Hall–Littlewood polynomials

The Hall–Littlewood polynomial P is defined by

${\displaystyle P_{\lambda }(x_{1},\ldots ,x_{n};t)=\left(\prod _{i\geq 0}\prod _{j=1}^{m(i)}{\frac {1-t}{1-t^{j}}}\right){\sum _{w\in S_{n}}w\left(x_{1}^{\lambda _{1}}\cdots x_{n}^{\lambda _{n}}\prod _{i<j}{\frac {x_{i}-tx_{j}}{x_{i}-x_{j}}}\right)},}$

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,

${\displaystyle P_{42}(x_{1},x_{2};t)=x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}+(1-t)x_{1}^{3}x_{2}^{3}}$

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

${\displaystyle s_{\lambda }(x)=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x,t)}$

where ${\displaystyle K_{\lambda \mu }(t)}$ are the Kostka–Foulkes polynomials. Note that as ${\displaystyle t=1}$, these reduce to the ordinary Kostka coefficients.

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

${\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{\mathrm {charge} (T)}}$

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.

• Hall polynomial

• 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.

• Weisstein, Eric W. "Hall–Littlewood Polynomial". MathWorld.