Hall–Littlewood polynomials
# 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

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

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,

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

### Specializations

We have that $P_{\lambda }(x;1)=m_{\lambda }(x)$, $P_{\lambda }(x;0)=s_{\lambda }(x)$ and $P_{\lambda }(x;-1)=P_{\lambda }(x)$ where the latter is the Schur P polynomials.

## Properties

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

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

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

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

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

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.

