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 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.