Hall–Littlewood polynomial

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).

The Hall–Littlewood polynomial P is defined by

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,

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.

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