- Timestamp:
- Apr 28, 2005, 6:32:06 PM (19 years ago)
- Branches:
- (u'spielwiese', '2a584933abf2a2d3082034c7586d38bb6de1a30a')
- Children:
- fefe03c337c2d780101b98e7858eb2538ed41587
- Parents:
- fdebd3cb4759f0739bae5afd99a64ab39ae4efb4
- Location:
- Singular/LIB
- Files:
-
- 3 edited
-
ncdecomp.lib (modified) (5 diffs)
-
nctools.lib (modified) (18 diffs)
-
qmatrix.lib (modified) (3 diffs)
Legend:
- Unmodified
- Added
- Removed
-
Singular/LIB/ncdecomp.lib
rfdebd3 rcfc9c0 1 1 /////////////////////////////////////////////////////////////////////////////// 2 version="$Id: ncdecomp.lib,v 1. 7 2005-03-25 18:38:46 levandov Exp $";2 version="$Id: ncdecomp.lib,v 1.8 2005-04-28 16:32:06 levandov Exp $"; 3 3 category="Noncommutative"; 4 4 info=" … … 105 105 /////////////////////////////////////////////////////////////////////////////// 106 106 proc CentralQuot(module I, ideal G) 107 "USAGE: CentralQuot(M, T), for a module M and an ideal T, 108 RETURN: module of the central quotient I:G, 107 "USAGE: CentralQuot(M, G), for a module M and an ideal G, 108 PURPOSE: compute the central quotient I:G 109 RETURN: module 109 110 NOTE: the output module is not necessarily a Groebner basis, 110 111 SEE ALSO: CentralSaturation, CenCharDec … … 139 140 proc CentralSaturation(module M, ideal T) 140 141 "USAGE: CentralSaturation(M, T), for a module M and an ideal T, 141 RETURN: module of the central saturation of M by T (also denoted by M:T^{\infty}), 142 PURPOSE: compute the central saturation of M by T (also denoted by M:T^{\infty}) 143 RETURN: module 142 144 NOTE: the output module is not necessarily a Groebner basis, 143 145 SEE ALSO: CentralQuot, CenCharDec … … 178 180 proc CenCharDec(module I, def #) 179 181 "USAGE: CenCharDec(I, C); I a module, C an ideal/list of generators of the center; 180 PURPOSE: compute a finite central character decomposition (or point outthat there is no finite one),182 PURPOSE: compute a finite central character decomposition (or determine that there is no finite one), 181 183 RETURN: a list L, where each entry consists of three records: 182 184 @* L[*][1] ('ideal' type), the central character as the maximal ideal in the center, 183 185 @* L[*][2] ('module' type), the Groebner basis of the weight module, corresponding to the character, 184 @* L[*][3] ('int' type) is the K-dimension of the weight module (-1 is returned for aninfinite dimension);185 NOTE: some modules have no finite decomposition (in such case one186 186 @* L[*][3] ('int' type) is the K-dimension of the weight module (-1 is returned in the case of infinite dimension); 187 NOTE: actual decomposition is a sum of L[i][2]; 188 @* some modules have no finite decomposition (in such case one gets warning message) 187 189 SEE ALSO: CentralQuot, CentralSaturation 188 190 EXAMPLE: example CenCharDec; shows examples … … 327 329 "USAGE: IntersectWithSub(M,Z), M an ideal, Z an ideal/list of pairwise commutative elements 328 330 PURPOSE: computes an intersection of M with the subalgebra, generated by Z 329 RETURN: ideal (of two- -sided generators, not a Groebner basis!)331 RETURN: ideal (of two-sided generators, not a Groebner basis!) 330 332 NOTE: usually one puts generators of the center into Z 331 333 EXAMPLE: example IntersectWithSub; shows an example -
Singular/LIB/nctools.lib
