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).
The Hall–Littlewood polynomial P is defined by
where λ is a partition of at most n with elements λi, and m(i) elements equal to i, and Sn is the symmetric group of order n!.
As an example,
We have that , and where the latter is the Schur P polynomials.
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.
- 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.