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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra rep1.spad}
+\author{Holger Gollan, Johannes Grabmeier, Thorsten Werther}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{package REP1 RepresentationPackage1}
+<<package REP1 RepresentationPackage1>>=
+)abbrev package REP1 RepresentationPackage1
+++ Authors: Holger Gollan, Johannes Grabmeier, Thorsten Werther
+++ Date Created: 12 September 1987
+++ Date Last Updated: 24 May 1991
+++ Basic Operations: antisymmetricTensors,symmetricTensors,
+++   tensorProduct, permutationRepresentation 
+++ Related Constructors: RepresentationPackage1, Permutation
+++ Also See: IrrRepSymNatPackage 
+++ AMS Classifications:
+++ Keywords: representation, symmetrization, tensor product
+++ References:
+++   G. James, A. Kerber: The Representation Theory of the Symmetric
+++    Group. Encycl. of Math. and its Appl. Vol 16., Cambr. Univ Press 1981;
+++   J. Grabmeier, A. Kerber: The Evaluation of Irreducible
+++    Polynomial Representations of the General Linear Groups
+++    and of the Unitary Groups over Fields of Characteristic 0,
+++    Acta Appl. Math. 8 (1987), 271-291;
+++   H. Gollan, J. Grabmeier: Algorithms in Representation Theory and
+++    their Realization in the Computer Algebra System Scratchpad,
+++    Bayreuther Mathematische Schriften, Heft 33, 1990, 1-23
+++ Description: 
+++   RepresentationPackage1 provides functions for representation theory
+++   for finite groups and algebras.
+++   The package creates permutation representations and uses tensor products
+++   and its symmetric and antisymmetric components to create new
+++   representations of larger degree from given ones.
+++   Note: instead of having parameters from \spadtype{Permutation}
+++   this package allows list notation of permutations as well:
+++   e.g. \spad{[1,4,3,2]} denotes permutes 2 and 4 and fixes 1 and 3.
+ 
+RepresentationPackage1(R): public == private where
+ 
+   R   : Ring
+   OF    ==> OutputForm
+   NNI   ==> NonNegativeInteger
+   PI    ==> PositiveInteger
+   I     ==> Integer
+   L     ==> List
+   M     ==> Matrix
+   P     ==> Polynomial
+   SM    ==> SquareMatrix
+   V     ==> Vector
+   ICF   ==> IntegerCombinatoricFunctions Integer
+   SGCF  ==> SymmetricGroupCombinatoricFunctions
+   PERM  ==> Permutation
+ 
+   public ==> with
+ 
+     if R has commutative("*") then
+       antisymmetricTensors : (M R,PI) ->  M R
+         ++ antisymmetricTensors(a,n) applies to the square matrix
+         ++ {\em a} the irreducible, polynomial representation of the
+         ++ general linear group {\em GLm}, where m is the number of
+         ++ rows of {\em a}, which corresponds to the partition
+         ++ {\em (1,1,...,1,0,0,...,0)} of n.
+         ++ Error: if n is greater than m.
+         ++ Note: this corresponds to the symmetrization of the representation
+         ++ with the sign representation of the symmetric group {\em Sn}.
+         ++ The carrier spaces of the representation are the antisymmetric
+         ++ tensors of the n-fold tensor product.
+     if R has commutative("*") then
+       antisymmetricTensors : (L M R, PI) -> L M R
+         ++ antisymmetricTensors(la,n) applies to each 
+         ++ m-by-m square matrix in
+         ++ the list {\em la} the irreducible, polynomial representation
+         ++ of the general linear group {\em GLm} 
+         ++ which corresponds
+         ++ to the partition {\em (1,1,...,1,0,0,...,0)} of n.
+         ++ Error: if n is greater than m.
+         ++ Note: this corresponds to the symmetrization of the representation
+         ++ with the sign representation of the symmetric group {\em Sn}.
+         ++ The carrier spaces of the representation are the antisymmetric
+         ++ tensors of the n-fold tensor product.
+     createGenericMatrix : NNI -> M P R
+       ++ createGenericMatrix(m) creates a square matrix of dimension k
+       ++ whose entry at the i-th row and j-th column is the
+       ++ indeterminate {\em x[i,j]} (double subscripted).
+     symmetricTensors : (M R, PI) -> M R
+       ++ symmetricTensors(a,n) applies to the m-by-m 
+       ++ square matrix {\em a} the
+       ++ irreducible, polynomial representation of the general linear
+       ++ group {\em GLm}
+       ++ which corresponds to the partition {\em (n,0,...,0)} of n.
+       ++ Error: if {\em a} is not a square matrix.
+       ++ Note: this corresponds to the symmetrization of the representation
+       ++ with the trivial representation of the symmetric group {\em Sn}.
+       ++ The carrier spaces of the representation are the symmetric
+       ++ tensors of the n-fold tensor product.
+     symmetricTensors : (L M R, PI)  ->  L M R
+       ++ symmetricTensors(la,n) applies to each m-by-m square matrix in the
+       ++ list {\em la} the irreducible, polynomial representation
+       ++ of the general linear group {\em GLm}
+       ++ which corresponds
+       ++ to the partition {\em (n,0,...,0)} of n.
