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 which acts on the finite set . A common task in computational group theory is to compute the orbit of some element under G. At the same time, one can record a Schreier vector for . This vector can then be used to find an element satisfying , for any . 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 is a vector such that:
For (the manner in which the are chosen will be made clear in the next section) for
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,
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 is not in orbit: append to orbit set
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
return g
References