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 $C_{p_{i}^{\lambda }}$ where $\lambda =(\lambda _{1},\lambda _{2},\ldots )$ is a partition of $n$ called the type of M. Let $g_{\mu ,\nu }^{\lambda }(p)$ be the number of subgroups N of M such that N has type $\nu$ and the quotient M/N has type $\mu$. 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 $u_{\lambda }$ a generators and multiplication given by the $g_{\mu ,\nu }^{\lambda }$ as structure constants

$u_{\mu }u_{\nu }=\sum _{\lambda }g_{\mu ,\nu }^{\lambda }u_{\lambda }$

which is freely generated by the $u_{\mathbf {1} _{n}}$ corresponding to the elementary p-groups. The map from $H(p)$ to the algebra of symmetric functions $e_{n}$ given by $u_{\mathbf {1} _{n}}\mapsto p^{-n(n-1)}e_{n}$ 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