Hall polynomial
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The Hall polynomials in mathematics were developed by Philip Hall in the 1950s in the study of group representations.
A finite abelian p-group M is a direct sum of cyclic p-power components 
where 
is a partition of n called the type of M. Let 
be the number of subgroups N of M such that N has type ν and the quotient M/N has type μ. Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.
Hall next constructs an algebra H(p) with symbols uλ 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 H(p) to the algebra of symmetric functions en 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.
[edit] References
- Ian G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9
