1pwkond(1) User Commands pwkond(1)
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6 pwkond - peak word condensation
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9 pwkond [OPTIONS] <Name> [<Name> ...]
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12 After the irreducible constituents of a module, or a number of modules,
13 have been found with chop(1), this program can be used:
14 · to calulate peak words for the constituents,
16 · to condense the module using the peak words,
18 · to transform the generators on the constituents to the standard basis
20 as defined by the peak word kernel, and
21 · to calculate a basis reflecting the direct decomposition of the mod‐
23 ule, if the module is semisimple.
24 By definition, a "peak word" for the ith constituent is an algebra ele‐
26 ment which has minimal nullity on the ith constituent and which oper‐
27 ates regularly (i.e., with nullity 0) on the other constituents. Also
28 for identifying words (see chop(1)), the nullity of a peak word on its
29 constituent is equal to the degree of the splitting field for that con‐
30 stituent.
31 When more than one module is specified on the command line, the peak
33 words found by pwkond are "global", i.e., each peak word selects
34 exactly one of the constituents of each module. Running pwkond succes‐
35 sively on two modules does not generally produce global peak words,
36 since a peak word found for module M may have a non-zero nullity on a
37 different constituent that occurs in another module N but not in M.
38 The -e option can be used to exclude certain words from the search.
40 List is a list of integers or ranges of integers, for example "-e
41 57,82-112,289". Using -i, you can specify a list of words which will
42 be tested first. This can significantly reduce computation time if you
43 already know one or more peak words for a given module. The -n option
44 disables the condensation phase. If this option is used, the program
45 stops after the peak words have been found. If the -t option is speci‐
46 fied, pwkond transforms the generators of all irreducible constituents
47 to the standard basis defined by the peak word.
48 For each composition factor there are several output files. If, for
50 example, one composition factor is X10a, pwkond will produce the fol‐
51 lowing files:
52 X10a.std.1 and X10a.std.1
54 The operation of the generators on the constituent with respect
55 to the standard basis defined by the peak word. These files are
56 created only if the -t option is used.
57 X10a.op
59 Spin-up script for the standard basis. See zsb(1) for details.
60 X10a.1k and X10a.2k
62 The action of the generators on the condensed module.
63 X10a.np
65 Condensed peak word. This is a nilpotent matrix.
66 X10a.im
68 Image of the peak word.
69 X10a.k Kernel of the peak word.
71 The .cfinfo file is written each time a peak word is found. So, if the
73 program does not terminate or dies unexpectedly the information about
74 the peak words found so far is not lost.
75 If the module is semisimple, pwkond can calculate a basis that respects
77 the decomposition into irreducible constituents. With respect to this
78 basis, the generators are in block diagonal form, where the blocks
79 occur in the order determined by chop(1). All blocks corresponding to
80 the same constituent are equal, not only equivalent, and the blocks
81 occur in their "natural" order (as defined by chop(1)). This is essen‐
82 tial for the tensor condensation procedure (see precond(1)). To cal‐
83 culate the semisimplicity basis, use the -b option. The basis is writ‐
84 ten to Name.ssb. Using -b with a module that is not semisimple pro‐
85 duces undefined results. Most probably, pwkond will stop with the
86 error message "row index out of range", or it will write a singular
87 matrix to Name.ssb.
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90 -Q Quiet, no messages.
91 -V Verbose, more messages.
93 -T <MaxTime>
95 Set CPU time limit
96 -G Produce output in GAP format. This option implies -Q.
98 -n Find peak words only; do not condense.
100 -p Use full polynomials in peak word search.
102 -i <List>
104 Words to try first; e.g., -i 100,20-35.
105 -e <List>
107 Exclude words from search; e.g., -e 3,20-99.
108 -t Transform generators into standard basis.
110 -b Calculate a semisimplicity basis.
112 -k Compute kernel of peak words.
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116 Internally, a peak word is represented by a pair (n,p) where n is the
117 canonical number of the word (see zmw(1)), and p is a polynomial. The
118 peak word represented by this pair is p(Wn), Wn being the nth word.
119 Without -p, pwkond considers only linear polynomials. If the -p option
120 is used, pwkond can find polynomials of any degree.
121 Whenever a peak word is found, the generalized condensation is calcu‐
123 lated as follows: The peakword is caculated as a matrix acting on V,
124 which is then repeatedly raised to higher powers until the nullity sta‐
125 bilizes. The stable nullity equals the multiplicity k of the con‐
126 stituent times the degree [E:F] of the splitting field extension. Hav‐
127 ing a power w^N of the peakword with stable nullity, the condensation
128 onto its kernel, i.e., the projection of V onto V/w^N(V), is determined
129 in the same way as in the zqt(1) program.
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132 Name.cfinfo
133 Constituent info file.
134 Name.{1,2,...}
136 Generators.
137 Name<Cf>.{1,2,...}
139 Generators on the constituents.
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142 Name.cfinfo
143 Constituent info file.
144 Name<Cf>.{1,2,...}k
146 Condensed generators.
147 Name<Cf>.{1,2,...}.std
149 Condensed generators in standard basis (with -t).
150 Name<Cf>.op
152 Spin-up script for standard basis (with -t).
153 Name<Cf>.np
155 Condensed peak word.
156 Name<Cf>.im
158 Image used for condensation.
159 Name<Cf>.k
161 Peakword kernel (with -k or without -n).
162 Name.ssb
164 Semisimplicity basis (with -b).
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167 chop(1), precond(1), zmw(1), zqt(1), zsb(1)
168MeatAxe 2.4.24 pwkond(1)