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
![{\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)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af45e3e9c8849883a5e7bbe7039b00b783a20a7)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5c7b649a075c4d9342fe744558823c0dff20a2)
Specializations
We have that
,
and
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca154ee786f7d63ad79f98a6c9fd330f10061037)
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,
![{\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{\mathrm {charge} (T)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41a18c9937185b56ead52a56e4bcdbec0132bdb3)
where "charge" is a certain combinatorial statistic on semistandard Young tableaux,
and the sum is taken over the set
of all semi-standard Young tableaux T with shape λ and type μ.
See also
References
External links