Attachment 'DemoCombinat.txt'
Download 1 *----* MuPAD Pro 4.0.0 -- The Open Computer Algebra System
2 /| /|
3 *----* | Copyright (c) 1997 - 2006 by SciFace Software
4 | *--|-* All rights reserved.
5 |/ |/
6 *----* Licensed to: MuPAD Combinat Developer
7
8
9 +---+
10 | T | MuPAD-Combinat 1.3.3 (development)
11 +---+---+
12 | A | K | an open source MuPAD package for
13 +---+---+---+
14 | I | N | research in Algebraic Combinatorics
15 +---+---+
16
17 This package provides or extends the following libraries:
18 combinat, examples, Dom, Cat, output, experimental, IPC, operators
19
20 For quick information on a particular library, please type:
21 info(library) or ?library (requires MuPAD >= 4.0.0)
22
23 For the full html documentation, please browse through:
24 http://mupad-combinat.sf.net/ (project web page)
25 file:/media/AdvDisk/Combinat//index.html (local mirror)
26
27 -- Interface:
28 packages::Combinat::dotCategories, packages::Combinat::help,
29 packages::Combinat::viewDot, packages::Combinat::viewDotTeX,
30 packages::Combinat::viewTeX
31 >> TEXTWIDTH:=84:
32 >> 1+1
33
34 2
35 Time: 0 msec
36 >> export(combinat):
37 >> partitions::list(5)
38
39 [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
40 Time: 68 msec
41 >> partitions::count(10)
42
43 42
44 Time: 4 msec
45 >> trees::list(5)
46
47 -- o , o , o , o , o , o , o , o , o , o , o , o , o , o --
48 | // \\ /|\ /|\ / \ / \ /|\ / \ / \ / \ | | | | | |
49 | | | /\ | | | | /\ | /|\ / \ / \ | | |
50 | | | | | / \ | |
51 -- | --
52 Time: 392 msec
53 >> trees::count(6)
54
55 42
56 Time: 0 msec
57 >> trees::random(50)
58
59 o
60 |
61 / / \\
62 / \| /\
63 / / |\\ |
64 // \ \
65 | | |
66 / \ | /|\
67 / \ |
68 |
69 / \
70 /|\ |
71 |/ \
72 | / \
73 /|\
74 |
75 Time: 96 msec
76
77 Sin
78
79 >> print(eq) $ eq in binaryTrees::grammar::specification
80
81 Tree = Union(LeafNil, Node)
82
83 Node = Alias(NodeRaw, x -> dom([op(x, 1), op(op(x, 2))]))
84
85 NodeRaw = Prod(NodeLabel, ChildList)
86
87 NodeLabel = Atom(NIL)
88
89 ChildList = Prod(Tree, Tree)
90
91 Leaf = Alias(LeafLabel, combinat::binaryTrees@DOM_LIST)
92
93 LeafLabel = Atom(NIL)
94
95 LeafNil = Epsilon(.)
96 >> r := binaryTrees::grammar::recurrenceRelation():
97 >> assume(n>0):
98 >> u(n) = factor(op(solve(r, u(n)),1))
99
100 2 u(n - 1) (2 n - 1)
101 u(n) = --------------------
102 n + 1
103 Time: 136 msec
104 >> domain Trees
105 >> inherits combinat::trees;
106 >> category Cat::TreesClass(Ordered = FALSE);
107 >> axiom Ax::normalRep, Ax::canonicalRep;
108 >>
109 >> info := "A domain for rooted unordered trees";
110 >>
111 >> grafts :=
112 >> proc(x:dom, y:dom) : Type::ListOf(dom)
113 >> local i;
114 >> begin
115 >> [sort(x::append(y))] . // Base case
116 >> (map( (x[i])::grafts(y), // Recursively
117 >> z -> sort(x::subsop(i=z))) // graft back each result in turn
118 >> $ i=1..nops(x)) // on all childs
119 >> end_proc;
120 >> end_domain:
121 >> T := Trees::list(4);
122
123 -- o, o , o , o --
124 | | | / \ /|\ |
125 | | / \ | |
126 -- | --
127 Time: 100 msec
128 >> t1 := T[2]; t2 := subsop(Trees::list(5)[7], 0=0)
129
130 o
131 |
132 / \
133 Time: 4 msec
134
135 0
136 / \
137 | |
138 Time: 52 msec
139 >> t1::grafts(t2)
140
141 -- o , o , o , o --
142 | / \ | | | |
143 | /\ 0 /|\ / \ / \ |
144 | / \ 0 | | |
145 | | | / \ 0 0 |
146 | | | / \ / \ |
147 -- | | | | --
148 Time: 36 msec
149 >> domain preLieAlgebra(R = Dom::ExpressionField(): Cat::Ring)
150 >> inherits Dom::FreeModule(R, Trees);
151 >> category Cat::AlgebraWithBasis(R); // Non Associative...
