Math.Combinat.RootSystems

Naive (very inefficient) algorithm to generate the irreducible (Dynkin) root systems

Based on https://en.wikipedia.org/wiki/Root_system

Half-integers

newtype HalfInt

The type of half-integers (internally represented by their double)

TODO: refactor this into its own module

Constructors

HalfInt Int

Instances

Num HalfInt

Defined in Math.Combinat.RootSystems

Methods

(+) :: HalfInt -> HalfInt -> HalfInt

(-) :: HalfInt -> HalfInt -> HalfInt

(*) :: HalfInt -> HalfInt -> HalfInt

negate :: HalfInt -> HalfInt

abs :: HalfInt -> HalfInt

signum :: HalfInt -> HalfInt

fromInteger :: Integer -> HalfInt

Num HalfVec

Defined in Math.Combinat.RootSystems

Methods

(+) :: HalfVec -> HalfVec -> HalfVec

(-) :: HalfVec -> HalfVec -> HalfVec

(*) :: HalfVec -> HalfVec -> HalfVec

negate :: HalfVec -> HalfVec

abs :: HalfVec -> HalfVec

signum :: HalfVec -> HalfVec

fromInteger :: Integer -> HalfVec

Show HalfInt

Defined in Math.Combinat.RootSystems

Methods

showsPrec :: Int -> HalfInt -> ShowS

show :: HalfInt -> String

showList :: [HalfInt] -> ShowS

Eq HalfInt

Defined in Math.Combinat.RootSystems

Methods

(==) :: HalfInt -> HalfInt -> Bool

(/=) :: HalfInt -> HalfInt -> Bool

Ord HalfInt

Defined in Math.Combinat.RootSystems

Methods

compare :: HalfInt -> HalfInt -> Ordering

(<) :: HalfInt -> HalfInt -> Bool

(<=) :: HalfInt -> HalfInt -> Bool

(>) :: HalfInt -> HalfInt -> Bool

(>=) :: HalfInt -> HalfInt -> Bool

max :: HalfInt -> HalfInt -> HalfInt

min :: HalfInt -> HalfInt -> HalfInt

half :: HalfInt

divByTwo :: Int -> HalfInt

mulByTwo :: HalfInt -> Int

scaleBy :: Int -> HalfInt -> HalfInt

Vectors of half-integers

type HalfVec = [HalfInt]

scaleVec :: Int -> HalfVec -> HalfVec

negateVec :: HalfVec -> HalfVec

safeZip :: (a -> b -> c) -> [a] -> [b] -> [c]

Dynkin diagrams

data Dynkin

Constructors

A !Int

B !Int

C !Int

D !Int

E6

E7

E8

F4

G2

Instances

Show Dynkin

Defined in Math.Combinat.RootSystems

Methods

showsPrec :: Int -> Dynkin -> ShowS

show :: Dynkin -> String

showList :: [Dynkin] -> ShowS

Eq Dynkin

Defined in Math.Combinat.RootSystems

Methods

(==) :: Dynkin -> Dynkin -> Bool

(/=) :: Dynkin -> Dynkin -> Bool

The roots of root systems

ambientDim :: Dynkin -> Int

The ambient dimension of (our representation of the) system (length of the vector)

simpleRootsOf :: Dynkin -> [HalfVec]

positiveRootsOf :: Dynkin -> Set HalfVec

negativeRootsOf :: Dynkin -> Set HalfVec

allRootsOf :: Dynkin -> Set HalfVec

Positive roots

findPositiveHyperplane :: [HalfVec] -> [Double]

Finds a vector, which is hopefully not orthognal to any root (generated by the given simple roots), and has positive dot product with each of them.

positiveRoots :: [HalfVec] -> Set HalfVec

basisOfPositives :: Set HalfVec -> [HalfVec]

Operations on half-integer vectors

bracket :: HalfVec -> HalfVec -> HalfInt

bracket b a = (a,b)/(a,a)

mirror :: HalfVec -> HalfVec -> HalfVec

mirror b a = b - 2*(a,b)/(a,a) * a

cartanMatrix :: [HalfVec] -> Array (Int, Int) Int

Cartan matrix of a list of (simple) roots

printMatrix :: Show a => Array (Int, Int) a -> IO ()

Mirroring

mirrorClosure :: [HalfVec] -> Set HalfVec

We mirror stuff until there is no more things happening (very naive algorithm, but seems to work)

mirrorStep :: Set HalfVec -> Set HalfVec

E6, E7 and E8

simpleRootsE6_123 :: [HalfVec]

This is a basis of E6 as the subset of the even E8 root system where the first three coordinates agree (they are consolidated into the first coordinate here)

simpleRootsE7_12 :: [HalfVec]

This is a basis of E8 as the subset of the even E8 root system where the first two coordinates agree (they are consolidated into the first coordinate here)

simpleRootsE7_diag :: [HalfVec]

This is a basis of E7 as the subset of the even E8 root system for which the sum of coordinates sum to zero

simpleRootsE8_even :: [HalfVec]

simpleRootsE8_odd :: [HalfVec]