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)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)$SGCF
-- end of while
-- end of j-loop
-- end of i-loop
c
symmetricTensors(la : L M R, k : PI) ==
[symmetricTensors (ma, k) for ma in la]
tensorProduct(a: M R, b: M R) ==
n : NNI := nrows a
m : NNI := nrows b
nc : NNI := ncols a
mc : NNI := ncols b
c : M R := zero(n * m, nc * mc)
indexr : NNI := 1 -- row index
for i in 1..n repeat
for k in 1..m repeat
indexc : NNI := 1 -- column index
for j in 1..nc repeat
for l in 1..mc repeat
c(indexr,indexc) := a(i,j) * b(k,l)
indexc := indexc + 1
indexr := indexr + 1
c
tensorProduct (la: L M R, lb: L M R) ==
[tensorProduct(la.i, lb.i) for i in 1..maxIndex la]
tensorProduct(a : M R) == tensorProduct(a, a)
tensorProduct(la : L M R) ==
tensorProduct(la :: L M R, la :: L M R)
permutationRepresentation (p : PERM I, n : I) ==
-- permutations are assumed to permute {1,2,...,n}
a : M I := zero(n :: NNI, n :: NNI)
for i in 1..n repeat
a(p.i,i) := 1
a
permutationRepresentation (p : L I) ==
-- permutations are assumed to permute {1,2,...,n}
n : I := #p
a : M I := zero(n::NNI, n::NNI)
for i in 1..n repeat
a(p.i,i) := 1
a
permutationRepresentation(listperm : L PERM I, n : I) ==
-- permutations are assumed to permute {1,2,...,n}
[permutationRepresentation(perm, n) for perm in listperm]
permutationRepresentation (listperm : L L I) ==
-- permutations are assumed to permute {1,2,...,n}
[permutationRepresentation perm for perm in listperm]
createGenericMatrix(m) ==
res : M P R := new(m,m,0$(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
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