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 $C_{p^\lambda_i}$ where $\lambda=(\lambda_1,\lambda_2,\ldots)$ is a partition of $n$ called the type of M. Let $g^\lambda_{\mu,\nu}(p)$ 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 $u λ$ a generators and multiplication given by the $g^\lambda_{\mu,\nu}$ as structure constants

$u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda$

which is freely generated by the $u_{\mathbf1_n}$ 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.