Changeset fd5013 in git for Singular/LIB/finvar.lib
- Timestamp:
- Aug 2, 2006, 5:40:53 PM (18 years ago)
- Branches:
- (u'spielwiese', 'fe61d9c35bf7c61f2b6cbf1b56e25e2f08d536cc')
- Children:
- f6f1dbfc1e8487ccbf1ca0d7a3d2600069315a9f
- Parents:
- d1932987874a4523d2a42efe3046b953ed52af76
- File:
-
- 1 edited
-
Singular/LIB/finvar.lib (modified) (28 diffs)
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Singular/LIB/finvar.lib
rd193298 rfd5013 1 1 /////////////////////////////////////////////////////////////////////////////// 2 version="$Id: finvar.lib,v 1.5 0 2006-07-18 15:48:13Singular Exp $"2 version="$Id: finvar.lib,v 1.51 2006-08-02 15:40:47 Singular Exp $" 3 3 category="Invariant theory"; 4 4 info=" … … 138 138 RETURN: a <list>, the first list element will be a gxn <matrix> representing 139 139 the Reynolds operator if we are in the non-modular case; if the 140 characteristic is >0, minpoly==0 and the finite group non-cyclic the140 characteristic is >0, minpoly==0 and the finite group is non-cyclic the 141 141 second list element is an <int> giving the lowest common multiple of 142 142 the matrix group elements' order (used in molien); in general all … … 148 148 (or the generators themselves during the first run). All the ones that 149 149 have been generated before are thrown out and the program terminates 150 when no new elements found in one run. Additionally each time a new150 when no new elements are found in one run. Additionally each time a new 151 151 group element is found the corresponding ring mapping of which the 152 152 Reynolds operator is made up is generated. They are stored in the rows … … 358 358 elements generated by group_reynolds(), lcm is the second return value 359 359 of group_reynolds() 360 RETURN: in case of characteristic 0 a 1x2 <matrix> giving enumerator and360 RETURN: in case of characteristic 0 a 1x2 <matrix> giving numerator and 361 361 denominator of Molien series; in case of prime characteristic a ring 362 362 with the name `ringname` of characteristic 0 is created where the same … … 908 908 new group element is found the corresponding ring mapping of which the 909 909 Reynolds operator is made up is generated. They are stored in the rows 910 of the first return value. In characteristic 0 the term s1/det(1-xE)910 of the first return value. In characteristic 0 the term 1/det(1-xE) 911 911 is computed whenever a new element E is found. In prime characteristic 912 912 a Brauer lift is involved and the terms are only computed after the 913 913 entire matrix group is generated (to avoid the modular case). The 914 returned matrix gives enumerator and denominator of the expanded914 returned matrix gives numerator and denominator of the expanded 915 915 version where common factors have been canceled. 916 916 EXAMPLE: example reynolds_molien; shows an example … … 2064 2064 ASSUME: REY is the first return value of group_reynolds or reynolds_molien and 2065 2065 M the one of molien or the second one of reynolds_molien 2066 DISPLAY: information about the various stages of the program meif v does not2066 DISPLAY: information about the various stages of the program if v does not 2067 2067 equal 0 2068 2068 RETURN: primary invariants (type <matrix>) of the invariant ring … … 2207 2207 ringname gives the name of a ring of characteristic 0 that has been 2208 2208 created by molien or reynolds_molien 2209 DISPLAY: information about the various stages of the program meif v does not2209 DISPLAY: information about the various stages of the program if v does not 2210 2210 equal 0 2211 2211 RETURN: primary invariants (type <matrix>) of the invariant ring … … 2360 2360 <int> 2361 2361 ASSUME: REY is the first return value of group_reynolds or reynolds_molien 2362 DISPLAY: information about the various stages of the program meif v does not2362 DISPLAY: information about the various stages of the program if v does not 2363 2363 equal 0 2364 2364 RETURN: primary invariants (type <matrix>) of the invariant ring and an … … 2503 2503 <int> 2504 2504 ASSUME: REY is the first return value of group_reynolds or reynolds_molien 2505 DISPLAY: information about the various stages of the program meif v does not2505 DISPLAY: information about the various stages of the program if v does not 2506 2506 equal 0 2507 2507 RETURN: primary invariants (type <matrix>) of the invariant ring and an … … 2644 2644 G1,G2,...: <matrices> generating a finite matrix group, v: an optional 2645 2645 <int> 2646 DISPLAY: information about the various stages of the program meif v does not2646 DISPLAY: information about the various stages of the program if v does not 2647 2647 equal 0 2648 2648 RETURN: primary invariants (type <matrix>) of the invariant ring … … 2784 2784 G1,G2,...: <matrices> generating a finite matrix group, flags: an 2785 2785 optional <intvec> with three entries, if the first one equals 0 (also 2786 the default), the program meattempts to compute the Molien series and2787 Reynolds operator, if it equals 1, the program meis told that the2786 the default), the program attempts to compute the Molien series and 2787 Reynolds operator, if it equals 1, the program is told that the 2788 2788 Molien series should not be computed, if it equals -1 characteristic 0 2789 2789 is simulated, i.e. the Molien series is computed as if the base field 2790 2790 were characteristic 0 (the user must choose a field of large prime 2791 characteristic, e.g. 32003) and if the first one is anything else, it2791 characteristic, e.g. 32003), and if the first one is anything else, it 2792 2792 means that the characteristic of the base field divides the group 2793 2793 order, the second component should give the size of intervals between … … 2797 2797 common factors should always be canceled when the expansion is simple 2798 2798 (the root of the extension field occurs not among the coefficients) 2799 DISPLAY: information about the various stages of the program meif the third2799 DISPLAY: information about the various stages of the program if the third 2800 2800 flag does not equal 0 2801 2801 RETURN: primary invariants (type <matrix>) of the invariant ring and if … … 2892 2892 else 2893 2893 { if (v) 2894 { " Since it is impossible for this program meto calculate the Molien series for";2894 { " Since it is impossible for this program to calculate the Molien series for"; 2895 2895 " invariant rings over extension fields of prime characteristic, we have to"; 2896 2896 " continue without it."; … … 3199 3199 ASSUME: REY is the first return value of group_reynolds or reynolds_molien and 3200 3200 M the one of molien or the second one of reynolds_molien 3201 DISPLAY: information about the various stages of the program meif v does not3201 DISPLAY: information about the various stages of the program if v does not 3202 3202 equal 0 3203 3203 RETURN: primary invariants (type <matrix>) of the invariant ring … … 3346 3346 ringname gives the name of a ring of characteristic 0 that has been 3347 3347 created by molien or reynolds_molien 3348 DISPLAY: information about the various stages of the program meif v does not3348 DISPLAY: information about the various stages of the program if v does not 3349 3349 equal 0 3350 3350 RETURN: primary invariants (type <matrix>) of the invariant ring … … 3499 3499 bases elements, v: an optional <int> 3500 3500 ASSUME: REY is the first return value of group_reynolds or reynolds_molien 3501 DISPLAY: information about the various stages of the program meif v does not3501 DISPLAY: information about the various stages of the program if v does not 3502 3502 equal 0 3503 3503 RETURN: primary invariants (type <matrix>) of the invariant ring and an … … 3646 3646 bases elements, v: an optional <int> 3647 3647 ASSUME: REY is the first return value of group_reynolds or reynolds_molien 3648 DISPLAY: information about the various stages of the program meif v does not3648 DISPLAY: information about the various stages of the program if v does not 3649 3649 equal 0 3650 3650 RETURN: primary invariants (type <matrix>) of the invariant ring and an … … 3792 3792 where -|r| to |r| is the range of coefficients of the random 3793 3793 combinations of bases elements, v: an optional <int> 3794 DISPLAY: information about the various stages of the program meif v does not3794 DISPLAY: information about the various stages of the program if v does not 3795 3795 equal 0 3796 3796 RETURN: primary invariants (type <matrix>) of the invariant ring … … 3943 3943 where -|r| to |r| is the range of coefficients of the random 3944 3944 combinations of bases elements, flags: an optional <intvec> with three 3945 entries, if the first one equals 0 (also the default), the program me3945 entries, if the first one equals 0 (also the default), the program 3946 3946 attempts to compute the Molien series and Reynolds operator, if it 3947 equals 1, the program meis told that