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

$P_{\lambda }(x_1,\dots ,x_n;t)=\left(\prod _{i\ge 0}\prod _{j=1}^{m(i)}\frac{1-t}{1-t^j}\right)\sum _{w\in S_n}w(x_1^{{\lambda }_1}\cdots x_n^{{\lambda }_n}\prod _{i<j}\frac{x_i-t{x}_j}{x_i-x_j}),$

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,

$K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}{t}^{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}(T)}$

where "charge" is a certain combinatorial statistic on semistandard Young tableaux, and the sum is taken over the set $SSYT(\lambda ,\mu )$ of all semi-standard Young tableaux T with shape λ and type μ.

See also

• Hall polynomial