HaskellForMaths-0.4.9: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Algebras.GroupAlgebra

Contents

Description

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]]

Synopsis

# Documentation

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

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]].

# Orphan instances

 (Eq k, Num k) => HopfAlgebra k (Permutation Int) Source # Instance details Methodsantipode :: Vect k (Permutation Int) -> Vect k (Permutation Int) Source # (Eq k, Num k) => Bialgebra k (Permutation Int) Source # Instance details (Eq k, Num k) => Coalgebra k (Permutation Int) Source # Instance details Methodscounit :: Vect k (Permutation Int) -> k Source #comult :: Vect k (Permutation Int) -> Vect k (Tensor (Permutation Int) (Permutation Int)) Source # (Eq k, Num k) => Algebra k (Permutation Int) Source # Instance details Methodsunit :: k -> Vect k (Permutation Int) Source #mult :: Vect k (Tensor (Permutation Int) (Permutation Int)) -> Vect k (Permutation Int) Source # (Eq k, Num k) => Module k (Permutation Int) Int Source # Instance details Methodsaction :: Vect k (Tensor (Permutation Int) Int) -> Vect k Int Source # (Eq k, Num k) => Module k (Permutation Int) [Int] Source # Instance details Methodsaction :: Vect k (Tensor (Permutation Int) [Int]) -> Vect k [Int] Source # Source # Note that the inverse of a group algebra element can only be efficiently calculated if the group generated by the non-zero terms is very small (eg <100 elements). Instance details Methods