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 λ ( x 1 , … , x n ; t ) = ( ∏ i ≥ 0 ∏ j = 1 m ( i ) 1 − t 1 − t j ) ∑ w ∈ S n w ( x 1 λ 1 ⋯ x n λ n ∏ i < j x i − t x j x i − x j ) , {\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 λi, and m(i) elements equal to i, and Sn 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 {\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 P λ ( x ; 1 ) = m λ ( x ) {\displaystyle P_{\lambda }(x;1)=m_{\lambda }(x)} , P λ ( x ; 0 ) = s λ ( x ) {\displaystyle P_{\lambda }(x;0)=s_{\lambda }(x)} and
P λ ( x ; − 1 ) = P λ ( x ) {\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
s λ ( x ) = ∑ μ K λ μ ( t ) P μ ( x , t ) {\displaystyle s_{\lambda }(x)=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x,t)}
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