en>Jfmantis: Cleaned up using AutoEd

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en>Philip Trueman: Reverted edits by Ryanne jwood13 (talk) to last version by A3 nm

2013-04-09T17:25:35Z

Reverted edits by Ryanne jwood13 (talk) to last version by A3 nm

← Older revision		Revision as of 19:25, 9 April 2013
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	~~Nice to meet you~~, ~~my title~~ is ~~Figures Held though I don~~'~~t truly like being called like~~ that. ~~Puerto Rico~~ is ~~exactly where he~~ and ~~his wife live~~. ~~To gather coins~~ is ~~what his family~~ and ~~him enjoy~~. ~~Since she was 18 she~~'~~s been operating~~ as a ~~receptionist~~ but ~~her promotion~~ by ~~no means arrives~~.<br><br>~~Feel free~~ to ~~visit my web blog~~ ... [http://www.~~pponline.Co~~.uk/~~user~~/~~miriamlinswkucrd at home std test~~]		In [[mathematics]], a '''Molien series''' is a [[generating function]] attached to a [[linear representation]] ρ of a [[group (mathematics)\|group]] ''G'' on a [[finite-dimensional vector space]] ''V''. It counts the [[homogeneous polynomial]]s of a given [[total degree]] ''d'' that are [[Invariant (mathematics)\|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''<sub>''d''</sub> 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''<sub>''d''</sub> is its dimension.

			The Molien series is then by definition the [[formal power series]]

			:<math>M(t) = \sum_d n_d t^d.</math>

			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''<sub>''d''</sub> 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 group]]s is most often studied.

			==Formula==
			[[Theodor Molien\|Molien]] showed that
			:<math> M(t) = \frac{1}{\|G\|} \sum_{g\in G} \frac{1}{\det(I-tg)} </math>
			This means that the coefficient of ''t<sup>d</sup>'' in this series is the dimension ''n''<sub>''d''</sub> defined above.
			It assumes that the characteristic of the field does not divide \|''G''\| (but even without this assumption, Molien's formula in the form <math> \left\|G\right\| \cdot M\left(t\right) = \sum_{g\in G} \frac{1}{\det(I-tg)}</math> is valid, although it does not help with computing ''M(t)'').

			==Example==
			Consider <math>S_3</math> acting on '''R'''<sup>3</sup> by permuting the coordinates. Note that <math>\det(I-tg)</math> is constant on conjugacy classes, so it is enough to take one from each of the three classes in <math>S_3</math>; so
			<math>\det(I-te) = (1-t)^3,\det(I-t\sigma_2) = (1-t)(1-t^2)</math> and <math> \det(1-t\sigma_3) = (1-t^3)</math>
			where <math>\sigma_2=(1,2)</math> and <math>\sigma_3 = (1,2,3)</math>.

			Then
			:<math>
			M(t) = \frac16\left(\frac{1}{(1-t)^3} + \frac3{(1-t)(1-t^2)} + \frac{2}{1-t^3}\right)
			= \frac{1}{(1-t)(1-t^2)(1-t^3)}
			</math>

			==References==
			*David A. Cox, John B. Little, Donal O'Shea (2005), ''Using Algebraic Geometry'', pp. 295–8
			*{{cite journal
			\| last = Molien
			\| first = Th.
			\| author1-link= Theodor Molien
			\| year = 1897
			\| title = Uber die Invarianten der linearen Substitutionsgruppen.
			\| journal = Sitzungber. Konig. Preuss. Akad. Wiss. (J. Berl. Ber.)
			\| volume = 52
			\| pages = 1152–1156
			\| jfm = 28.0115.01
			\| url = http://books.google.com/?id=EIxK-opAmJYC&pg=PA1152
			}}
			*{{cite book
			\| last = Mukai
			\| first = S.
			\| year = 2002
			\| title = An introduction to invariants and moduli
			\| series = Cambridge Studies in Advanced Mathematics
			\| volume = 81
			\| isbn = 978-0-521-80906-1
			\| url = http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521809061
			}}

			[[Category:Invariant theory]]
			[[Category:Representation theory of groups]]

en>DoriSmith: Reverted edits by Mkarlesky (talk) to last version by Ehamberg

2012-07-14T05:36:27Z

Reverted edits by Mkarlesky (talk) to last version by Ehamberg

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Nice to meet you, my title is Figures Held though I don't truly like being called like that. Puerto Rico is exactly where he and his wife live. To gather coins is what his family and him enjoy. Since she was 18 she's been operating as a receptionist but her promotion by no means arrives.<br><br>Feel free to visit my web blog ... [http://www.pponline.Co.uk/user/miriamlinswkucrd at home std test]

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en>Jfmantis: Cleaned up using AutoEd

en>Philip Trueman: Reverted edits by Ryanne jwood13 (talk) to last version by A3 nm

en>DoriSmith: Reverted edits by Mkarlesky (talk) to last version by Ehamberg