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Schreier vector

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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 orbits of a permutation group.

Overview

Suppose G is a finite group with generating sequence X = { x 1 , x 2 , . . . , x r } {\displaystyle X=\{x_{1},x_{2},...,x_{r}\}} which acts on the finite set Ω = { 1 , 2 , . . . , n } {\displaystyle \Omega =\{1,2,...,n\}} . A common task in computational group theory is to compute the orbit of some element ω ∈ Ω {\displaystyle \omega \in \Omega } under G. At the same time, one can record a Schreier vector for ω {\displaystyle \omega } . This vector can then be used to find an element g ∈ G {\displaystyle g\in G} satisfying ω g = α {\displaystyle \omega ^{g}=\alpha } , for any α ∈ ω G {\displaystyle \alpha \in \omega ^{G}} . Use of Schreier vectors to perform this requires less storage space and time complexity than storing these g explicitly.

Formal definition

All variables used here are defined in the overview.

A Schreier vector for ω ∈ Ω {\displaystyle \omega \in \Omega } is a vector v = ( v [ 1 ] , v [ 2 ] , . . . , v [ n ] ) {\displaystyle \mathbf {v} =(v,v,...,v)} such that:

  1. v [ ω ] = − 1 {\displaystyle v=-1}
  2. For α ∈ ω G ∖ { ω } , v [ α ] ∈ { 1 , . . . , r } {\displaystyle \alpha \in \omega ^{G}\setminus \{{\omega }\},v\in \{1,...,r\}} (the manner in which the v [ α ] {\displaystyle v} are chosen will be made clear in the next section)
  3. v [ α ] = 0 {\displaystyle v=0} for α ∉ ω G {\displaystyle \alpha \notin \omega ^{G}}

Use in algorithms

Here we illustrate, using pseudocode, the use of Schreier vectors in two algorithms

  • Algorithm to compute the orbit of ω under G and the corresponding Schreier vector
Input: ω in Ω, X = { x 1 , x 2 , . . . , x r } {\displaystyle X=\{x_{1},x_{2},...,x_{r}\}}
for i in { 0, 1, …, n }:
set v = 0
set orbit = { ω }, v = −1
for α in orbit and i in { 1, 2, …, r }:
if α x i {\displaystyle \alpha ^{x_{i}}} is not in orbit:
append α x i {\displaystyle \alpha ^{x_{i}}} to orbit
set v [ α x i ] = i {\displaystyle v=i}
return orbit, v
  • Algorithm to find a g in G such that ω = α for some α in Ω, using the v from the first algorithm
Input: v, α, X
if v = 0:
return false
set g = e, and k = v (where e is the identity element of G)
while k ≠ −1:
set g = x k g , α = α x k − 1 , k = v [ α ] {\displaystyle g={x_{k}}g,\alpha =\alpha ^{x_{k}^{-1}},k=v}
return g

References

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