HaskellForMaths-0.4.8: 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 :: (Num k, Eq k) => [k] -> Octonion k
- toList :: Num a => Octonion a -> [a]
- expose :: Octonion t -> [(Int, t)]
- nf :: (Ord t, Ord t1, Num t) => [(t1, t)] -> [(t1, t)]
- m :: (Integral a1, Num a) => (a1, a) -> (a1, a) -> (a1, a)
- conj :: Num k => Octonion k -> Octonion k
- sqnorm :: Num a => Octonion a -> a
- isOrthogonal :: (Num a, Eq a) => Octonion a -> Octonion a -> Bool
- antiCommutes :: (Num a, Eq a) => a -> a -> Bool
- octonions :: (Eq k, Num k) => [k] -> [Octonion k]
- isUnit :: (Num a, Eq 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 # |
Methods (==) :: Octonion k -> Octonion k -> Bool # (/=) :: Octonion k -> Octonion k -> Bool #
(Ord k, Num k, Fractional k) => Fractional (Octonion k) Source # |
Methods (/) :: Octonion k -> Octonion k -> Octonion k # recip :: Octonion k -> Octonion k # fromRational :: Rational -> Octonion k #
(Ord k, Num k) => Num (Octonion k) Source # |
Methods (+) :: Octonion k -> Octonion k -> Octonion k # (-) :: Octonion k -> Octonion k -> Octonion k # (*) :: Octonion k -> Octonion k -> Octonion k # negate :: Octonion k -> Octonion k # abs :: Octonion k -> Octonion k # signum :: Octonion k -> Octonion k # fromInteger :: Integer -> Octonion k #
Ord k => Ord (Octonion k) Source # |
Methods compare :: Octonion k -> Octonion k -> Ordering # (<) :: Octonion k -> Octonion k -> Bool # (<=) :: Octonion k -> Octonion k -> Bool # (>) :: Octonion k -> Octonion k -> Bool # (>=) :: Octonion k -> Octonion k -> Bool # max :: Octonion k -> Octonion k -> Octonion k # min :: Octonion k -> Octonion k -> Octonion k #
Show k => Show (Octonion k) Source # |
Methods showsPrec :: Int -> Octonion k -> ShowS # show :: Octonion k -> String # showList :: [Octonion k] -> ShowS #

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

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

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

nf :: (Ord t, Ord t1, Num t) => [(t1, t)] -> [(t1, t)] Source #

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

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

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

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

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

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

isUnit :: (Num a, Eq 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 #