2,055 research outputs found
Schur polynomials, banded Toeplitz matrices and Widom's formula
We prove that for arbitrary partitions and integers the sequence of Schur
polynomials for sufficiently large, satisfy a
linear recurrence. The roots of the characteristic equation are given
explicitly. These recurrences are also valid for certain sequences of minors of
banded Toeplitz matrices.
In addition, we show that Widom's determinant formula from 1958 is a special
case of a well-known identity for Schur polynomials
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
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