Christoffel–Darboux formula: Difference between revisions

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In [[mathematics]], the '''generalized symmetric group''' is the [[wreath product]] <math>S(m,n) := Z_m \wr S_n</math> of the [[cyclic group]] of order ''m'' and the [[symmetric group]] on ''n'' letters.
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== Examples ==
* For <math>m=1,</math> the generalized symmetric group is exactly the ordinary symmetric group: <math>S(1,n) = S_n.</math>
* For <math>m=2,</math> one can consider the cyclic group of order 2 as positives and negatives (<math>Z_2 \cong \{\pm 1\}</math>) and identify the generalized symmetric group <math>S(2,n)</math> with the [[signed symmetric group]].
 
== Representation theory ==
There is a natural representation of <math>S(m,n)</math> as [[generalized permutation matrices]], where the nonzero entries are ''m''th [[roots of unity]]: <math>Z_m \cong \mu_m.</math>
 
The representation theory has been studied since {{Harv|Osima|1954}}; see references in {{Harv|Can|1996}}. As with the symmetric group, the representations can be constructed in terms of [[Specht module]]s; see {{Harv|Can|1996}}.
 
== Homology ==
The first [[group homology]] group (concretely, the [[abelianization]]) is <math>Z_m \times Z_2</math> (for ''m'' odd this is isomorphic to <math>Z_{2m}</math>): the <math>Z_m</math> factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to <math>Z_m</math> (concretely, by taking the product of all the <math>Z_m</math> values), while the sign map on the symmetric group yields the <math>Z_2.</math> These are independent, and generate the group, hence are the abelianization.
 
The second homology group (in classical terms, the [[Schur multiplier]]) is given by {{Harv|Davies|Morris|1974}}:
:<math>H_2(S(2k+1,n)) = \begin{cases} 1 & n < 4\\
\mathbf{Z}/2 & n \geq 4.\end{cases}</math>
:<math>H_2(S(2k+2,n)) = \begin{cases} 1 & n = 0, 1\\
\mathbf{Z}/2 & n = 2\\
(\mathbf{Z}/2)^2 & n = 3\\
(\mathbf{Z}/2)^3 & n \geq 4.
\end{cases}</math>
Note that it depends on ''n'' and the sign of ''m:'' <math>H_2(S(2k+1,n)) \approx H_2(S(1,n))</math> and <math>H_2(S(2k+2,n)) \approx H_2(S(2,n)),</math> which are the Schur multipliers of the symmetric group and signed symmetric group.
 
== References ==
{{reflist}}
{{refbegin}}
* {{citation | first1 = J. W. | last1 = Davies | first2 = A. O. | last2 = Morris | title = The Schur Multiplier of the Generalized Symmetric Group | year = 1974 | journal = J. London Math. Soc. | series = 2 | volume = 8 | pages = 615–620 | url = http://jlms.oxfordjournals.org/cgi/reprint/s2-8/4/615.pdf}}
* {{citation | first = Himmet | last = Can | title = Representations of the Generalized Symmetric Groups | journal = Contributions to Algebra and Geometry | volume = 37 | year = 1996 | number = 2 | pages = 289–307 | url = http://www.emis.de/journals/BAG/vol.37/no.2/b37h2can.ps.gz }}
* {{citation | first = M. | last = Osima | title=On the representations of the generalized symmetric group | journal = Math. J. Okayama Univ. | volume = 4 | year = 1954 | pages = 39–54}}
{{refend}}
 
[[Category:Permutation groups]]

Latest revision as of 17:09, 12 March 2014

Idea of Specwog tactical combat knife originated from studying the distinctive manner and methods of fighting used in Dean Rostohar's Tactical Technical Combat System - T.T.C.S. (T.T.B.S.). A will need for a knife having special, distinctive type emerged. If you beloved this information in addition to you want to be given guidance relating to Cold Steel Recon Knife Reviews kindly check out our web site. Dean Rostohar decided to style the knife that brings the most effective results in combat situations.

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