Hall polynomial

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 [math]\displaystyle{ C_{p^\lambda_i} }[/math] where [math]\displaystyle{ \lambda=(\lambda_1,\lambda_2,\ldots) }[/math] is a partition of [math]\displaystyle{ n }[/math] called the type of M. Let [math]\displaystyle{ g^\lambda_{\mu,\nu}(p) }[/math] be the number of subgroups N of M such that N has type [math]\displaystyle{ \nu }[/math] and the quotient M/N has type [math]\displaystyle{ \mu }[/math]. Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.

Hall next constructs an algebra [math]\displaystyle{ H(p) }[/math] with symbols [math]\displaystyle{ u_\lambda }[/math] a generators and multiplication given by the [math]\displaystyle{ g^\lambda_{\mu,\nu} }[/math] as structure constants

[math]\displaystyle{ u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda }[/math]

which is freely generated by the [math]\displaystyle{ u_{\mathbf1_n} }[/math] corresponding to the elementary p-groups. The map from [math]\displaystyle{ H(p) }[/math] to the algebra of symmetric functions [math]\displaystyle{ e_n }[/math] given by [math]\displaystyle{ u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n }[/math] 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.

References

• I.G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9

{(math-stub}}