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In mathematics, a Molien series is a generating function attached to a linear representation ρ of a group ''G'' on a finite-dimensional vector space ''V''. It counts the homogeneous polynomials of a given total degree ''d'' that are invariants for ''G''. It is named for Theodor Molien. ==Formulation== More formally, there is a vector space of such polynomials, for each given value of ''d'' = 0, 1, 2, ..., and we write ''n''''d'' for its vector space dimension, or in other words the number of linearly independent homogeneous invariants of a given degree. In more algebraic terms, take the ''d''-th symmetric power of ''V'', and the representation of ''G'' on it arising from ρ. The invariants form the subspace consisting of all vectors fixed by all elements of ''G'', and ''n''''d'' is its dimension. The Molien series is then by definition the formal power series : This can be looked at another way, by considering the representation of ''G'' on the symmetric algebra of ''V'', and then the whole subalgebra ''R'' of ''G''-invariants. Then ''n''''d'' is the dimension of the homogeneous part of ''R'' of dimension ''d'', when we look at it as graded ring. In this way a Molien series is also a kind of Hilbert function. Without further hypotheses not a great deal can be said, but assuming some conditions of finiteness it is then possible to show that the Molien series is a rational function. The case of finite groups is most often studied. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Molien series」の詳細全文を読む スポンサード リンク
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