```-- Copyright (c) David Amos, 2008. All rights reserved.

{-# OPTIONS_GHC -fglasgow-exts #-}

module Math.Projects.RootSystem where

import Data.Ratio
import qualified Data.List as L
import qualified Data.Set as S

import Math.Algebra.LinearAlgebra
import Math.Algebra.Group.PermutationGroup hiding (elts, order, closure)
import Math.Algebra.Group.SchreierSims as SS
import Math.Algebra.Group.StringRewriting as SG

import Math.Algebra.Field.Base -- for Q

data Type = A | B | C | D | E | F | G

-- Humphreys, Reflection Groups and Coxeter Groups

-- SIMPLE SYSTEMS
-- sometimes called fundamental systems

-- The ith basis vector in K^n
basisElt :: Int -> Int -> [Q] -- this type signature determines all the rest
basisElt n i = replicate (i-1) 0 ++ 1 : replicate (n-i) 0
-- We need to work over the rationals to ensure that arithmetic is exact
-- So long as our simple systems are rational, then reflection matrices are rational

-- A simple system is like a basis for the root system (see Humphreys p8 for full definition)
-- simpleSystem :: Type -> Int -> [[Q]]
simpleSystem A n | n >= 1 = [e i <-> e (i+1) | i <- [1..n]]
where e = basisElt (n+1)
simpleSystem B n | n >= 2 = [e i <-> e (i+1) | i <- [1..n-1]] ++ [e n]
where e = basisElt n
simpleSystem C n | n >= 2 = [e i <-> e (i+1) | i <- [1..n-1]] ++ [2 *> e n]
where e = basisElt n
simpleSystem D n | n >= 4 = [e i <-> e (i+1) | i <- [1..n-1]] ++ [e (n-1) <+> e n]
where e = basisElt n
simpleSystem E n | n `elem` [6,7,8] = take n simpleroots
where e = basisElt 8
simpleroots = ((1/2) *> (e 1 <-> e 2 <-> e 3 <-> e 4 <-> e 5 <-> e 6 <-> e 7 <+> e 8))
: (e 1 <+> e 2)
: [e (i-1) <-> e (i-2) | i <- [3..8]]
simpleSystem F 4 = [e 2 <-> e 3, e 3 <-> e 4, e 4, (1/2) *> (e 1 <-> e 2 <-> e 3 <-> e 4)]
where e = basisElt 4
simpleSystem G 2 = [e 1 <-> e 2, ((-2) *> e 1) <+> e 2 <+> e 3]
where e = basisElt 3

-- ROOT SYSTEMS
-- Calculating the full root system from the fundamental roots

-- Humphreys p3
-- Weyl group element corresponding to a root
-- w r is the reflection in the hyperplane orthogonal to r
w r s = s <-> (2 * (s <.> r) / (r <.> r)) *> r

-- Given a simple system, return the full root system
-- The closure of a set of roots under reflection
closure rs = S.toList \$ closure' S.empty (S.fromList rs) where
closure' interior boundary
| S.null boundary = interior
| otherwise =
let interior' = S.union interior boundary
boundary' = S.fromList [w r s | r <- rs, s <- S.toList boundary] S.\\ interior'
in closure' interior' boundary'

-- WEYL GROUP
-- The finite reflection group generated by the root system

-- Generators of the Weyl group as permutation group on the roots
weylPerms t n =
let rs = simpleSystem t n
xs = closure rs
toPerm r = fromPairs [(x, w r x) | x <- xs]
in map toPerm rs

-- Generators of the Weyl group as a matrix group
weylMatrices t n = map wMx (simpleSystem t n)

-- The Weyl group element corresponding to a root, represented as a matrix
wMx r = map (w r) [e i | i <- [1..d]] -- matrix for reflection in hyperplane orthogonal to r
where d = length r -- dimension of the space
e = basisElt d
-- the images of the basis elts form the columns of the matrix
-- however, reflection matrices are symmetric, so they also form the rows

-- CARTAN MATRIX, DYNKIN DIAGRAM, COXETER SYSTEM

cartanMatrix t n = [[2 * (ai <.> aj) / (ai <.> ai) | aj <- roots] | ai <- roots]
where roots = simpleSystem t n
-- Note: The Cartan matrices for A, D, E systems are symmetric.
