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

Safe Haskell | None
Language | Haskell98

Math.Projects.ChevalleyGroup.Exceptional

Synopsis

- newtype Octonion k = O [(Int, k)]
- i0 :: Octonion Q
- i1 :: Octonion Q
- i2 :: Octonion Q
- i3 :: Octonion Q
- i4 :: Octonion Q
- i5 :: Octonion Q
- i6 :: Octonion Q
- fromList :: (Eq k, Num k) => [k] -> Octonion k
- toList :: Num a => Octonion a -> [a]
- expose :: Octonion k -> [(Int, k)]
- nf :: (Num b, Ord a, Ord b) => [(a, b)] -> [(a, b)]
- m :: (Num a2, Integral a1) => (a1, a2) -> (a1, a2) -> (a1, a2)
- conj :: Num k => Octonion k -> Octonion k
- sqnorm :: Num a => Octonion a -> a
- isOrthogonal :: (Eq a, Num a) => Octonion a -> Octonion a -> Bool
- antiCommutes :: (Eq a, Num a) => a -> a -> Bool
- octonions :: (Eq k, Num k) => [k] -> [Octonion k]
- isUnit :: (Eq a, Num a) => Octonion a -> Bool
- unitImagOctonions :: (Eq a, Num a) => [a] -> [Octonion a]
- autFrom :: (Ord a, Num a) => Octonion a -> Octonion a -> Octonion a -> [[a]]
- (%^) :: (Eq k, Num k) => Octonion k -> [[k]] -> Octonion k
- alpha3 :: [[F3]]
- beta3 :: [[F3]]
- gamma3s :: [Octonion F3]
- gamma3 :: [[F3]]
- alpha3' :: Permutation (Octonion F3)
- beta3' :: Permutation (Octonion F3)
- gamma3' :: Permutation (Octonion F3)
- g2_3 :: [Permutation (Octonion F3)]
- alpha4 :: [[F4]]
- beta4 :: [[F4]]
- gamma4s :: [Octonion F4]
- gamma4 :: [[F4]]
- alpha4' :: Permutation (Octonion F4)
- beta4' :: Permutation (Octonion F4)
- gamma4' :: Permutation (Octonion F4)

# Documentation

newtype Octonion k Source #

Constructors

O [(Int, k)] |

Instances

Eq k => Eq (Octonion k) Source # |
Instance details Defined in Math.Projects.ChevalleyGroup.Exceptional Methods (==) :: Octonion k -> Octonion k -> Bool Source # (/=) :: Octonion k -> Octonion k -> Bool Source #
(Ord k, Num k, Fractional k) => Fractional (Octonion k) Source # |
Instance details Defined in Math.Projects.ChevalleyGroup.Exceptional Methods (/) :: Octonion k -> Octonion k -> Octonion k Source # recip :: Octonion k -> Octonion k Source # fromRational :: Rational -> Octonion k Source #
(Ord k, Num k) => Num (Octonion k) Source # |

Instance details

Defined in Math.Projects.ChevalleyGroup.Exceptional

Methods

(+) :: Octonion k -> Octonion k -> Octonion k Source #

(-) :: Octonion k -> Octonion k -> Octonion k Source #

(*) :: Octonion k -> Octonion k -> Octonion k Source #

negate :: Octonion k -> Octonion k Source #

abs :: Octonion k -> Octonion k Source #

signum :: Octonion k -> Octonion k Source #

fromInteger :: Integer -> Octonion k Source #

Ord k => Ord (Octonion k) Source # |

Instance details

Defined in Math.Projects.ChevalleyGroup.Exceptional

Methods

compare :: Octonion k -> Octonion k -> Ordering Source #

(<) :: Octonion k -> Octonion k -> Bool Source #

(<=) :: Octonion k -> Octonion k -> Bool Source #

(>) :: Octonion k -> Octonion k -> Bool Source #

(>=) :: Octonion k -> Octonion k -> Bool Source #

max :: Octonion k -> Octonion k -> Octonion k Source #

min :: Octonion k -> Octonion k -> Octonion k Source #

Show k => Show (Octonion k) Source # |
Instance details Defined in Math.Projects.ChevalleyGroup.Exceptional Methods showsPrec :: Int -> Octonion k -> ShowS Source # show :: Octonion k -> String Source # showList :: [Octonion k] -> ShowS Source #

i0 :: Octonion Q Source #

i1 :: Octonion Q Source #

i2 :: Octonion Q Source #

i3 :: Octonion Q Source #

i4 :: Octonion Q Source #

i5 :: Octonion Q Source #

i6 :: Octonion Q Source #

fromList :: (Eq k, Num k) => [k] -> Octonion k Source #

toList :: Num a => Octonion a -> [a] Source #

expose :: Octonion k -> [(Int, k)] Source #

nf :: (Num b, Ord a, Ord b) => [(a, b)] -> [(a, b)] Source #

m :: (Num a2, Integral a1) => (a1, a2) -> (a1, a2) -> (a1, a2) Source #

conj :: Num k => Octonion k -> Octonion k Source #

sqnorm :: Num a => Octonion a -> a Source #

isOrthogonal :: (Eq a, Num a) => Octonion a -> Octonion a -> Bool Source #

antiCommutes :: (Eq a, Num a) => a -> a -> Bool Source #

octonions :: (Eq k, Num k) => [k] -> [Octonion k] Source #

isUnit :: (Eq a, Num a) => Octonion a -> Bool Source #

unitImagOctonions :: (Eq a, Num a) => [a] -> [Octonion a] Source #

autFrom :: (Ord a, Num a) => Octonion a -> Octonion a -> Octonion a -> [[a]] Source #

(%^) :: (Eq k, Num k) => Octonion k -> [[k]] -> Octonion k Source #

alpha3 :: [[F3]] Source #

beta3 :: [[F3]] Source #

gamma3s :: [Octonion F3] Source #

gamma3 :: [[F3]] Source #

alpha3' :: Permutation (Octonion F3) Source #

beta3' :: Permutation (Octonion F3) Source #

gamma3' :: Permutation (Octonion F3) Source #

g2_3 :: [Permutation (Octonion F3)] Source #

Generators for G2(3), a finite simple group of order 4245696, as a permutation group on the 702 unit imaginary octonions over F3

alpha4 :: [[F4]] Source #

beta4 :: [[F4]] Source #

gamma4s :: [Octonion F4] Source #

gamma4 :: [[F4]] Source #

alpha4' :: Permutation (Octonion F4) Source #

beta4' :: Permutation (Octonion F4) Source #

gamma4' :: Permutation (Octonion F4) Source #