+       ++ Error: if the matrices in {\em la} are not square matrices.
+       ++ Note: this corresponds to the symmetrization of the representation
+       ++ with the trivial representation of the symmetric group {\em Sn}.
+       ++ The carrier spaces of the representation are the symmetric
+       ++ tensors of the n-fold tensor product.
+     tensorProduct : (M R, M R) -> M R
+       ++ tensorProduct(a,b) calculates the Kronecker product
+       ++ of the matrices {\em a} and b.
+       ++ Note: if each matrix corresponds to a group representation
+       ++ (repr. of  generators) of one group, then these matrices
+       ++ correspond to the tensor product of the two representations.
+     tensorProduct : (L M R, L M R) -> L M R
+       ++ tensorProduct([a1,...,ak],[b1,...,bk]) calculates the list of
+       ++ Kronecker products of the matrices {\em ai} and {\em bi}
+       ++ for {1 <= i <= k}.
+       ++ Note: If each list of matrices corresponds to a group representation
+       ++ (repr. of generators) of one group, then these matrices
+       ++ correspond to the tensor product of the two representations.
+     tensorProduct : M R -> M R
+       ++ tensorProduct(a) calculates the Kronecker product
+       ++ of the matrix {\em a} with itself.
+     tensorProduct : L M R -> L M R
+       ++ tensorProduct([a1,...ak]) calculates the list of
+       ++ Kronecker products of each matrix {\em ai} with itself
+       ++ for {1 <= i <= k}.
+       ++ Note: If the list of matrices corresponds to a group representation
+       ++ (repr. of generators) of one group, then these matrices correspond
+       ++ to the tensor product of the representation with itself.
+     permutationRepresentation : (PERM I, I) -> M I
+       ++ permutationRepresentation(pi,n) returns the matrix
+       ++ {\em (deltai,pi(i))} (Kronecker delta) for a permutation
+       ++ {\em pi} of {\em {1,2,...,n}}.
+     permutationRepresentation : L I -> M I
+       ++ permutationRepresentation(pi,n) returns the matrix
+       ++ {\em (deltai,pi(i))} (Kronecker delta) if the permutation
+       ++ {\em pi} is in list notation and permutes {\em {1,2,...,n}}.
+     permutationRepresentation : (L PERM I, I) -> L M I
+       ++ permutationRepresentation([pi1,...,pik],n) returns the list
+       ++ of matrices {\em [(deltai,pi1(i)),...,(deltai,pik(i))]}
+       ++ (Kronecker delta) for the permutations {\em pi1,...,pik} 
+       ++ of {\em {1,2,...,n}}.
+     permutationRepresentation : L L I -> L M I
+       ++ permutationRepresentation([pi1,...,pik],n) returns the list
+       ++ of matrices {\em [(deltai,pi1(i)),...,(deltai,pik(i))]}
+       ++ if the permutations {\em pi1},...,{\em pik} are in 
+       ++ list notation and are permuting {\em {1,2,...,n}}.
+ 
+   private ==> add
+ 
+     -- import of domains and packages
+ 
+     import OutputForm
+ 
+     -- declaration of local functions:
+ 
+ 
+     calcCoef : (L I, M I) -> I
+       -- calcCoef(beta,C) calculates the term
+       -- |S(beta) gamma S(alpha)| / |S(beta)|
+ 
+ 
+     invContent : L I -> V I
+       -- invContent(alpha) calculates the weak monoton function f with
+       -- f : m -> n with invContent alpha. f is stored in the returned
+       -- vector
+ 
+ 
+     -- definition of local functions
+ 
+ 
+     calcCoef(beta,C) ==
+       prod : I := 1
+       for i in 1..maxIndex beta repeat
+         prod := prod * multinomial(beta(i), entries row(C,i))$ICF
+       prod
+ 
+ 
+     invContent(alpha) ==
+       n : NNI := (+/alpha)::NNI
+       f : V I  := new(n,0)
+       i : NNI := 1
+       j : I   := - 1
+       for og in alpha repeat
+         j := j + 1
+         for k in 1..og repeat
+           f(i) := j
+           i := i + 1
+       f
+ 
+ 
+     -- exported functions:
+ 
+ 
+ 
+     if R has commutative("*") then
+       antisymmetricTensors ( a : M R , k : PI ) ==
+ 
+         n      : NNI   := nrows a
+         k = 1 => a
+         k > n =>
+           error("second parameter for antisymmetricTensors is too large")
+         m      :   I   := binomial(n,k)$ICF
+         il     : L L I   := [subSet(n,k,i)$SGCF for i in 0..m-1]
+         b      :  M R   := zero(m::NNI, m::NNI)
+         for i in 1..m repeat
+           for j in 1..m repeat
+             c : M R := zero(k,k)
+             lr: L I := il.i
+             lt: L I := il.j
+             for  r in 1..k repeat
+               for t in 1..k repeat
+                 rr : I := lr.r
+                 tt : I := lt.t
+                 --c.r.t := a.(1+rr).(1+tt)
+                 setelt(c,r,t,elt(a, 1+rr, 1+tt))
+             setelt(b, i, j, determinant c)
+         b
+ 
+ 
+     if R has commutative("*") then
