This paper deals with a formula satisfied by ‘r-reduced’
Schur functions. Schur functions originally appear as irreducible characters of general linear
group over the complex number field. In this paper they are considered as weighted homogeneous
polynomials with respect to the power sum symmetric functions. More precisely, for
a Young diagram λ of size n, the Schur function indexed by λ reads
where χλ(v) is the character value of the
irreducible representation Sλ of the group
algebra ℚ[Sfr ]n, evaluated at the conjugacy class of the
cycle type v =
Setting tjr = 0 for j = 1, 2, … in
Sλ(t), we have the r-reduced Schur function
The set of all r-reduced Schur functions spans the polynomial ring
P(r) = ℚ[tj;
j[nequiv ]0 (mod r)]. We show that a good choice
of basis elements leads to an explicit
description of all other r-reduced Schur functions involving the
The formula has not only a purely combinatorial meaning, but also nice
implications in two different fields. One is about the basic representation of the affine
Lie algebra A(1)r−1.
We show that the basis in the main theorem gives in turn a weight
basis of the basic A(1)r−1-module realised in
P(r). The other implication is about modular
representations of the symmetric group. Our explicit formula implies that the
determination of the decomposition matrices reduces to that for the basic set we give
in this paper.
The paper is organised as follows. In Section 1 we introduce generalised Maya
diagrams and associated r-reduced Schur functions. In Section 2 we discuss
combinatorics of Young diagrams. Section 3 is devoted to the main theorem. In
Section 4 we describe weight vectors of the basic A(1)r−1-module.
In Section 5 the formula is translated into that in the modular representation theory.