152 >>
153 >> v := dom::term(Trees([NIL]));
154 >>
155 >> mult2Basis :=
156 >> proc(t1 : Trees,
157 >> t2 : Trees)
158 >> local t;
159 >> begin
160 >> dom::plus(dom::term(t) $ t in Trees::grafts(t1,t2));
161 >> end;
162 >> end_domain:
163 >> pr := preLieAlgebra():
164 >> d := pr::v
165
166 B(o)
167 Time: 4 msec
168 >> d := d * pr::v
169
170 B o
171 (|)
172 Time: 8 msec
173 >> d := d * pr::v
174
175 B o + B/ o \
176 (/ \) | | |
177 \ | /
178 Time: 4 msec
179 >> d := d * pr::v
180
181 3 B/ o \ + B/ o \ + B/ o \ + B o
182 | / \ | | | | | | | (/|\)
183 \ | / \ / \ / | | |
184 \ | /
185 Time: 12 msec
186 >> d := d * pr::v;
187
188 B o + 6 B/ o \ + 4 B/ o \ + 3 B/ o \ + B/ o \ + B/ o \ + 4 B/ o \ +
189 (// \\) | /|\ | | / \ | | | | | | | | | | | / \ |
190 \ | / | | | | / \ | | | | | | | \ /\ /
191 \ | / \ | / \ / \ / | | |
192 \ | /
193
194 B/ o \ + 3 B/ o \
195 | | | | / \ |
196 \ /|\ / \ | | /
197 Time: 40 msec
198 >> _plus(coeff(d))
199
200 24
201 Time: 0 msec
202 >> d := d * pr::v;
203
204 15 B/ o \ + 10 B/ o \ + 10 B/ o \ + 5 B/ o \ + B/ o \ + 6 B/ o \ +
205 | /|\ | | / \ | | / \ | | / \ | | | | | | |
206 \ || / \ | /\ / | | | | \ /|\ / \ // \\ / | /|\ |
207 \ | / \ | /
208
209 10 B/ o \ + 15 B/ o \ + 5 B/ o \ + 4 B/ o \ + 3 B/ o \ + B/ o \ +
210 | /|\ | | / \ | | / \ | | | | | | | | | |
211 \ /\ / | /\ | | | | | / \ | | | | | | |
212 \ | / | | | | | | | / \ | | | |
213 \ | / \ | / \ | / \ / \ /
214
215 B/ o \ + 5 B/ o \ + 4 B/ o \ + B/ o \ + 3 B/ o \ + 10 B/ o \ +
216 | | | | / \ | | | | | | | | | | | /|\ |
217 | | | | | | | / \ | | | | | / \ | | | |
218 | | | \ / \ / \ /\ / \ /|\ / \ | | / \ | /
219 | | |
220 \ | /
221
222 10 B/ o \ + B o
223 | // \\ | (//|\\)
224 \ | /
225 Time: 88 msec
226 >> _plus(coeff(d))
227
228 120
229 Time: 0 msec
230 >> d := d * pr::v;
231
232 B o + 15 B/ o \ + 45 B/ o \ + 20 B/ o \ + 20 B/ o \ +
233 (/// \\\) | //|\\ | | // \\ | | // \\ | | // \\ |
234 \ | / \ || / \ /\ / | | |
235 \ | /
236
237 60 B/ o \ + 45 B/ o \ + 15 B/ o \ + 15 B/ o \ + 18 B/ o \ +
238 | /|\ | | /|\ | | /|\ | | /|\ | | / \ |
239 | || | | /\ | | | | | | | | /\ |
240 \ | / \ | / \ / \ / | | | \ || /
241 \ | /
242
243 15 B/ o \ + 10 B/ o \ + 10 B/ o \ + 6 B/ o \ + 5 B/ o \ +
244 | | | | | | | | | | / \ | | | |
245 | /|\ | | / \ | | / \ | | | | | / \ |
246 \ || / \ | /\ / | | | | \ /|\ / \ /|\ /
247 \ | /
248
249 B/ o \ + 6 B/ o \ + 24 B/ o \ + 10 B/ o \ + 15 B/ o \ +
250 | | | | | | | / \ | | | | | | |
251 | | | | | | | /\ | | /|\ | | / \ |
252 \ // \\ / | /|\ | \ /\ / \ /\ / | /\ |
253 \ | / \ | /
254
255 15 B/ o \ + 18 B/ o \ + 6 B/ o \ + 5 B/ o \ + 4 B/ o \ +
256 | / \ | | / \ | | / \ | | | | | | |
257 | | | | | | | | | | | / \ | | | |
258 \ / \ / | / \ | | | | | | | | / \ |
259 \ | / | | | | | | | | |
260 \ | / \ | / \ | /
261
262 3 B/ o \ + B/ o \ + B/ o \ + 6 B/ o \ + 5 B/ o \ + 4 B/ o \ +
