From 75929417aeca94fed8cb3a5d75d01c8aa5d9d4d6 Mon Sep 17 00:00:00 2001
From: Friedrich Rober <friedrich.rober@rwth-aachen.de>
Date: Thu, 8 Nov 2018 13:22:50 +0100
Subject: Add comments

---
 gap/subgroupladders.gi | 103 ++++++++++++++++++++++++++++++++-----------------
 1 file changed, 67 insertions(+), 36 deletions(-)

(limited to 'gap')

diff --git a/gap/subgroupladders.gi b/gap/subgroupladders.gi
index aba383a..7c373d5 100644
--- a/gap/subgroupladders.gi
+++ b/gap/subgroupladders.gi
@@ -4,78 +4,97 @@
 # Implementations
 #
 
-
+#
+# Generate a Young subgroup of a partition part = (p_1, ..., p_k) of some positive integer n.
+# Every p_i is a list of positive integers such that the union of the p_i is disjoint and equals {1,...n}.
+# The Young subgroup is then the direct product of symmetric groups on the p_i.
+#
 InstallGlobalFunction( YoungGroupFromPartition,
 function(part)
 	local
-		i,
-		P,
-		G,
-		grps,
-		olds,
-		news,
-		perms,
-		info,
-		generators;
-
+		p,				# loop variable over the partition.
+		Y,				# the Young subgroup formed over the partition.
+		generators,		# the generators of Y.
+		G,				# the symmetric group on p.
+		grps,			# record entry of info.
+		olds,			# record entry of info.
+		news,			# record entry of info.
+		perms,			# record entry of info.
+		info;			# record for the DirectProductInfo of Y.
+
+
+	# Initialize the variables.
 	grps := [];
 	olds := [];
 	news := [];
 	perms := [];
 	generators := [];
 
-	for i in part do
-		G := SymmetricGroup(i);
+	# Generate the record entries of info.
+	for p in part do
+		G := SymmetricGroup(p);
 		Append(generators, GeneratorsOfGroup(G));
 		Add(grps, G);
-		Add(olds, i);
-		Add(news, i);
+		Add(olds, p);
+		Add(news, p);
 		Add(perms, ());
 	od;
 
-	P := Group(generators);
+	# Generate the Young subgroup Y and add the DirectProductInfo info to Y.
+	Y := Group(generators);
 	info := rec( groups := grps,
 	             olds := olds,
 	             news := news,
 	             perms := perms,
 	             embeddings := [],
 	             projections := [] );
-	SetDirectProductInfo(P, info);
+	SetDirectProductInfo(Y, info);
 
-	return P;
+	return Y;
 end);
 
 
 InstallGlobalFunction( SubgroupLadder,
 function(G)
 	local
-		orb,
-		i,
-		k,
-		n,
-		pair,
-		ladder,
-		directfactors,
-		generators,
-		mapping,
-		output,
-		partition,
-		FindPos;
-
-	FindPos := function(list, x)
-		local n, i;
-		n := Length(list);
+		orb, 			# the orbits of the permutation group G
+		n,				# largest moved point of G
+		i,				# loop variable 
+		k,				# size of current partition
+		ladder,			# the ladder is a list containing pairs [partition, mapping]
+		partition,		# partition is a list of positive integers (a_1, ..., a_k) such that
+						# a_1 is the biggest integer of the list.
+		mapping, 		# mapping is a list of positive integers (m_1, ..., m_n) such that
+						# 1 <= m_i <= k for all i.
+		pair,			# a entry of the ladder [partition, mapping] used to construct a young subgroup
+		output,			# a list of groups forming the ladder of G into S_n
+		FindPos;		# a local function
+
+	#
+	# l = [l_1, ..., l_n] is a list of lists.
+	# Check wether x in contained in some list l_i of l.
+	# If it is, return the smallest integer i such that x in l_i.
+	# Otherwise return 0.
+	#
+	FindPos := function(l, x)
+		local 
+			n, 			# length of l
+			i;			# loop variable over l
+		n := Length(l);
 		for i in [1..n] do
-			if x in list[i] then
+			if x in l[i] then
 				return i;
 			fi;
 		od;
+		return 0;
 	end;
 
+	# Check if G is a permuatation group
 	if (not IsPermGroup(G)) then
 		ErrorNoReturn("the argument must be a permutation group!\n");
 	fi;
 
+	# Initialize the variables
 	n := LargestMovedPoint(G);
 
 	orb := List(Orbits(G, [1..n]));
@@ -91,12 +110,19 @@ function(G)
 	mapping := List([1..n], x -> FindPos(orb, x));
 	ladder := [[List(partition), List(mapping)]];
 
+	# Start the iterative construction of the ladder
 	while (Length(partition) <> 1 or partition[1] < n) do
-		if (Length(partition) = 1 and partition[1] < n) then
+		# This is the case where partition = (a) and the current young subgroup is S_{a}.
+		# Add the group S_{a+1} to the ladder 
+		# and continue iteration with this group.
+		if (Length(partition) = 1) then
 			mapping[Position(mapping, 0)] := 1;
 			partition[1] := partition[1] + 1;
 			Add(ladder, [List(partition), List(mapping)]);
 		else
+			# This is the case where partition = (a_1, a_2, a_3, ..., a_k) and a_2 = 1.
+			# Add the young subgroup with partition (a_1, a_3, ..., a_k)
+			# and continue iteration with this group.
 			if (partition[2] = 1) then
 				Remove(partition, 2);
 				for i in [1..n] do
@@ -106,6 +132,10 @@ function(G)
 				od;
 				partition[1] := partition[1] + 1;
 				Add(ladder, [List(partition), List(mapping)]);
+			# This is the case where partition = (a_1, a_2, a_3, ..., a_k) and a_2 > 1.
+			# First add the young subgroup with partition (a_1, 1, a_2 - 1, a_3, ..., a_k).
+			# Then add the young subgroup with partition (a_1 + 1, a_2 - 1, a_3, ..., a_k)	
+			# and continue iteration with this group.
 			else
 				mapping[Position(mapping, 2)] := Length(partition)+1;
 				partition[2] := partition[2] - 1;
@@ -120,6 +150,7 @@ function(G)
 		fi;
 	od;
 
+	# Construct the young subgroups of the pairs [partition, mapping] stored in ladder
 	for pair in ladder do
 		k := Length(pair[1]);
 		mapping := pair[2];
-- 
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