# 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
where
is a partition of called the *type* of *M*. Let be the number of
subgroups *N* of *M* such that *N* has type and the quotient *M/N* has type . Hall showed that the functions *g* are polynomial functions of *p* with integer coefficients: these are the *Hall polynomials*.

Hall next constructs an algebra with symbols a generators and multiplication given by the as structure constants

which is freely generated by the corresponding to the elementary *p*-groups.
The map from to the algebra of symmetric functions given
by 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