263 | | | | | | | | | | / \ | | | | | | |
264 | | | | | | | | | | | | | / \ | | | |
265 | | | | | | | | | | | | | | | | / \ |
266 | / \ | | | | | | | \ / \ / \ / \ / \ /\ /
267 \ | / \ / \ / | | |
268 \ | /
269
270 B/ o \ + 3 B/ o \ + 24 B/ o \ + 10 B/ o \ + 15 B/ o \ +
271 | | | | | | | / \ | | | | | / \ |
272 | | | | | | | /\ | | /|\ | | | | |
273 | | | | / \ | | | | | | | | | |
274 \ /|\ / \ | | / \ | / \ | / \ | /
275
276 45 B/ o \ + 36 B/ o \ + 60 B/ o \ + 15 B/ o \ + 10 B/ o \ +
277 | / \ | | / \ | | /|\ | | /| \ | | | |
278 | | /\ | | /|\ | \ |/\ / \ /|\ / | // \\ |
279 \ | / \ | / \ | /
280
281 6 B/ o \ + B/ o \ + 15 B/ o \ + 20 B/ o \ + 10 B/ o \ +
282 | / \ | | | | | / \ | | / \ | | / \ |
283 \ //\\ / \ //|\\ / \ |/|\ / | | /\ | | | | |
284 \ | / \ | | /
285
286 10 B/ o \ + 15 B/ o \
287 | / \ | | /|\ |
288 \ /\ /\ / \ ||| /
289 Time: 188 msec
290 >> _plus(coeff(d))
291
292 720
293 Time: 0 msec
294 >> // Identité Prélie :
295 >> bool( (x*y)*z - x*(y*z) = (x*z)*y - x*(z*y) )
296 >> $ x in pr::basis(3) $ y in pr::basis(3) $ z in pr::basis(3);
297
298 TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE
299 Time: 120 msec
300 >> read("experimental/2005-09-08-David.mu"):
301 //////////////////////////////////////////////////////////////////////
302 Loading worksheet: Twisted Kac algebras
303 Cf. p. 715 of '2-cocycles and twisting of Kac algebras'
304
305 Version: $Id: 2005-09-08-David.mu 7495 2008-04-30 20:01:44Z nthiery $
306 To update to the latest version, go to the MuPAD-Combinat directory and type:
307 svn update
308 Content:
309 G := DihedralGroup(4)
310 SkewTensorProduct(A, B) -- Skew tensor product of A and B (A being the dual of B)
311 coidealDual([ p ]) -- Basis of the dual of the left coideal generated by p
312
313 TwistedDihedralGroupAlgebra:
314 KD4 := TwistedDihedralGroupAlgebra(4):
315 KD4 := KD(4): -- shortcut
316 KD4::G = KD4::group -- KD4 expressed on group elements
317 KD4::G([3,1]) -- a^3 b
318 KD4::M = KD4::matrix -- KD4 expressed as block diagonal matrices
319 KD4::G::tensorSquare -- the tensor product KD4::G # KD4::G
320 KD4::M::tensorSquare --
321
322 KD4::coeffRing -- the coefficient field
323 KD4::coeffRing::primitiveUnitRoot(4)-- the complex value I
324
325 KD4::M(x), KD4::G(x) -- conversions between bases
326 KD4::e(1), KD4::e(2,2,1) -- matrix units
327 KD4::p(2,2,j), KD4::r(2,2,j) -- some projectors of the j-th block
328 KD4::p1, KD4::p2, KD4::q1, KD4::q1 -- some projectors
329 KD4::G::Omega -- Omega in the group basis
330 KD4::M::tensorSquare( KD4::G::Omega )-- Omega in the matrix basis
331 KD4::M::coproductAsMatrix(e(1)) -- the coproduct of e(1) as a matrix
332
333 KD4::automorphismReverseOddBlocks -- some (potential) automorphisms
334 KD4::automorphismTransposeEvenBlocks--
335 KD4::automorphismTransposeOddBlocks -- (not an automorphism for KD4!)
