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NAME:
dimension_symmetrization
SYNOPSIS:
INT dimension_symmetrization(OP n,part,a)
DESCRIPTION:
computes the dimension of the degree of a irreducible
representation of the GL_n, n is a INTEGER object, labeled
by the PARTITION object a.
RETURN:
OK if no error occured.
EXAMPLE:
#include "def.h"
#include "macro.h"
main()
{
OP a,b,c;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
printeingabe("Enter the degree of the linear group:");
scan(INTEGER,b);
printeingabe("Enter the part labeling the irrep of the linear group:");
scan(PARTITION,a);
printeingabe("The degree of the irrep is:");
dimension_symmetrization(b,a,c);
println(c);
freeall(a); freeall(b); freeall(c);
ende();
}
NAME:
bdg
SYNOPSIS:
INT bdg(part,perm,D); OP part,perm,D;
DESCRIPTION:
Calculates the irreduzible matrix representation
D^part(perm), whose entries are of integral numbers.
reference: H. Boerner:
Darstellungen von Gruppen, Springer 1955.
pp. 104-107.
NAME:
sdg
SYNOPSIS:
INT sdg(part,perm,D); OP part,perm,D;
DESCRIPTION:
Calculates the irreduzible matrix representation
D^part(perm), which consists of rational numbers.
reference: G. James/ A. Kerber:
Representation Theory of the Symmetric Group.
Addison/Wesley 1981.
pp. 124-126.
NAME:
odg
SYNOPSIS:
INT odg(part,perm,D); OP part,perm,D;
DESCRIPTION:
Calculates the irreduzible matrix representation
D^part(perm), which consists of real numbers.
reference: G. James/ A. Kerber:
Representation Theory of the Symmetric Group.
Addison/Wesley 1981.
pp. 127-129.
COMMENT:
4. Polynomial Representations of GLm(C):
---------------------------------------
GENERAL HINT:
For the routines below are just for constructing small examples
and very time-and space-consuming, the user should take care
that his calculations do not exceed the following limits:
i) Group size <= 1000.
ii) Matrix size (m^n) <= 256.
a) Decomposition of the n-fold tensorproduct of the identical
representation of Glm(C) onto itself.
NAME:
glmndg
SYNOPSIS:
INT glmndg(m,n,M,VAR); OP m,n,M; INT VAR;
DESCRIPTION:
If VAR is equal to 0L the orthogonal representation
is used for the decomposition, otherwise, if VAR
equals 1L, the natural representation is considered.
The result is the VECTOR-Object M, consisting of
components of type MATRIX, representing the several
irreducible matrix representations of GLm(C) with
part_1' <= m, where part is a partition of n.
COMMENT:
b) Calculation of only one polynomial representation <<part>>
of GLm(C).
NAME:
glpdg
SYNOPSIS:
INT glpdg(m,part,M); OP m,part,M;
DESCRIPTION:
part has to be an PARTITION object with not more
than m parts.
For this partition, the program calculates the
polynomial irreducible representation
<<part>> of GLm(C), which ist stored in the
MATRIX-Object M.
reference: 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 Applicandae Mathematicae 8 (1987).
(Describes a method different from
the one implemented here, but gives a lot
of theoratical background.)
COMMENT:
5. Checking Homomorphy of Representations of GLm(C):
------------------------------------------------
NAME:
glm_homtest
SYNOPSIS:
INT glm_homtest(m,M); OP m,M;
DESCRIPTION:
The relation D(A)*D(B) = D(A*B) is verified
with two random integer matrices.
In case of M not being a representation, the
procedure displays an error message to stdout.
COMMENT:
/* Documentation of routines, concerning the calculation of
symmetry adapted bases for general finite permutation groups
1. Calculating of a general symmetry adapted Basis:
--------------------------------------------------
SYNOPSIS: sab_input(S,SMat,M); OP S,SMat,M;
group_gen(S,SMat,D,Di); OP S,SMat,D,Di;
sab(Di,D,B,M,mpc); OP Di,D,B,M,mpc;
The procedure sab_input reads the necessary input from the
standard-input.
The input-format is as follows:
--------------------------------------------------------------
nr of generators of G | orderS
(INTEGER ) |
set S of generators of G | S
(VECTOR of PERMUTATIONS of |
length n, where G <= Sn) |
nr. of irred. representations | anz_irr
(INTEGER ) |
matrices of irr.representations |
for the elements s in S | SMat
(VECTOR of VECTOR of MATRIX) |
symmetric operator M | M
(MATRIX) |
--------------------------------------------------------------
With this input, group_gen calulates the whole symmetry group G.
The group elements are stored in D the first line of their
irreducible matrix representations are stored in Di in the
order of the invers elements. D has the same type as S and Di
is a threedimensional VECTOR structure.
Finally sab can be called, which calculates the symmetry adapted
basis in B and the decomposed Operator in M as a vector of
matrices representing the blockdiagonal structure of M.
Every block occures once, its multiplicity ist stored in the
vector mpc.
REFERENCE: E.Stiefel/A.Faessler:
Gruppentheoretische Methoden und ihre Anwendung
Teubner, 1979.
*/
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