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symmetrica 2.0+ds-6
 `123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195` ``````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(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(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(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(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 <> of GLm(C). NAME: glpdg SYNOPSIS: INT glpdg(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 <> 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(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. */ ``````