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]].

Orphan instances

(Eq k, Num k) => HopfAlgebra k (Permutation Int)

Methods

antipode :: Vect k (Permutation Int) -> Vect k (Permutation Int)

(Eq k, Num k) => Bialgebra k (Permutation Int)

(Eq k, Num k) => Coalgebra k (Permutation Int)

Methods

counit :: Vect k (Permutation Int) -> k

comult :: Vect k (Permutation Int) -> Vect k (Tensor (Permutation Int) (Permutation Int))

(Eq k, Num k) => Algebra k (Permutation Int)

Methods

unit :: k -> Vect k (Permutation Int)

mult :: Vect k (Tensor (Permutation Int) (Permutation Int)) -> Vect k (Permutation Int)

(Eq k, Num k) => Module k (Permutation Int) Int

Methods

action :: Vect k (Tensor (Permutation Int) Int) -> Vect k Int

(Eq k, Num k) => Module k (Permutation Int) [Int]

Methods

action :: Vect k (Tensor (Permutation Int) [Int]) -> Vect k [Int]

HasInverses (GroupAlgebra Q)

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).

Methods

inverse :: GroupAlgebra Q -> GroupAlgebra Q