rfdebd3 rcfc9c0 1 1 /////////////////////////////////////////////////////////////////////////////// 2 version="$Id: nctools.lib,v 1.1 0 2005-02-23 18:10:46 levandov Exp $";2 version="$Id: nctools.lib,v 1.11 2005-04-28 16:32:06 levandov Exp $"; 3 3 category="Noncommutative"; 4 4 info=" … … 16 16 RootOfUnity(n); the minimal polynomial for the n-th primitive root of unity, 17 17 Weyl([p]); creates Weyl algebra structure in a basering (two different realizations), 18 CreateWeyl(n, [p]); returns n-th Weyl algebra in (x(i),D(i)) presentation ; in char p, if an integer p is given,19 Heisenberg(N, [p,d]); returns N-th Heisenberg algebra in (x(i),y(i),h) realization,18 CreateWeyl(n, [p]); returns n-th Weyl algebra in (x(i),D(i)) presentation, 19 Heisenberg(N, [p,d]); returns n-th Heisenberg algebra in (x(i),y(i),h) realization, 20 20 Exterior(); returns qring, the exterior algebra of a basering, 21 Fin_dim_algebra(matrix M, list #); creates finite dimensional algebra structure from the basering and the multiplication matrix M,21 fin_dim_algebra(matrix M, list #); creates finite dimensional algebra structure from the basering and the multiplication matrix M, 22 22 23 23 AUXILIARY PROCEDURES: 24 Newton(f); Newton diagram of a polynomial f,24 Newton(f); Newton diagram of a polynomial, 25 25 NCRelations(r); recovers the non-commutative relations of a G-algebra, 26 IsCentral(p,[v]); check for the commutativity of polynomial p withthe G-algebra,26 IsCentral(p,[v]); check for the commutativity of a polynomial in the G-algebra, 27 27 Is_NC(); checks whether basering is noncommutative, 28 28 UpOneMatrix(N); returns NxN matrix with 1's in the whole upper triagle, … … 34 34 /////////////////////////////////////////////////////////////////////////////// 35 35 36 // This procedure computes a weights vector for a G-Algebra 37 // r must be a G-algebra 36 // This procedure computes a weights vector for a G-algebra r 38 37 39 38 proc Gweights(def r) 40 39 "USAGE: Gweights(r); r a ring or a square matrix 41 RETURN: aweight vector for the G-algebra:42 r itself, if it is of the type ring,40 PURPOSE: compute the weight vector for the G-algebra: 41 either r itself, if it is of the type ring, 43 42 or for a G-algebra, defined by the square matrix r 43 RETURN: intvec 44 44 NOTE: with Gweights you obtain a vector, which must be used to redefine the G-Algebra. If the input is a matrix and the output is the zero vector then there is not a G-algebra structure associated to these relations with respect to the given variables. Another possibility is to use wRing or weightedRing to obtain directly the G-Algebra. 45 45 EXAMPLE: example Gweights; shows examples … … 122 122 proc weightedRing(def r) 123 123 "USAGE: weightedRing(r); r a ring 124 RETURN: ring with the relations of r with order changed to comply with 125 the ordering condition for G-algebras 126 NOTE: you have to activate this ring with the "setring" command 124 PURPOSE: equip a ring with such weights,that the relations of a new ring 125 comply with the ordering condition for G-algebras 126 RETURN: ring 127 NOTE: you have to activate this ring with the \"setring\" command 127 128 EXAMPLE: example weightedRing; shows examples 128 129 SEE ALSO: wRing, Gweights … … 337 338 proc Newton(poly f) 338 339 "USAGE: Newton(f); f a poly 339 RETURN: intmat, representing the Newton diagram of f 340 PURPOSE: compute the Newton diagram of f 341 RETURN: intmat 340 342 NOTE: each row is the exponent of a monomial of f 341 343 EXAMPLE: example Newton; shows examples … … 373 375 proc NCRelations(def r) 374 376 "USAGE: NCRelations(r); r a ring 375 