Math.Algebras.GroupAlgebra

A module for doing arithmetic in the group algebra.

Group elements are represented as permutations of the integers, and are entered and displayed using a Haskell-friendly version of cycle notation. For example, the permutation (1 2 3)(4 5) would be entered as p [[1,2,3],[4,5]], and displayed as [[1,2,3],[4,5]].

Given a field K and group G, the group algebra KG is the free K-vector space over the elements of G. Elements of the group algebra consist of arbitrary K-linear combinations of elements of G. For example, p [[1,2,3]] + 2 * p [[1,2],[3,4]]

Documentation

type GroupAlgebra k = Vect k (Permutation Int)

p :: [[Int]] -> GroupAlgebra Q

Construct a permutation, as an element of the group algebra, from a list of cycles. For example, p [[1,2],[3,4,5]] constructs the permutation (1 2)(3 4 5), which is displayed as [[1,2],[3,4,5]].