# Hall–Littlewood polynomials

In mathematics, the **Hall–Littlewood polynomials** are symmetric functions depending on a parameter *t* and a partition λ. They are Schur functions when *t* is 0 and monomial symmetric functions when *t* is 1 and are special cases of Macdonald polynomials.
They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961).

## Definition

The Hall–Littlewood polynomial *P* is defined by

where λ is a partition of at most *n* with elements λ_{i}, and *m*(*i*) elements equal to *i*, and *S*_{n} is the symmetric group of order *n*!.

As an example,

### Specializations

We have that , and
where the latter is the Schur *P* polynomials.

## Properties

Expanding the Schur polynomials in terms of the Hall–Littlewood polynomials, one has

where are the Kostka–Foulkes polynomials. Note that as , these reduce to the ordinary Kostka coefficients.

A combinatorial description for the Kostka–Foulkes polynomials was given by Lascoux and Schützenberger,

where "charge" is a certain combinatorial statistic on semistandard Young tableaux,
and the sum is taken over all semi-standard Young tableaux with shape *λ* and type *μ*.

## See also

## References

- I.G. Macdonald (1979).
*Symmetric Functions and Hall Polynomials*. Oxford University Press. pp. 101–104. ISBN 0-19-853530-9. - D.E. Littlewood (1961). "On certain symmetric functions".
*Proceedings of the London Mathematical Society*.**43**: 485–498. doi:10.1112/plms/s3-11.1.485.