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 $$C_{p^\lambda_i}$$ 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 $$\nu$$ 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 $$g^\lambda_{\mu,\nu}$$ 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.