Schur functions: Theme and variations.

Abstract

this article we shall survey various generalizations, analogues and deformations of Schur functions --- some old, some new --- that have been proposed at various times. We shall present these as a sequence of variations on a theme and (unlike e.g. Bourbaki) we shall proceed from the particular to the general. Thus Variations 1 and 2 are included in Variation 3 ; Variations 4 and 5 are particular cases of Variation 6 ; and in their turn Variations 6, 7 and 8 (in part) are included in Variation 9. To introduce our theme, we recall [M 1 , Ch. I, x 3] that the Schur function s (x 1 ; : : : ; xn ) (where x 1 , : : : , xn are independent indeterminates and = ( 1 ; : : : ; n ) is a partition of length n) may be defined as the quotient of two alternants : (0:1) s (x 1 ; : : : ; xn ) = det