Initial population.

1 files changed, 207 insertions, 0 deletions

diff --git a/src/hyper/pages/grpthry.ht b/src/hyper/pages/grpthry.ht
new file mode 100644
index 00000000..d75c0e4e
--- /dev/null
+++ b/src/hyper/pages/grpthry.ht
@@ -0,0 +1,207 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% @(#)grpthry.ht        1.2 90/08/22 14:54:14
+
+% Group Theory Page
+% authors: H. Gollan, J. Grabmeier, August 1989
+% @(#)grpthry.ht        2.6 90/05/21 14:44:11
+\begin{page}{GroupTheoryPage}{Group Theory}
+\beginscroll
+\Language{} can work with individual permutations, permutation
+groups and do representation theory.
+\horizontalline
+\newline
+\beginmenu
+\menulink{Info on Group Theory}{InfoGroupTheoryPage}
+
+
+%\menulink{Permutation}{PermutationXmpPage}
+%Calculating within symmetric groups.
+
+%\menulink{Permutation Groups}{PermutationGroupXmpPage}
+%Working with subgroups of a symmetric group.
+
+%\menulink{Permutation Group Examples}{PermutationGroupExampleXmpPage}
+%Working with permutation groups, predefined in the system as Rubik's group.
+
+\menulink{Info on Representation Theory}{InfoRepTheoryPage}
+
+%\menulink{Irreducible Representations of Symmetric Groups}{IrrRepSymNatXmpPage}
+%Alfred Young's natural form for these representations.
+
+%\menulink{Representations of Higher Degree}{RepresentationPackage1XmpPage}
+%Constructing new representations by symmetric and antisymmetric
+%tensors.
+
+%\menulink{Decomposing Representations}{RepresentationPackage2XmpPage}
+%Parker's `Meat-Axe', working in prime characteristics.
+
+\menulink{Representations of \texht{$A_6$}{A6}}{RepA6Page}
+The irreducible representations of the alternating group \texht{$A_6$}{A6} over fields
+of characteristic 2.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+% RepA6Page
+% author: J. Grabmeier,  08/08/89
+
+\begin{page}{RepA6Page}{Representations of \texht{$A_6$}{A6}}
+\beginscroll
+In what follows you'll see how to use \Language{} to get all the irreducible
+representations of the alternating group \texht{$A_6$}{A6} over the field with two
+elements (GF 2).
+First, we generate \texht{$A_6$}{A6} by a three-cycle: x = (1,2,3)
+and a 5-cycle: y = (2,3,4,5,6).  Next we have \Language{} calculate
+the permutation representation over the integers and over GF 2:
+\spadpaste{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]] \bound{genA6}}
+\spadpaste{pRA6 := permutationRepresentation (genA6, 6) \bound{pRA6} \free{genA6}
+}
+Now we apply Parker's 'Meat-Axe' and split it:
+\spadpaste{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2)) \free{pRA6} \bound{sp0}}
+We have found the trivial module as a quotient module
+and a 5-dimensional sub-module.
+Try to split again:
+\spadpaste{sp1 := meatAxe sp0.1 \bound{sp1}}
+and we find a 4-dimensional sub-module and the trivial one again.
+Now we can test if this representaton is absolutely irreducible:
+\spadpaste{isAbsolutelyIrreducible? sp1.2 }
+and we see that this 4-dimensional representation is absolutely irreducible.
+So, we have found a second irreducible representation.
+Now, we construct a representation by reducing an irreducible one
+of the symmetric group S_6 over the integers mod 2.
+We take the one labelled by the partition [2,2,1,1] and
+restrict it to \texht{$A_6$}{A6}:
+\spadpaste{d2211 := irreducibleRepresentation ([2,2,1,1],genA6) \bound{d2211} }
+Now split it:
+\spadpaste{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2 \free{d2211}
+\bound{sp2}}
+This gave both a five and a four dimensional representation.
+Now we take the 4-dimensional one
+and we shall see that it is absolutely irreducible:
+\spadpaste{isAbsolutelyIrreducible? sp2.1}
+The two 4-dimensional representations are not equivalent:
+\spadpaste{areEquivalent? (sp1.2, sp2.1)}
+So we have found a third irreducible representation.
+Now we construct a new representation using the tensor product
+and try to split it:
+\spadpaste{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16 \bound{dA6d16}}
+The representation is irreducible, but it may be not absolutely irreducible.
