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 $\lambda =({\lambda }_1,{\lambda }_2,\dots )$ 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

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