336
337 // To get shorter notations:
338 export(KD4, Alias, e, p1, p2, q1, q2):
339 alias(view = KD4::M::coproductAsMatrix):
340 // Then you can do:
341 e(2,2,1) ...
342 view(e(1))
343
344 TwistedQuaternionGroupAlgebra(N)
345 KQ4 := TwistedDihedralGroupAlgebra(4):
346 KQ4 := KD(4): -- shortcut
347 Same usage as for KD(N)
348
349 algebraClosure([a,b,c])
350 coidealClosure([a,b,c])
351 coidealAndAlgebraClosure([a,b,c])
352 echelonForm([a,b,c], Reduced)
353 Isomorphism KD(2N) <-> KQ(2N)
354 The most natural isomorphism, in the G basis:
355 KQ4::G(KD4::G([1,0])): -- The image of a of KD4 in KQ4
356 KD4::G(KQ4::G([0,1])): -- The image of b of KQ4 in KD4
357
358 The 8 possible isomorphisms in the M basis:
359 phi := isomorphismKDMKQM(4, 3, TRUE)-- isomorphism KD(4)::M -> KQ(4)::M
360 KD4::M::isHopfAlgebraMorphism(f);
361 inv := KD4::M::inverseOfModuleMorphism(phi);
362 KQ4::M::isHopfAlgebraMorphism(inv);
363
364
365 A sample computation:
366 M := KQ(4):
367 Fbasis := coidealAndAlgebraClosure([M::e(1) + M::e(2)]):
368 F := Dom::SubFreeModule(Fbasis, [Cat::FiniteDimensionalHopfAlgebraWithBasis(M::coeffRing)]):
369 Fdual := Dom::DualOfFreeModule(F):
370 G := Fdual::intrisicGroup():
371 G::list(); // C'est le groupe dihedral D4
372
373 //////////////////////////////////////////////////////////////////////
374 >> KD3 := KD(3):
375 >> KD3::categories
376
377 [Cat::HopfAlgebraWithSeveralBases(Q(II, epsilon)),
378
379 TwistedDihedralOrQuaternionGroupAlgebra(3),
380
381 Cat::AlgebraWithSeveralBases(Q(II, epsilon)), Cat::Algebra(Q(II, epsilon)),
382
383 Cat::ModuleWithSeveralBases(Q(II, epsilon)), Cat::Ring,
384
385 Cat::Module(Q(II, epsilon)), Cat::DomainWithSeveralRepresentations, Cat::Rng,
386
387 Cat::SemiRing, Cat::LeftModule(KD(3, Q(II, epsilon))),
388
389 Cat::LeftModule(Q(II, epsilon)), Cat::RightModule(Q(II, epsilon)),
390
391 Cat::UseOverloading, Cat::FacadeDomain, Cat::SemiRng, Cat::Monoid,
392
393 Cat::AbelianGroup, Cat::SemiGroup, Cat::CancellationAbelianMonoid,
394
395 Cat::AbelianMonoid, Cat::AbelianSemiGroup, Cat::Object, Cat::BaseCategory]
396 Time: 4 msec
397 >> [aa,bb] := KD3::group::algebraGenerators::list()
398
399 [B(a), B(b)]
400 Time: 4 msec
401 >> bb^2
402
403 B(1)
404 Time: 12 msec
405 >> aa^2, aa^6, bb*aa
406
407 2 5
408 B(a ), B(1), B(a b)
409 Time: 4 msec
410 >> (1 - aa^3)*(bb + aa^3) + 1/2*bb*aa^3
411
412 3 3
413 -1 B(1) + B(b) + B(a ) + -1/2 B(a b)
414 Time: 24 msec
415 >> KD3::M(aa + 2*bb)
416
417 +- -+
418 | 3, 0, 0, 0, 0, 0, 0, 0 |
419 | |
420 | 0, -1, 0, 0, 0, 0, 0, 0 |
421 | |
422 | 0, 0, -3, 0, 0, 0, 0, 0 |
423 | |
424 | 0, 0, 0, 1, 0, 0, 0, 0 |
425 | |
426 | 0, 0, 0, 0, epsilon, 2, 0, 0 |
427 | |
428 | 0, 0, 0, 0, 2, 1 - epsilon, 0, 0 |
429 | |
430 | 0, 0, 0, 0, 0, 0, epsilon - 1, 2 |
431 | |
432 | 0, 0, 0, 0, 0, 0, 2, -epsilon |
433 +- -+
