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| In [[mathematics]], especially the field of [[computational group theory]], a '''Schreier vector''' is a tool for reducing the time and space complexity required to calculate [[orbit (group theory)|orbits]] of a [[permutation group]].
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| ==Overview==
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| Suppose G is a finite [[Group (mathematics)|group]] with generating sequence <math>X = \{x_1,x_2,...,x_r\}</math> which [[Group action|acts]] on the finite set <math>\Omega = \{1,2,...,n\}</math>. A common task in computational group theory is to compute the [[Orbit (group theory)|orbit]] of some element <math>\omega \in \Omega</math> under G. At the same time, one can record a Schreier vector for <math>\omega</math>. This vector can then be used to find an element <math>g \in G</math> satisfying <math>\omega^g = \alpha</math>, for any <math>\alpha \in \omega^G</math>. Use of Schreier vectors to perform this requires less storage space and time complexity than storing these g explicitly.
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| ==Formal definition==
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| All variables used here are defined in the overview.
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| A Schreier vector for <math>\omega \in \Omega</math> is a vector <math>\mathbf{v} = (v[1],v[2],...,v[n])</math> such that:
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| # <math>v[\omega] = -1</math>
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| # For <math>\alpha \in \omega^G \setminus \{ {\omega} \} , v[\alpha] \in \{1,...,r\}</math> (the manner in which the <math>v[\alpha]</math> are chosen will be made clear in the next section)
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| # <math>v[\alpha] = 0</math> for <math>\alpha \notin \omega^G</math>
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| ==Use in algorithms==
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| Here we illustrate, using [[pseudocode]], the use of Schreier vectors in two algorithms
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| * Algorithm to compute the orbit of ''ω'' under ''G'' and the corresponding Schreier vector
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| :Input: ''ω'' in ''Ω'', <math>X = \{x_1,x_2,...,x_r\}</math>
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| :for ''i'' in { 0, 1, …, ''n'' }:
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| ::set ''v''[''i''] = 0
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| :set ''orbit'' = { ''ω'' }, ''v''[''ω''] = −1
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| :for ''α'' in ''orbit'' and ''i'' in { 1, 2, …, ''r'' }:
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| ::if <math>\alpha^{x_i}</math> is not in ''orbit'':
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| :::append <math>\alpha^{x_i}</math> to ''orbit''
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| :::set <math>v[\alpha^{x_i}] = i</math>
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| :return ''orbit'', ''v''
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| * Algorithm to find a ''g'' in ''G'' such that ''ω''<sup>''g''</sup> = ''α'' for some ''α'' in ''Ω'', using the ''v'' from the first algorithm
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| :Input: ''v'', ''α'', ''X''
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| :if ''v''[''α''] = 0:
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| ::return false
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| :set ''g'' = ''e'', and ''k'' = ''v''[''α''] (where ''e'' is the identity element of ''G'')
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| :while ''k'' ≠ −1:
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| ::set <math>g = {x_k}g, \alpha = \alpha^{x_k^{-1}}, k = v[\alpha]</math>
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| :return ''g''
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| ==References==
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| {{refimprove|date=February 2008}}
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| *{{Citation | last1=Butler | first1=G. | title=Fundamental algorithms for permutation groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Computer Science | isbn=978-3-540-54955-0 | id={{MathSciNet | id = 1225579}} | year=1991 | volume=559}}
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| *{{Citation | last1=Holt | first1=Derek F. | title=A Handbook of Computational Group Theory | publisher=[[CRC Press]] | location=London | isbn=978-1-58488-372-2 | year=2005}}
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| *{{Citation | last1=Seress | first1=Ákos | title=Permutation group algorithms | publisher=[[Cambridge University Press]] | series=Cambridge Tracts in Mathematics | isbn=978-0-521-66103-4 | id={{MathSciNet | id = 1970241}} | year=2003 | volume=152}}
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| [[Category:Computational group theory]]
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| [[Category:Permutation groups]]
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Nice to meet you, my title is Refugia. To play baseball is the hobby he will by no means quit doing. Managing people has been his working day occupation for a whilst. South Dakota is where I've usually been residing.
Feel free to surf to my web page; exchange.craftemergency.org