From e82050aeb53d6aabcce7a8f27659e2e8bb8fb4bc Mon Sep 17 00:00:00 2001
From: Jeshiba <baconp@gmail.com>
Date: Mon, 17 Jul 2017 16:18:14 -0400
Subject: documentation and prop_longestElement.

- Added more documentation in RootSystem.hs
- Added a test for longestElement.
---
 Math/Combinatorics/RootSystem.hs | 13 +++++++++++++
 1 file changed, 13 insertions(+)

(limited to 'Math')

diff --git a/Math/Combinatorics/RootSystem.hs b/Math/Combinatorics/RootSystem.hs
index 9ed7c89..716819c 100644
--- a/Math/Combinatorics/RootSystem.hs
+++ b/Math/Combinatorics/RootSystem.hs
@@ -66,10 +66,12 @@ s :: [Q] -> [Q] -> [Q]
 s alpha beta = beta <-> (dynkinIndex alpha beta) *> alpha
 
 
+-- |The indices of the simple roots in the reduced decomposition of the longest elements
 longestElementIndex :: SimpleSystem -> [Int]
 longestElementIndex ss = (+1) <$> fromJust <$> flip elemIndex ss <$> longestElement ss
 
 
+-- |The reduced decomposition of the longest elements
 longestElement :: SimpleSystem -> [[Q]]
 longestElement ss = longestElement' [] posRoots
     where 
@@ -95,6 +97,8 @@ positiveRoots' cm = positiveRoots'' [] (iMx $ length cm)
               | null npr = pr
               | otherwise = positiveRoots'' (pr ++ npr) (S.toList . S.fromList $ mconcat $ zipWith newRootIndices npr (pIndex pr cm <$> npr))
 
+-- |calculate the successors of ri in the simple string 
+-- |given root index ri and its p-vector pi
 newRootIndices :: [Q] -> [Q] -> [[Q]]
 newRootIndices ri pi = (ri <+>) <$> (go pi [] 1)
     where go :: [Q] -> [Int] -> Int -> [[Q]]
@@ -102,9 +106,12 @@ newRootIndices ri pi = (ri <+>) <$> (go pi [] 1)
           go (x:xs) ys k = go xs (if x == 0 then ys else ys ++ [k]) (k + 1)
 
 
+-- |given the already-found positive roots allRootIndices, 
+-- |the transposed Cartan Matrix cm and the root index, calculate its p-vector
 pIndex :: [[Q]] -> [[Q]] -> [Q] -> [Q]
 pIndex allRootIndices cm rootIndex = (mIndex rootIndex allRootIndices) <-> (dynkinIndex' rootIndex cm)
 
+-- |given the root index and all already-found positive roots, calculate its m-vector
 mIndex :: [Q] -> [[Q]] -> [Q]
 mIndex rootIndex allRootIndices = 
     if isMultBasis rootIndex 
@@ -151,6 +158,7 @@ numRoots F 4 = 48
 numRoots G 2 = 12
 
 
+-- |Auxilieary function to transform an Int to a Root system for quick check
 int2TypeInt :: Int -> (Type, Int)
 int2TypeInt n
   | n > 108 = int2TypeInt (n `mod` 108 + 1)
@@ -196,6 +204,11 @@ allRoots ss = S.toList $ closure S.empty (S.fromList ss) where
                 boundary' = S.fromList [s alpha beta | alpha <- ss, beta <- S.toList boundary] S.\\ interior'
             in closure interior' boundary'
 
+-- | Test that the length of the longest element is half of the number of roots
+prop_longestElement :: Int -> Bool
+prop_longestElement n = let (t, m) = int2TypeInt n in
+     numRoots t m == 2 * (length $ longestElement $ simpleSystem t m)
+
 -- Old code in HaskellForMaths
 {--
 -- WEYL GROUP
-- 
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