Schur Functions : Theme and Variations
نویسنده
چکیده
In 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 [M1, Ch. I, § 3] that the Schur function sλ(x1, . . . , xn) (where x1, . . . , xn are independent indeterminates and λ = (λ1, . . . , λn) is a partition of length ≤ n) may be defined as the quotient of two alternants :
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