the Molien series should not be3947 equals 1, the program is told that the Molien series should not be 3948 3948 computed, if it equals -1 characteristic 0 is simulated, i.e. the 3949 3949 Molien series is computed as if the base field were characteristic 0 3950 3950 (the user must choose a field of large prime characteristic, e.g. 3951 32003) and if the first one is anything else, it means that the3951 32003), and if the first one is anything else, it means that the 3952 3952 characteristic of the base field divides the group order, the second 3953 3953 component should give the size of intervals between canceling common … … 3957 3957 always be canceled when the expansion is simple (the root of the 3958 3958 extension field does not occur among the coefficients) 3959 DISPLAY: information about the various stages of the program meif the third3959 DISPLAY: information about the various stages of the program if the third 3960 3960 flag does not equal 0 3961 3961 RETURN: primary invariants (type <matrix>) of the invariant ring and if … … 4060 4060 else 4061 4061 { if (v) 4062 { " Since it is impossible for this program meto calculate the Molien series for";4062 { " Since it is impossible for this program to calculate the Molien series for"; 4063 4063 " invariant rings over extension fields of prime characteristic, we have to"; 4064 4064 " continue without it."; … … 5954 5954 Molien series is computed as if the base field were characteristic 0 5955 5955 (the user must choose a field of large prime characteristic, e.g. 5956 32003) and if the first one is anything else, it means that the5956 32003), and if the first one is anything else, it means that the 5957 5957 characteristic of the base field divides the group order (i.e. it will 5958 5958 not even be attempted to compute the Reynolds operator or Molien … … 6051 6051 else 6052 6052 { if (v) 6053 { " Since it is impossible for this program meto calculate the Molien6053 { " Since it is impossible for this program to calculate the Molien 6054 6054 series for"; 6055 6055 " invariant rings over extension fields of prime characteristic, we … … 6171 6171 i.e. the Molien series is computed as if the base field were 6172 6172 characteristic 0 (the user must choose a field of large prime 6173 characteristic, e.g. 32003) and if the first one is anything else,6173 characteristic, e.g. 32003), and if the first one is anything else, 6174 6174 then the characteristic of the base field divides the group order 6175 6175 (i.e. we will not even attempt to compute the Reynolds operator or … … 6286 6286 else 6287 6287 { if (v) 6288 { " Since it is impossible for this program meto calculate the Molien6288 { " Since it is impossible for this program to calculate the Molien 6289 6289 series for"; 6290 6290 " invariant rings over extension fields of prime characteristic, we … … 6403 6403 THEORY: The ideal of algebraic relations of the invariant ring generators is 6404 6404 calculated, then the variables of the original ring are eliminated and 6405 the polynomials that are left over define the orbit variety 6405 the polynomials that are left over define the orbit variety. 6406 6406 EXAMPLE: example orbit_variety; shows an example 6407 6407 " … … 6620 6620 @* s: a <string> giving a name for a new ring 6621 6621 RETURN: The procedure ends with a new ring named s. 6622 It contains a Groebner basis 6623 (type <ideal>, named G) for the ideal defining the 6624 relative orbit variety with respect to I in the new ring. 6622 It contains a Groebner basis (type <ideal>, named G) for the ideal 6623 defining the relative orbit variety with respect to I in the new ring. 6625 6624 THEORY: A Groebner basis of the ideal of algebraic relations of the invariant 6626 6625 ring generators is calculated, then one of the basis elements plus the … … 6707 6706 @* F: a 1xm <matrix> defining an invariant ring of some matrix group 6708 6707 RETURN: The <ideal> defining the image under that group of the variety defined 6709 by I 6708 by I. 6710 6709 THEORY: rel_orbit_variety(I,F) is called and the newly introduced 6711 6710 @* variables in the output are replaced by the generators of the 6712 6711 @* invariant ring. This ideal in the original variables defines the image 6713 @* of the variety defined by I 6712 @* of the variety defined by I. 6714 6713 EXAMPLE: example image_of_variety; shows an example 6715 6714 "
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