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[edit]

Suppose G is a finite group with generating sequence X = \{x_1,x_2,...,x_r\} which acts on the finite set \Omega = \{1,2,...,n\}. A common task in computational group theory is to compute the orbit of some element \omega \in \Omega under G. At the same time, one can record a Schreier vector for \omega. This vector can then be used to find an element g \in G satisfying \omega^g = \alpha, for any \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[edit]

All variables used here are defined in the overview.

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

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

Use in algorithms[edit]

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

References[edit]