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Hall-Littlewood polynomial : ウィキペディア英語版
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 .
==Definition==
The Hall–Littlewood polynomial ''P'' is defined by
:P_\lambda(x_1,\ldots,x_n;t) = \left( \prod_ \prod_^ \fracw\left(x_1^\cdots x_n^\prod_\frac\right)},

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_(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

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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