8,700 research outputs found
Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
We study the combinatorial structure of the irreducible characters of the
classical groups , ,
, and the
"non-classical" odd symplectic group , finding new
connections to the probabilistic model of Last Passage Percolation (LPP).
Perturbing the expressions of these characters as generating functions of
Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that
interpolate between characters of and and between characters of
and . We identify the first family as a
one-parameter specialization of Koornwinder polynomials, for which we thus
provide a novel combinatorial structure; on the other hand, the second family
appears to be new. We next develop a method of Gelfand-Tsetlin pattern
decomposition to establish identities between all these polynomials that, in
the case of characters, can be viewed as describing the decomposition of
irreducible representations of the groups when restricted to certain subgroups.
Through these formulas we connect orthogonal and symplectic characters, and
more generally the interpolating polynomials, to LPP models with various
symmetries, thus going beyond the link with classical Schur polynomials
originally found by Baik and Rains [BR01a]. Taking the scaling limit of the LPP
models, we finally provide an explanation of why the Tracy-Widom GOE and GSE
distributions from random matrix theory admit formulations in terms of both
Fredholm determinants and Fredholm Pfaffians.Comment: 60 pages, 11 figures. Typos corrected and a few remarks adde
Cumulants, lattice paths, and orthogonal polynomials
A formula expressing free cumulants in terms of the Jacobi parameters of the
corresponding orthogonal polynomials is derived. It combines Flajolet's theory
of continued fractions and Lagrange inversion. For the converse we discuss
Gessel-Viennot theory to express Hankel determinants in terms of various
cumulants.Comment: 11 pages, AMS LaTeX, uses pstricks; revised according to referee's
suggestions, in particular cut down last section and corrected some wrong
attribution
Linearization coefficients for orthogonal polynomials using stochastic processes
Given a basis for a polynomial ring, the coefficients in the expansion of a
product of some of its elements in terms of this basis are called linearization
coefficients. These coefficients have combinatorial significance for many
classical families of orthogonal polynomials. Starting with a stochastic
process and using the stochastic measures machinery introduced by Rota and
Wallstrom, we calculate and give an interpretation of linearization
coefficients for a number of polynomial families. The processes involved may
have independent, freely independent or q-independent increments. The use of
noncommutative stochastic processes extends the range of applications
significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier
and Rogers and continuous big q-Hermite polynomials. We also show that the
q-Poisson process is a Markov process.Comment: Published at http://dx.doi.org/10.1214/009117904000000757 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Double Schubert polynomials for the classical groups
For each infinite series of the classical Lie groups of type B,C or D, we
introduce a family of polynomials parametrized by the elements of the
corresponding Weyl group of infinite rank. These polynomials represent the
Schubert classes in the equivariant cohomology of the appropriate flag variety.
They satisfy a stability property, and are a natural extension of the (single)
Schubert polynomials of Billey and Haiman, which represent non-equivariant
Schubert classes. They are also positive in a certain sense, and when indexed
by maximal Grassmannian elements, or by the longest element in a finite Weyl
group, these polynomials can be expressed in terms of the factorial analogues
of Schur's Q- or P-functions defined earlier by Ivanov.Comment: 41 pages, 2 tables; comments welcom
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