge when the Hilbert
space is projected onto the lowest Landau level of the magnetic field. In the
presence of an additional harmonic potential, the N-body spectrum depends
linearly on the coupling (statistics) parameter. We calculate the second virial
coefficient and find that in the strong magnetic field limit it develops a
step-wise behavior as a function of the statistics parameter, in contrast to
the linear dependence in the case of Abelian anyons. For small enough values of
the statistics parameter we relate the N-body partition functions in the lowest
Landau level to those of SU(2) bosons and find that the cluster (and virial)
coefficients dependence on the statistics parameter cancels.Comment: 35 pages, revtex, 3 eps figures include
Skew Schubert functions and the Pieri formula for flag manifolds
We show the equivalence of the Pieri formula for flag manifolds and certain
identities among the structure constants, giving new proofs of both the Pieri
formula and of these identities. A key step is the association of a symmetric
function to a finite poset with labeled Hasse diagram satisfying a symmetry
condition. This gives a unified definition of skew Schur functions, Stanley
symmetric function, and skew Schubert functions (defined here). We also use
algebraic geometry to show the coefficient of a monomial in a Schubert
polynomial counts certain chains in the Bruhat order, obtaining a new
combinatorial construction of Schubert polynomials.Comment: 24 pages, LaTeX 2e, with epsf.st
Grand canonical partition functions for multi level para Fermi systems of any order
A general formula for the grand canonical partition function for a para Fermi
system of any order and of any number of levels is derived.Comment: 9 pages, latex, no figure
Raising operators and the Littlewood-Richardson polynomials
We use Young's raising operators to derive a Pieri rule for the ring
generated by the indeterminates given in Macdonald's 9th Variation of
the Schur functions. Under an appropriate specialisation of , we
derive the Pieri rule for the ring \La(a) of double symmetric functions,
which has a basis consisting of the double Schur functions. Together with a
suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us
to obtain a new proof of a rule to calculate the Littlewood--Richardson
polynomials, which gives a multiplication rule for the double Schur functions
Zonal polynomials via Stanley's coordinates and free cumulants
We study zonal characters which are defined as suitably normalized
coefficients in the expansion of zonal polynomials in terms of power-sum
symmetric functions. We show that the zonal characters, just like the
characters of the symmetric groups, admit a nice combinatorial description in
terms of Stanley's multirectangular coordinates of Young diagrams. We also
study the analogue of Kerov polynomials, namely we express the zonal characters
as polynomials in free cumulants and we give an explicit combinatorial
interpretation of their coefficients. In this way, we prove two recent
conjectures of Lassalle for Jack polynomials in the special case of zonal
polynomials.Comment: 45 pages, second version, important change
Universal optimality of Patterson's crossover designs
We show that the balanced crossover designs given by Patterson [Biometrika 39
(1952) 32--48] are (a) universally optimal (UO) for the joint estimation of
direct and residual effects when the competing class is the class of connected
binary designs and (b) UO for the estimation of direct (residual) effects when
the competing class of designs is the class of connected designs (which
includes the connected binary designs) in which no treatment is given to the
same subject in consecutive periods. In both results, the formulation of UO is
as given by Shah and Sinha [Unpublished manuscript (2002)]. Further, we
introduce a functional of practical interest, involving both direct and
residual effects, and establish (c) optimality of Patterson's designs with
respect to this functional when the class of competing designs is as in (b)
above.Comment: Published at http://dx.doi.org/10.1214/009053605000000723 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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