1chop(1) User Commands chop(1)
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6 chop - find irreducible constituents of a matrix
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9 chop [OPTIONS] <Name>
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12 This program calculates the irreducible constituents of a given matrix
13 representation. The representing matrices of the generators are read
14 from input files, see "INPUT FILES" below. Unless a different number
15 of generators has been specified with -g, two generators are expected.
16 However, if the -i option is used, and the file Name.cfinfo exists,
17 chop takes the number of generators from this file and ignores the -g
18 option.
19 For each composition factor chop writes the action of the generators to
21 CFName.1, CFName.2, .... CFName is the name of the composition factor,
22 which is constructed by appending the dimension and a letter to the
23 module name. For example, "X10a.1" is the action of the first genera‐
24 tor on the first composition factor of dimension 10 of the module X.
25 If a second, inequivalent composition factor of dimension 10 was found,
26 it would be named "X10b" and so on. Chop also creates the file
27 Name.cfinfo containing a list of all composition factors. This file is
28 used by subsequent programs such as pwkond(1).
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31 -Q Quiet, no messages.
32 -V Verbose, more messages.
34 -T <MaxTime>
36 Set CPU time limit
37 -G, --gap
39 Produce output in GAP format. This option implies -Q.
40 -g <NGen>
42 Set the number of generators. Default: 2.
43 -s <Word>
45 Start with word number Word. Default: 1.
46 -n <MaxNul>
48 Set limit on nullity. Default: 3.
49 -d <MaxDeg>
51 Set limit on degrees of polynomials. Default: 5.
52 -i, --read-cfinfo
54 Read Name.cfinfo, if it exists
55
57 Chop repeatedly splits a module into submodule and quotient until it
58 arrives at the irreducible constituents. Thus, it finds a composition
59 series. The program assumes that the algebra generated by the input
60 matrices contains the unit matrix.
61 In order to split a given module or to prove its irreducibility the
63 algorithm needs an element of the algebra with a nontrivial but low-
64 dimensional kernel. Such elements are searched by taking linear combi‐
65 nations of certain products of the generators ("words"). By default,
66 chop tries all words in the order defined by the word generator. The
67 -s option may be used to make chop start with a word different from 1.
68 For each word A generated in this way, the program calculates its char‐
70 acteristic polynomial and examines the irreducible factors. If p(x) is
71 an irreducible factor, p(A) has a nontrivial kernel. Then, one vector
72 of the kernel is chosen and the smallest submodule containing this vec‐
73 tor is calculated. If the vector spans a proper submodule, the action
74 of the generators on this submodule as well as on the quotient are cal‐
75 culated and the same procedure is applied recursively to both submodule
76 and quotient.
77 To avoid expensive matrix multiplications in the calculation of p(A),
79 there is a limit on the degree of p(x). This limit can be set with the
80 -d option and defaults to 5.
81 If a module cannot be split by the program, it may be irreducible. In
83 order to prove this, chop uses Norton's criterion. This requires, how‐
84 ever, to find an algebra element with a small kernel, because up to
85 scalar multiples each vector in the kernel must be examined to see
86 whether it spins up to the whole module. For this reason a "nullity
87 threshold" m is maintained by the program. Initially, m is set to 3 or
88 to the value given in the -n option. Each algebra element that has a
89 nullity less then or equal to m is used for the Norton test.
90 In some cases the algorithm described is not able to split the module
92 although it is reducible. These exceptional cases are treated with an
93 alternative strategy described in Lux and Ivanyos, "Treating the excep‐
94 tional cases of the Meataxe", 19 Feb 1998 (unpublished).
95 Algebra elements with trivial kernel are useless for the algorithm, so
97 an attempt is made to avoid unnecessary computation of such elements.
98 Once an element is known to have a trivial kernel on a given module M,
99 the program will mark it as invertible and ignore it for all con‐
100 stituents of M.
101 If a constituent is irreducible but not absolutely irreducible, the
103 nullity of any element in the algebra will be a multiple of [E:F],
104 where F is the ground field and E the splitting field. This situation
105 is recognized by calculating the greatest common divisor d of all nul‐
106 lities which occur during the search. In order to prove that the
107 splitting field degree is equal to d, the following method is used:
108 Take a word with nullity d and two vectors v1, v2 in its null-space.
109 Use these vectors as seeds for a standard basis algorithm. If the
110 resulting representations are different, [E:F] is less than d, and the
111 word is discarded. Otherwise, the linear map which transforms one
112 standard basis into the other is an endomorphism e of the module. If
113 v1, under the action of e, spins up to the whole null space, then
114 [E:F]=d. Otherwise, take a third vector not in the span and repeat the
115 procedure above. Again, this yields an endomorphism, or it turns out
116 that [E:F]<d. These steps are repeated until a word with nullity [E:F]
117 is found.
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120 Name.{1,2,...}
121 Generators.
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124 Name.cfinfo
125 Constituent info file
126 CFName.{1,2,...}
128 Generators on the constituents.
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131 cfcomp(1), pseudochop(q), pwkond(1)
132MeatAxe 2.4.24 chop(1)