1,921 research outputs found
Shifted symmetric functions and multirectangular coordinates of Young diagrams
In this paper, we study shifted Schur functions , as well as a
new family of shifted symmetric functions linked to Kostka
numbers. We prove that both are polynomials in multi-rectangular coordinates,
with nonnegative coefficients when written in terms of falling factorials. We
then propose a conjectural generalization to the Jack setting. This conjecture
is a lifting of Knop and Sahi's positivity result for usual Jack polynomials
and resembles recent conjectures of Lassalle. We prove our conjecture for
one-part partitions.Comment: 2nd version: minor modifications after referee comment
Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit
This paper surveys eight classes of polynomials associated with -type and
-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and
interpolation (or shifted) Jack and Macdonald polynomials and their -type
extensions. Among these the -type interpolation Jack polynomials were
probably unobserved until now. Much emphasis is put on combinatorial formulas
and binomial formulas for (most of) these polynomials. Possibly new results
derived from these formulas are a limit from Koornwinder to Macdonald
polynomials, an explicit formula for Koornwinder polynomials in two variables,
and a combinatorial expression for the coefficients of the expansion of
-type Jacobi polynomials in terms of Jack polynomials which is different
from Macdonald's combinatorial expression. For these last coefficients in the
two-variable case the explicit expression in Koornwinder & Sprinkhuizen (1978)
is now obtained in a quite different way.Comment: v5: 27 pages, formulas (10.7) and (10.14) correcte
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