Changeset fc4d39 in git for Singular/LIB/finvar.lib


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Timestamp:
Jun 25, 2007, 8:14:48 PM (17 years ago)
Author:
Simon King <king@…>
Branches:
(u'spielwiese', 'fe61d9c35bf7c61f2b6cbf1b56e25e2f08d536cc')
Children:
c3e83ff6d02f8f4c286a0e7293ff8c8e13630af8
Parents:
39a4a17a3e58df1244f46113c77d22d50c8db2e5
Message:
e-mail updated. Comments updated.


git-svn-id: file:///usr/local/Singular/svn/trunk@10149 2c84dea3-7e68-4137-9b89-c4e89433aadc
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  • Singular/LIB/finvar.lib

    r39a4a17 rfc4d39  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: finvar.lib,v 1.68 2007-03-29 14:36:01 king Exp $"
     2version="$Id: finvar.lib,v 1.69 2007-06-25 18:14:48 king Exp $"
    33category="Invariant theory";
    44info="
    55LIBRARY:  finvar.lib    Invariant Rings of Finite Groups
    66AUTHOR: Agnes E. Heydtmann, email: agnes@math.uni-sb.de;
    7         Simon A. King, email: king@mfo.de
     7        Simon A. King, email: king@mathematik.uni-jena.de
    88OVERVIEW:
    99 A library for computing polynomial invariants of finite matrix groups and
    1010 generators of related varieties. The algorithms are based on B. Sturmfels,
    11  G. Kemper and W. Decker et al..
     11 G. Kemper, S. King and W. Decker et al..
    1212
    1313MAIN PROCEDURES:
    … …  
    1616 primary_invariants()              primary invariants (p.i.)
    1717 primary_invariants_random()       primary invariants, randomized alg.
    18  invariant_algebra_reynolds()      minimal (algebra) generating set for the invariant
     18 invariant_algebra_reynolds()      minimal generating set for the invariant
    1919                                   ring of a finite matrix group, in the non-modular case
    20  invariant_algebra_perm()          minimal (algebra) generating set for the invariant
     20 invariant_algebra_perm()          minimal generating set for the invariant
    2121                                   ring or a permutation group, in the non-modular case
    2222
    … …  
    7070// p_search_random                 searches a # of p.i., char p, randomized
    7171// concat_intmat                   concatenates two integer matrices
    72 // GetGroup                        disjoint cycle presentation of perm. group
    73 //                                 -> permuation matrices
     72// GetGroup                        gets disjoint cycle presentation of perm. group,
     73//                                 yields permuation matrices
    7474///////////////////////////////////////////////////////////////////////////////
    7575
    … …  
    44604460          of monomials) of the basering modulo the primary invariants, mapping
    44614461          those to invariants with the Reynolds operator. Among these images
    4462           or their power products we pick a maximal subset that is linearly
    4463           independent modulo the primary invariants using Groebner basis techniques
    4464           (see paper \"Fast Computation of Secondary Invariants\" by S. King).
     4462          or their power products we pick secondary invariants using Groebner
     4463          basis techniques (see S. King: Fast Computation of Secondary
     4464          Invariants).
    44654465          The size of this set can be read off from the Molien series.
    44664466NOTE:     Secondary invariants are not uniquely determined by the given data.
    … …  
    49124912          of monomials) of the basering modulo the primary invariants, mapping
    49134913          those to invariants with the Reynolds operator. Among these images
    4914           or their power products we pick a maximal subset that is linearly
    4915           independent modulo the primary invariants using Groebner basis techniques
    4916           (see paper \"Fast Computation of Secondary Invariants\" by S. King).
     4914          or their power products we pick secondary invariants using Groebner
     4915          basis techniques (see S. King: Fast Computation of Secondary Invariants).
    49174916          The size of this set can be read off from the Molien series. Here, only
    49184917          irreducible secondary invariants are explicitly computed, which saves time and
    … …  
    56445643          of monomials) of the basering modulo the primary invariants, mapping
    56455644          those to invariants with the Reynolds operator. Among these images
    5646           or their power products we pick a maximal subset that is linearly
    5647           independent modulo the primary invariants using Groebner basis techniques
    5648           (see paper \"Fast Computation of Secondary Invariants\" by S. King).
     5645          or their power products we pick secondary invariants using Groebner
     5646          basis techniques (see S. King: Fast Computation of Secondary Invariants).
    56495647          The size of this set can be read off from the Molien series.
    56505648EXAMPLE:  example secondary_charp; shows an example
    … …  
    61056103          hence the number of secondary invariants is the product of the degrees of
    61066104          primary invariants divided by the group order.