434 Time: 48 msec
435 >> coproduct(aa^3), coproduct(bb)
436
437 3 3
438 B(a ) # B(a ), B(b) # B(b)
439 Time: 496 msec
440 >> coproduct(aa)
441
442 2 2
443 7/16 B(a) # B(a) + 1/16 B(a) # B(a b) + -1/16 B(a) # B(a ) + 1/16 B(a) # B(a b) +
444
445 4 4 5
446 3/16 B(a) # B(a ) + 1/16 B(a) # B(a b) + 3/16 B(a) # B(a ) +
447
448 5
449 1/16 B(a) # B(a b) + 1/16 B(a b) # B(a) + 1/16 B(a b) # B(a b) +
450
451 2 2 / II \ 4
452 1/16 B(a b) # B(a ) + 1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
453 \ 8 /
454
455 / II \ 4 / II \ 5
456 | -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
457 \ 8 / \ 8 /
458
459 / II \ 5 2 2
460 | - -- - 1/16 | B(a b) # B(a b) + -1/16 B(a ) # B(a) + 1/16 B(a ) # B(a b) +
461 \ 8 /
462
463 2 2 2 2 2 4
464 -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
465
466 2 4 2 5 2 5
467 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) +
468
469 2 2 2 2
470 1/16 B(a b) # B(a) + 1/16 B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
471
472 2 2 / II \ 2 4
473 1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
474 \ 8 /
475
476 / II \ 2 4 / II \ 2 5
477 | - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
478 \ 8 / \ 8 /
479
480 / II \ 2 5 4
481 | -- - 1/16 | B(a b) # B(a b) + 3/16 B(a ) # B(a) +
482 \ 8 /
483
484 / II \ 4 4 2
485 | -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
486 \ 8 /
487
488 / II \ 4 2 4 4
489 | -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) +
490 \ 8 /
491
492 / II \ 4 4 4 5
493 | - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
494 \ 8 /
495
496 / II \ 4 5 4
497 | - -- - 1/16 | B(a ) # B(a b) + 1/16 B(a b) # B(a) +
498 \ 8 /
499
500 / II \ 4 4 2
501 | - -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
502 \ 8 /
503
504 / II \ 4 2 / II \ 4 4
505 | -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
506 \ 8 / \ 8 /
507
508 4 4 / II \ 4 5
509 1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
510 \ 8 /
511
512 4 5 5 / II \ 5
513 1/16 B(a b) # B(a b) + 3/16 B(a ) # B(a) + | - -- - 1/16 | B(a ) # B(a b) +
514 \ 8 /
515
516 5 2 / II \ 5 2 5 4
517 -1/16 B(a ) # B(a ) + | - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
518 \ 8 /
519
520 / II \ 5 4 5 5
521 | -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) +
522 \ 8 /
523
524 / II \ 5 5 5
525 | -- - 1/16 | B(a ) # B(a b) + 1/16 B(a b) # B(a) +
526 \ 8 /
527
528 / II \ 5 5 2
529 | -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
530 \ 8 /
531
532 / II \ 5 2 / II \ 5 4
533 | - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
534 \ 8 / \ 8 /
535
536 5 4 / II \ 5 5 5 5
537 1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) + 1/16 B(a b) # B(a b)
538 \ 8 /
539 Time: 260 msec
540 >> K := KD3::G: // un simple raccourci
541 >> checkAntipode := K::mu @ ( K::id # K::antipode ) @ K::coproduct:
542 >> checkAntipode(x) $ x in K::basis::list()
