A finite abelian p-group M is a direct sum of cyclic p-power components where is a partition of called the type of M. Let be the number of subgroups N of M such that N has type and the quotient M/N has type . Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.
Hall next constructs an algebra with symbols a generators and multiplication given by the as structure constants
which is freely generated by the corresponding to the elementary p-groups. The map from to the algebra of symmetric functions given by is a homomorphism and its image may be interpreted as the Hall-Littlewood polynomial functions. The theory of Schur functions is thus closely connected with the theory of Hall polynomials.
- I.G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9