RETURN: a list with two elements, both elements are of type matrix and 376 represent the matrices C,D defining the non-commutative relations 377 of the G-algebra r 377 PURPOSE: recover the noncommutative relations via matrices C and D from 378 a noncommutative ring 379 RETURN: list L with two elements, both elements are of type matrix: 380 L[1] = matrix of coefficients C, 381 L[2] = matrix of polynomials D 378 382 EXAMPLE: example NCRelations; shows examples 379 383 "{ … … 433 437 /////////////////////////////////////////////////////////////////////////////// 434 438 435 proc Fin_dim_algebra(matrix M, list #) 436 "USAGE: Fin_dim_algebra(M,[r]); M a matrix, r an optional ring. 437 RETURN: nothing. Creates finite dimensional algebra structure in a ring r 438 (if it is given) or in a basering (by default) from the matrix M. Exports the ideal called Quot for further qring definition. 439 NOTE: matrix M will be read according to relations x_j*x_i = M[i,j]; Quot is not given in its two-sided Groebner basis. 440 EXAMPLE: Fin_dim_algebra; shows examples 439 proc fin_dim_algebra(matrix M, list #) 440 "USAGE: fin_dim_algebra(M,[r]); M a matrix, r an optional ring. 441 PURPOSE: creates a finite dimensional algebra structure in a ring r 442 (if it is given) or in a basering (by default) from the matrix M. 443 Exports the ideal called Quot for further qring definition. 444 RETURN: nothing 445 NOTE: matrix M will be read according to relations Xj*Xi = M[i,j]; Quot is not given in its two-sided Groebner basis. 446 EXAMPLE: example fin_dim_algebra; shows examples 441 447 " 442 448 { … … 489 495 matrix S[3][3]; 490 496 S[2,3]=a*x(1); S[3,2]=-b*x(1); 491 Fin_dim_algebra(S);497 fin_dim_algebra(S); 492 498 Quot = twostd(Quot); 493 499 qring Qr = Quot; … … 498 504 499 505 proc IsCentral(poly p, list #) 500 "USAGE: IsCentral(p,[v]); p poly, v an integer (with v!=0 procedure will be verbose) 501 RETURN: integer (1 if p commutes with all variables, 0 otherwise) 506 "USAGE: IsCentral(p,[v]); p poly, v an integer (with v!=0, procedure will be verbose) 507 PURPOSE: check whether p is central in a basering (that is, commutes with everything) 508 RETURN: integer (1, if p commutes with all variables, 0 otherwise) 502 509 EXAMPLE: example IsCentral; shows examples 503 510 "{ … … 539 546 proc UpOneMatrix(int N) 540 547 "USAGE: UpOneMatrix(N); N an integer, the number of columns 541 RETURN: intmat, NxN matrix with 1's in the whole upper triagle 548 PURPOSE: compute an NxN matrix with 1's in the whole upper triagle 549 RETURN: intmat 542 550 NOTE: helpful for setting noncommutative algebras with complicated 543 551 coefficient matrices … … 567 575 /////////////////////////////////////////////////////////////////////////////// 568 576 proc ndc(list #) 569 "USAGE: ndc([v]); v an optional integer. If v!=0 procedure will be verbose 570 RETURN: ideal of non-degeneracy conditions of the basering 577 "USAGE: ndc([v]); v an optional integer (if v!=0, procedure will be verbose) 578 PURPOSE: compute the non-degeneracy conditions of the basering, 579 RETURN: ideal 571 580 EXAMPLE: example ndc; shows examples 572 581 " … … 633 642 proc RootOfUnity(int n) 634 643 "USAGE: RootOfUnity(n); n an integer 635 RETURN: number, the n-th primitive root of unity (for use as minpoly) 644 PURPOSE: compute the minimal polynomial for the n-th primitive root of unity 645 RETURN: number 636 646 NOTE: works only in field extensions by one element 637 647 EXAMPLE: example RootOfUnity; shows examples … … 711 721 proc Weyl(list #) 712 722 "USAGE: Weyl([p]); p an optional integer. 