Reduced Schur Functions and the Littlewood-Richardson Coefficients

Abstract

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 S λ ( t ) = ∑ v 1 + 2 v 2 + … = n χ λ ( v ) t 1 v 1 t 2 v 2 … v 1 ! v 2 ! … where χλ(v) is the character value of the irreducible representation Sλ of the group algebra QSn, evaluated at the conjugacy class of the cycle type v = ( 1 v 1 2 v 2 … n v n ) . Setting tjr = 0 for j = 1, 2, … in Sλ(t), we have the r-reduced Schur function S(r)λ(t). The set of all r-reduced Schur functions spans the polynomial ring P(r) = Q[tj; j≢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 Littlewood–Richardson coefficients. 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.