-- Those of B, C, F, G are not
-- Carter, Simple Groups of Lie Type, p44-5 gives the expected answers
-- They agree with our answers except for G2, which is the transpose
-- (So probably Carter defines the roots of G2 the other way round to Humphreys)

-- set the diagonal entries of (square) matrix mx to constant c
setDiag c mx@((x:xs):rs) = (c:xs) : zipWith (:) (map head rs) (setDiag c \$ map tail rs)
setDiag _ [[]] = [[]]

-- Carter, Segal, Macdonald p17-18
-- given a Cartan matrix, derive the corresponding matrix describing the Dynkin diagram
-- nij = Aij * Aji, nii = 0
dynkinFromCartan aij = setDiag 0 \$ (zipWith . zipWith) (*) aij (L.transpose aij)

dynkinDiagram t n = dynkinFromCartan \$ cartanMatrix t n

-- given the Dynkin diagram nij, derive the coefficients mij of the Coxeter group <si | si^2, (sisj)^mij> (so mii == 1)
-- using nij = 4 cos^2 theta_ij
-- nij == 0 <=> theta = pi/2
-- nij == 1 <=> theta = pi/3
-- nij == 2 <=> theta = pi/4
-- nij == 3 <=> theta = pi/6
coxeterFromDynkin nij = setDiag 1 \$ (map . map) f nij
where f 0 = 2; f 1 = 3; f 2 = 4; f 3 = 6

-- The mij coefficients of the Coxeter group <si | si^2, (sisj)^mij>, as a matrix
coxeterMatrix t n = coxeterFromDynkin \$ dynkinDiagram t n

-- Given the matrix of coefficients mij, return the Coxeter group <si | si^2, (sisj)^mij>
-- We assume but don't check that mii == 1 and mij == mji
fromCoxeterMatrix mx = (gs,rs) where
n = length mx
gs = map s_ [1..n]
rs = rules mx 1
rules [] _ = []
rules ((1:xs):rs) i = ([s_ i, s_ i],[]) : [powerRelation i j m | (j,m) <- zip [i+1..] xs] ++ rules (map tail rs) (i+1)
powerRelation i j m = (concat \$ replicate m [s_ i, s_ j],[])

-- Another presentation for the Coxeter group, using braid relations
fromCoxeterMatrix2 mx = (gs,rs) where
n = length mx
gs = map s_ [1..n]
rs = rules mx 1
rules [] _ = []
rules ((1:xs):rs) i = ([s_ i, s_ i],[]) : [braidRelation i j m | (j,m) <- zip [i+1..] xs] ++ rules (map tail rs) (i+1)
braidRelation i j m = (take m \$ cycle [s_ j, s_ i], take m \$ cycle [s_ i, s_ j])

coxeterPresentation t n = fromCoxeterMatrix \$ coxeterMatrix t n

eltsCoxeter t n = SG.elts \$ fromCoxeterMatrix2 \$ coxeterMatrix t n
-- it's just slightly faster to use the braid presentation

poincarePoly t n = map length \$ L.group \$ map length \$ eltsCoxeter t n

-- LIE ALGEBRAS

elemMx n i j = replicate (i-1) z ++ e j : replicate (n-i) z
where z = replicate n 0
e = basisElt n

lieMult x y = x*y - y*x

-- for gluing matrices together
(+|+) = zipWith (++) -- glue two matrices together side by side
(+-+) = (++)         -- glue two matrices together above and below

form D n = (zMx n +|+ idMx n)
+-+
(idMx n +|+ zMx n)
form C n = (2 : replicate (2*n) 0) :