    6107           <secondary_and_irreducibles_no_molien> should usually be faster and of more useful
    6108           functionality.
     6105NOTE:     <secondary_and_irreducibles_no_molien> should usually be faster and of
     6106          more useful functionality.
    61096107SEE ALSO: secondary_and_irreducibles_no_molien
    61106108EXAMPLE:  example secondary_no_molien; shows an example
    … …  
    63516349         monomials) of the basering modulo primary invariants, mapping those to
    63526350         invariants with the Reynolds operator. Among these images
    6353          or their power products we pick a maximal subset that is linearly
    6354          independent modulo the primary invariants using Groebner basis techniques
    6355          (see paper \"Fast Computation of Secondary Invariants\" by S. King).
     6351         or their power products we pick secondary invariants using Groebner
     6352         basis techniques (see S. King: Fast Computation of Secondary Invariants).
    63566353         We have the Reynolds operator, hence, we are in the non-modular case.
    63576354         Therefore, the invariant ring is Cohen-Macaulay, hence the number of
    63586355         secondary invariants is the product of the degrees of primary invariants
    63596356         divided by the group order.
     6357SEE ALSO: secondary_no_molien
    63606358EXAMPLE: example secondary_and_irreducibles_no_molien; shows an example
    63616359"
    … …  
    67666764         monomials) of the basering modulo primary and previously found secondary
    67676765         invariants, mapping those to invariants with the Reynolds operator. Among
    6768          these images we pick a maximal subset that is linearly independent modulo
    6769          the primary and previously found secondary invariants, using Groebner basis
     6766         these images we pick secondary invariants, using Groebner basis
    67706767         techniques.
     6768SEE ALSO: irred_secondary_char0
    67716769EXAMPLE: example irred_secondary_no_molien; shows an example
    67726770"
    … …  
    76477645               compatibility) the old procedure <relative_orbit_variety>
    76487646               is called, and in this case s gives the name of a new <ring>.
    7649 RETURN:   Without optional s, a list L of two rings is returned.
     7647RETURN:   Without optional string s, a list L of two rings is returned.
    76507648@*        The ring L[1] carries a weighted degree order with variables
    76517649            y(1..m), the weight of y(k) equal to the degree of the
    … …  
    76637661            application of <algebra_containment>
    76647662            (see @ref{algebra_containment}).
    7665 @*        For the case of optional s, see @ref{relative_orbit_variety}.
     7663@*        For the case of optional string s, the function is equivalent to
     7664          @ref{relative_orbit_variety}.
    76667665THEORY:   A Groebner basis of the ideal of algebraic relations of the invariant
    76677666          ring generators is calculated, then one of the basis elements plus
    … …  
    79367935///////////////////////////////////////////////////////////////////////////////
    79377936// Input: Disjoint cycle presentation of a permutation group
    7938 // Output: Permuation matrices representing that group
     7937// Output: Permutation matrices representing that group
    79397938///////////////////////////////////////////////////////////////////////////////
    79407939proc GetGroup(list GRP)
    … …  
    81318130ASSUME:   We are in the non-modular case, i.e., the characteristic of the basering
    81328131          does not divide the group order. Note that the function does not verify whether
    8133           the assumption holds or not
     8132          this assumption holds or not
    81348133DISPLAY:  Information on the progress of computations if v does not equal 0
    81358134THEORY:   We do an incremental search in increasing degree d. Generators of the invariant
    81368135          ring are found among the orbit sums of degree d. The generators are chosen by
    8137           Groebner basis techniques.
     8136          Groebner basis techniques (see S. King: Minimal generating sets of non-modular
     8137          invariant rings of finite groups).
     8138NOTE:     invariant_algebra_perm should not be used in rings with weighted orders.
    81388139SEE ALSO: invariant_algebra_reynolds
    81398140KEYWORDS: invariant ring minimal generating set permutation group
    … …  
    83128313THEORY:   We do an incremental search in increasing degree d. Generators of the invariant
    83138314          ring are found among the Reynolds images of monomials of degree d. The generators are
    8314           chosen by Groebner basis techniques.
     8315          chosen by Groebner basis techniques (see S. King: Minimal generating sets of
     8316          non-modular invariant rings of finite groups).
     8317NOTE:     invariant_algebra_reynolds should not be used in rings with weighted orders.
    83158318SEE ALSO: invariant_algebra_perm
    83168319KEYWORDS: invariant ring minimal generating set matrix group
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