543
544 B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1)
545 Time: 2020 msec
546 >> e := KD3::e:
547 >> K2basis := coidealAndAlgebraClosure([ e(1)+e(2) ])
548
549 -- +- -+ +- -+
550 | | 1, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
551 | | | | |
552 | | 0, 1, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
553 | | | | |
554 | | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 1, 0, 0, 0, 0, 0 |
555 | | | | |
556 | | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 1, 0, 0, 0, 0 |
557 | | |, | |,
558 | | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
559 | | | | |
560 | | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
561 | | | | |
562 | | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
563 | | | | |
564 | | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
565 -- +- -+ +- -+
566
567 +- -+ +- -+
568 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
569 | | | |
570 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
571 | | | |
572 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
573 | | | |
574 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
575 | |, | |,
576 | 0, 0, 0, 0, 0, -1, 0, 0 | | 0, 0, 0, 0, 1, 0, 0, 0 |
577 | | | |
578 | 0, 0, 0, 0, 1, 0, 0, 0 | | 0, 0, 0, 0, 0, 1, 0, 0 |
579 | | | |
580 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
581 | | | |
582 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
583 +- -+ +- -+
584
585 +- -+ +- -+ --
586 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
587 | | | | |
588 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
589 | | | | |
590 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
591 | | | | |
592 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
593 | |, | | |
594 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
595 | | | | |
596 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
597 | | | | |
598 | 0, 0, 0, 0, 0, 0, 1, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
599 | | | | |
600 | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 1 | |
601 +- -+ +- -+ --
602 Time: 13601 msec
603 >> K2 := Dom::SubFreeModule(K2basis,
604 >> [Cat::FiniteDimensionalHopfAlgebraWithBasis(KD3::coeffRing)]):
605 >> K2::isCommutative(), K2::isCocommutative()
606
607 TRUE, FALSE
608 Time: 56 msec
609 >> K2dual := K2::Dual():
610 >> K2dual::groupLikeElements()
611
612 _ _ _ _ _ _
613 [B([3, 3]), B([1, 1]), B([8, 8]), B([7, 7]), -II B([6, 5]) + B([5, 5]),
614
615 _ _
616 II B([6, 5]) + B([5, 5])]
617 Time: 1236 msec
618 >> G := K2dual::intrinsicGroup():
619 >> G::list()
620
621 [[], [1], [1, 1], [2], [1, 2], [1, 1, 2]]
622 Time: 0 msec
623 >> K2dual::isSemiSimple()
624
625 TRUE
626 Time: 56 msec
627 >> K2dual::simpleModulesDimensions()
628
629 [2, 1, 1]
630 Time: 704 msec
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