+       antisymmetricTensors(la: L M R, k: PI) ==
+         [antisymmetricTensors(ma,k) for ma in la]
+ 
+ 
+ 
+     symmetricTensors (a : M R, n : PI) ==
+ 
+       m : NNI := nrows a
+       m ^= ncols a =>
+         error("Input to symmetricTensors is no square matrix")
+       n = 1 => a
+ 
+       dim : NNI := (binomial(m+n-1,n)$ICF)::NNI
+       c : M R := new(dim,dim,0)
+       f : V I := new(n,0)
+       g : V I := new(n,0)
+       nullMatrix : M I := new(1,1,0)
+       colemanMatrix : M I
+ 
+       for i in 1..dim repeat
+         -- unrankImproperPartitions1 starts counting from 0
+         alpha := unrankImproperPartitions1(n,m,i-1)$SGCF
+         f := invContent(alpha)
+         for j in 1..dim repeat
+           -- unrankImproperPartitions1 starts counting from 0
+           beta := unrankImproperPartitions1(n,m,j-1)$SGCF
+           g := invContent(beta)
+           colemanMatrix := nextColeman(alpha,beta,nullMatrix)$SGCF
+           while colemanMatrix ^= nullMatrix repeat
+             gamma := inverseColeman(alpha,beta,colemanMatrix)$SGCF
+             help : R := calcCoef(beta,colemanMatrix)::R
+             for k in 1..n repeat
+               help := help * a( (1+f k)::NNI, (1+g(gamma k))::NNI )
+             c(i,j) := c(i,j) + help
+             colemanMatrix := nextColeman(alpha,beta,colemanMatrix)$SGCF
+           -- end of while
+         -- end of j-loop
+       -- end of i-loop
+ 
+       c
+ 
+ 
+     symmetricTensors(la : L M R, k : PI) ==
+       [symmetricTensors (ma, k) for ma in la]
+ 
+ 
+     tensorProduct(a: M R, b: M R) ==
+       n      : NNI := nrows a
+       m      : NNI := nrows b
+       nc     : NNI := ncols a
+       mc     : NNI := ncols b
+       c      : M R   := zero(n * m, nc * mc)
+       indexr : NNI := 1                             --   row index
+       for i in 1..n repeat
+          for k in 1..m repeat
+             indexc : NNI := 1                       --   column index
+             for j in 1..nc repeat
+                for l in 1..mc repeat
+                   c(indexr,indexc) := a(i,j) * b(k,l)
+                   indexc          := indexc + 1
+             indexr := indexr + 1
+       c
+ 
+ 
+     tensorProduct (la: L M R, lb: L M R) ==
+       [tensorProduct(la.i, lb.i) for i in 1..maxIndex la]
+ 
+ 
+     tensorProduct(a : M R) == tensorProduct(a, a)
+ 
+     tensorProduct(la : L M R) ==
+       tensorProduct(la :: L M R, la :: L M R)
+ 
+     permutationRepresentation (p : PERM I, n : I) ==
+       -- permutations are assumed to permute {1,2,...,n}
+       a : M I := zero(n :: NNI, n :: NNI)
+       for i in 1..n repeat
+          a(eval(p,i)$(PERM I),i) := 1
+       a
+ 
+ 
+     permutationRepresentation (p : L I) ==
+       -- permutations are assumed to permute {1,2,...,n}
+       n : I := #p
+       a : M I := zero(n::NNI, n::NNI)
+       for i in 1..n repeat
+          a(p.i,i) := 1
+       a
+ 
+ 
+     permutationRepresentation(listperm : L PERM I, n : I) ==
+       -- permutations are assumed to permute {1,2,...,n}
+       [permutationRepresentation(perm, n) for perm in listperm]
+ 
+     permutationRepresentation (listperm : L L I) ==
+       -- permutations are assumed to permute {1,2,...,n}
+       [permutationRepresentation perm for perm in listperm]
+ 
+     createGenericMatrix(m) ==
+       res : M P R := new(m,m,0$(P R))
+       for i in 1..m repeat
+         for j in 1..m repeat
+            iof : OF := coerce(i)$Integer
+            jof : OF := coerce(j)$Integer
+            le : L OF := cons(iof,list jof)
+            sy : Symbol := subscript(x::Symbol, le)$Symbol
+            res(i,j) := (sy :: P R)
+       res
+
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--All rights reserved.
+--
+--Redistribution and use in source and binary forms, with or without
+--modification, are permitted provided that the following conditions are
+--met:
+--
+--    - Redistributions of source code must retain the above copyright
+--      notice, this list of conditions and the following disclaimer.
+--
+--    - Redistributions in binary form must reproduce the above copyright
+--      notice, this list of conditions and the following disclaimer in
+--      the documentation and/or other materials provided with the
+--      distribution.
+--
+--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+--      names of its contributors may be used to endorse or promote products
+--      derived from this software without specific prior written permission.
+--
+--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+@
+<<*>>=
+<<license>>
+
+<<package REP1 RepresentationPackage1>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}