713 RETURN: nothing. Creates Weyl algebra structure in a basering. By default723 PURPOSE: create a Weyl algebra structure on a basering. By default 714 724 mimics (x(1..N),d(1..N)) realization. If p is given and is not zero, 715 725 uses (x(1),d(1),x(2),d(2),... ) realization. 726 RETURN: nothing 727 SEE ALSO: CreateWeyl 716 728 EXAMPLE: example Weyl; shows examples 717 729 " … … 769 781 proc Heisenberg(int N, list #) 770 782 "USAGE: Heisenberg(N, [p,d]); N an integer (setting 2*N+1 variables), p an optional integer (field characteristic), d an optional integer (power of h in the commutator) 771 RETURN: N-th Heisenberg algebra in x(i),y(i),h realization 772 NOTE: you have to activate this ring with the "setring" command 783 PURPOSE: create a N-th Heisenberg algebra in x(i),y(i),h realization 784 RETURN: nothing 785 NOTE: you have to activate this ring with the \"setring\" command 773 786 EXAMPLE: example Heisenberg; shows examples 774 787 " … … 805 818 "EXAMPLE:";echo=2; 806 819 def a = Heisenberg(2); 807 setring a; 808 a; 820 setring a; a; 821 def H3 = Heisenberg(3, 7, 2); 822 setring H3; H3; 809 823 } 810 824 … … 813 827 proc Exterior(list #) 814 828 "USAGE: Exterior(); 815 RETURN: qring, the exterior algebra of a basering 816 NOTE: you have to activate this qring with the "setring" command 829 PURPOSE: create the exterior algebra of a basering, 830 RETURN: qring 831 NOTE: you have to activate this qring with the \"setring\" command 817 832 EXAMPLE: example Exterior; shows examples 818 833 " … … 853 868 proc CreateWeyl(int n, list #) 854 869 "USAGE: CreateWeyl(n,[p]); n an integer, n>0; p an optional integer (field characteristic) 855 RETURN: a ring, describing n-th Weyl algebra 856 NOTE: You have to activate this ring with the "setring" command. 870 PURPOSE: create a n-th Weyl algebra 871 RETURN: ring 872 NOTE: you have to activate this ring with the \"setring\" command. 857 873 The presentation of n-th Weyl algebra is classical: 858 874 D(i)x(i)=x(i)D(i)+1 … … 896 912 proc Is_NC() 897 913 "USAGE: Is_NC(); 898 RETURN: 1, if basering is noncommutative; 0 otherwise. 914 PURPOSE: check whether a basering is commutative or not 915 RETURN: int (1, if basering is noncommutative; 0 otherwise) 899 916 EXAMPLE: example Is_NC; shows examples 900 917 "{ -
Singular/LIB/qmatrix.lib
rfdebd3 rcfc9c0 1 version="$Id: qmatrix.lib,v 1. 8 2005-02-23 18:10:46 levandov Exp $";1 version="$Id: qmatrix.lib,v 1.9 2005-04-28 16:32:06 levandov Exp $"; 2 2 category="Noncommutative"; 3 3 info=" … … 80 80 proc LengthSymElement(intvec v) 81 81 "USAGE: LengthSymElement(v); v an intvec representing an element of S(n) 82 RETURN: int, the length of v 82 PURPOSE: determine the length of v 83 RETURN: int 83 84 NOTE: if v doesn't represent an element of S(n), the output may have no sense 84 85 SEE ALSO: SymGroup, LengthSym … … 108 109 proc LengthSym(intmat M) 109 110 "USAGE: LengthSym(M); M an intmat representing a subset of S(n) (each row must be an element of S(n)) 110 RETURN: intvec, the i-th element is the length of the i-th row of M 111 PURPOSE: determine a vector, which i-th element is the length of the i-th row of M 112 RETURN: intvec 111 113 NOTE: If M is not a subset of S(n), the output may not have meaning 112 114 SEE ALSO: SymGroup, LengthSymElement
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