# 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 ${\displaystyle C_{p_{i}^{\lambda }}}$ where ${\displaystyle \lambda =(\lambda _{1},\lambda _{2},\ldots )}$ is a partition of ${\displaystyle n}$ called the type of M. Let ${\displaystyle g_{\mu ,\nu }^{\lambda }(p)}$ be the number of subgroups N of M such that N has type ${\displaystyle \nu }$ and the quotient M/N has type ${\displaystyle \mu }$. Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.

Hall next constructs an algebra ${\displaystyle H(p)}$ with symbols ${\displaystyle u_{\lambda }}$ a generators and multiplication given by the ${\displaystyle g_{\mu ,\nu }^{\lambda }}$ as structure constants

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

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