(map (0:) (form D n))
form B n = let id' = (-1) *>> idMx n
in (zMx n +|+ idMx n)
+-+
(id'  +|+ zMx n)

-- TESTING
-- The expected values of the root system, number of roots, order of Weyl group
-- for comparison against the calculated values

-- !! Not yet got root systems for E6,7,8, F4

-- Humphreys p41ff

-- The full root system
-- L.sort (rootSystem t n) == closure (simpleSystem t n)
-- rootSystem :: Type -> Int -> [[QQ]]
rootSystem A n | n >= 1 = [e i <-> e j | i <- [1..n+1], j <- [1..n+1], i /= j]
where e = basisElt (n+1)
rootSystem B n | n >= 2 = shortRoots ++ longRoots
where e = basisElt n
shortRoots = [e i | i <- [1..n]]
++ [[] <-> e i | i <- [1..n]]
longRoots  = [e i <+> e j | i <- [1..n], j <- [i+1..n]]
++ [e i <-> e j | i <- [1..n], j <- [i+1..n]]
++ [[] <-> e i <+> e j | i <- [1..n], j <- [i+1..n]]
++ [[] <-> e i <-> e j | i <- [1..n], j <- [i+1..n]]
rootSystem C n | n >= 2 = longRoots ++ shortRoots
where e = basisElt n
longRoots  = [2 *> e i | i <- [1..n]]
++ [[] <-> (2 *> e i) | i <- [1..n]]
shortRoots = [e i <+> e j | i <- [1..n], j <- [i+1..n]]
++ [e i <-> e j | i <- [1..n], j <- [i+1..n]]
++ [[] <-> e i <+> e j | i <- [1..n], j <- [i+1..n]]
++ [[] <-> e i <-> e j | i <- [1..n], j <- [i+1..n]]
rootSystem D n | n >= 4 =
[e i <+> e j | i <- [1..n], j <- [i+1..n]]
++ [e i <-> e j | i <- [1..n], j <- [i+1..n]]
++ [[] <-> e i <+> e j | i <- [1..n], j <- [i+1..n]]
++ [[] <-> e i <-> e j | i <- [1..n], j <- [i+1..n]]
where e = basisElt n
rootSystem G 2 = shortRoots ++ longRoots
where e = basisElt 3
shortRoots = [e i <-> e j | i <- [1..3], j <- [1..3], i /= j]
longRoots = concatMap (\r-> [r,[] <-> r]) [2 *> e i <-> e j <-> e k | i <- [1..3], [j,k] <- [[1..3] L.\\ [i]] ]

-- numRoots t n == length (closure \$ simpleSystem t n)
numRoots A n = n*(n+1)
numRoots B n = 2*n*n
numRoots C n = 2*n*n
numRoots D n = 2*n*(n-1)
numRoots E 6 = 72
numRoots E 7 = 126
numRoots E 8 = 240
numRoots F 4 = 48
numRoots G 2 = 12

-- The order of the Weyl group
-- orderWeyl t n == S.order (weylPerms t n)
orderWeyl A n = factorial (n+1)
orderWeyl B n = 2^n * factorial n
orderWeyl C n = 2^n * factorial n
orderWeyl D n = 2^(n-1) * factorial n
orderWeyl E 6 = 2^7 * 3^4 * 5
orderWeyl E 7 = 2^10 * 3^4 * 5 * 7
orderWeyl E 8 = 2^14 * 3^5 * 5^2 * 7
orderWeyl F 4 = 2^7 * 3^2
orderWeyl G 2 = 12

factorial n = product [1..toInteger n]

{-
-- now moved to TRootSystem
test1 = all (\(t,n) -> orderWeyl t n == L.genericLength (eltsCoxeter t n))
[(A,3),(A,4),(A,5),(B,3),(B,4),(B,5),(C,3),(C,4),(C,5),(D,4),(D,5),(F,4),(G,2)]

test2 = all (\(t,n) -> orderWeyl t n == SS.order (weylPerms t n))
[(A,3),(A,4),(A,5),(B,3),(B,4),(B,5),(C,3),(C,4),(C,5),(D,4),(D,5),(E,6),(F,4),(G,2)]
-}
```