Schreier vector

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:

2. For (the manner in which the are chosen will be made clear in the next section)
3. 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 ω^g = 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