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 Littlewood (1961).

Definition

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

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,

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

Specializations

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

Properties

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

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

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 were 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.