+\spadpaste{isAbsolutelyIrreducible? dA6d16}
+So let's try the same procedure over the field with 4 elements:
+\spadpaste{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2))) \bound{sp3}}
+Now we find two 8-dimensional representations, dA6d8a and dA6d8b.
+Both are absolutely irreducible...
+\spadpaste{isAbsolutelyIrreducible?  sp3.1}
+\spadpaste{isAbsolutelyIrreducible?  sp3.2}
+and they are not equivalent:
+\spadpaste{areEquivalent? (sp3.1,sp3.2)}
+So we have found five absolutely irreducible representations of \texht{$A_6$}{A6}
+in characteristic 2.
+General theory now tells us that there are no more irreducible ones.
+Here, for future reference are all the absolutely irreducible 2-modular
+representations of \texht{$A_6$}{A6}
+\spadpaste{sp0.2 \free{sp0}}
+\spadpaste{sp1.2 \free{sp1}}
+\spadpaste{sp2.1 \free{sp2}}
+\spadpaste{sp3.1 \free{sp3}}
+\spadpaste{sp3.2 \free{sp3}}
+And here again is the irreducible, but not absolutely irreducible
+representations of \texht{$A_6$}{A6} over GF 2
+\spadpaste{dA6d16 \free{dA6d16}}
+\endscroll
+\autobuttons \end{page}
+
+
+
+
+
+\begin{page}{InfoRepTheoryPage}{Representation Theory}
+\beginscroll
+\horizontalline
+Representation theory for finite groups studies finite groups by
+embedding them in a general linear group over a field or an
+integral domain.
+Hence, we are representing each element of the group by
+an invertible matrix.
+Two matrix representations of a given group are equivalent, if, by changing the
+basis of the underlying
+space, you can go from one to the other.  When you change bases, you
+transform the matrices that are the images of elements by
+conjugating them by an invertible matrix.
+\newline
+\newline
+If we can find a subspace which is fixed under the image
+of the group, then there exists a `base change' after which all the representing
+ matrices
+are in upper triangular block form. The block matrices on
+the main diagonal give a new representation of the group of lower degree.
+Such a representation is said to be `reducible'.
+\newline
+\beginmenu
+%\menulink{Irreducible Representations of Symmetric Groups}{IrrRepSymNatXmpPage}
+
+%Alfred Young's natural form for these representations.
+
+%\menulink{Representations of Higher Degree}{RepresentationPackage1XmpPage}
+%Constructing new representations by symmetric and antisymmetric
+%tensors.
+
+%\menulink{Decomposing Representations}{RepresentationPackage2XmpPage}
+%Parker's `Meat-Axe', working in prime characteristics.
+
+\menulink{Representations of \texht{$A_6$}{A6}}{RepA6Page}
+The irreducible representations of the alternating group \texht{$A_6$}{A6} over fields
+of characteristic 2.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+
+
+\begin{page}{InfoGroupTheoryPage}{Group Theory}
+%%
+%% Johannes Grabmeier 03/02/90
+%%
+\beginscroll
+A {\it group} is a set G together with an associative operation
+* satisfying the axioms of existence
+of a unit element and an inverse of every element of the group.
+The \Language{} category \spadtype{Group} represents this setting.
+Many data structures in \Language{} are groups and therefore there
+is a large variety of examples as fields and polynomials,
+although the main interest there is not the group structure.
+
+To work with and in groups in a concrete manner some way of
+representing groups has to be chosen. A group can be given
+as a list of generators and a set of relations. If there
+are no relations, then  we have a {\it free group}, realized
+in the domain \spadtype{FreeMonoid} which won't be discussed here.
+We consider {\it permutation groups}, where a group
+is realized as a subgroup of the symmetric group of a set, i.e.
+the group of all bijections of a set, the operation being the
+composition of maps.
+Indeed,  every group can be realized this way, although
+this may not be practical.
+
+Furthermore group elements can be given as invertible matrices.
+The group operation is reflected by matrix multiplication.
+More precise in representation theory group homomophisms
+from a group to general linear groups are contructed.
+Some algorithms are implemented in \Language{}.
+\newline
+%\beginmenu
+%\menulink{Permutation}{PermutationXmpPage}
+%Calculating within symmetric groups.
+
+%\menulink{Permutation Groups}{PermutationGroupXmpPage}
+%Working with subgroups of a symmetric group.
+
+%\menulink{Permutation Group Examples}{PermutationGroupExampleXmpPage}
+%Working with permutation groups, predefined in the system as Rubik's group.
+%\endmenu
+\endscroll
+\autobuttons \end{page}
+