*----* MuPAD Pro 4.0.0 -- The Open Computer Algebra System
/| /|
*----* | Copyright (c) 1997 - 2006 by SciFace Software
| *--|-* All rights reserved.
|/ |/
*----* Licensed to: MuPAD Combinat Developer
+---+
| T | MuPAD-Combinat 1.3.3 (development)
+---+---+
| A | K | an open source MuPAD package for
+---+---+---+
| I | N | research in Algebraic Combinatorics
+---+---+
This package provides or extends the following libraries:
combinat, examples, Dom, Cat, output, experimental, IPC, operators
For quick information on a particular library, please type:
info(library) or ?library (requires MuPAD >= 4.0.0)
For the full html documentation, please browse through:
http://mupad-combinat.sf.net/ (project web page)
file:/media/AdvDisk/Combinat//index.html (local mirror)
-- Interface:
packages::Combinat::dotCategories, packages::Combinat::help,
packages::Combinat::viewDot, packages::Combinat::viewDotTeX,
packages::Combinat::viewTeX
>> TEXTWIDTH:=84:
>> 1+1
2
Time: 0 msec
>> export(combinat):
>> partitions::list(5)
[[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
Time: 68 msec
>> partitions::count(10)
42
Time: 4 msec
>> trees::list(5)
-- o , o , o , o , o , o , o , o , o , o , o , o , o , o --
| // \\ /|\ /|\ / \ / \ /|\ / \ / \ / \ | | | | | |
| | | /\ | | | | /\ | /|\ / \ / \ | | |
| | | | | / \ | |
-- | --
Time: 392 msec
>> trees::count(6)
42
Time: 0 msec
>> trees::random(50)
o
|
/ / \\
/ \| /\
/ / |\\ |
// \ \
| | |
/ \ | /|\
/ \ |
|
/ \
/|\ |
|/ \
| / \
/|\
|
Time: 96 msec
Sin
>> print(eq) $ eq in binaryTrees::grammar::specification
Tree = Union(LeafNil, Node)
Node = Alias(NodeRaw, x -> dom([op(x, 1), op(op(x, 2))]))
NodeRaw = Prod(NodeLabel, ChildList)
NodeLabel = Atom(NIL)
ChildList = Prod(Tree, Tree)
Leaf = Alias(LeafLabel, combinat::binaryTrees@DOM_LIST)
LeafLabel = Atom(NIL)
LeafNil = Epsilon(.)
>> r := binaryTrees::grammar::recurrenceRelation():
>> assume(n>0):
>> u(n) = factor(op(solve(r, u(n)),1))
2 u(n - 1) (2 n - 1)
u(n) = --------------------
n + 1
Time: 136 msec
>> domain Trees
>> inherits combinat::trees;
>> category Cat::TreesClass(Ordered = FALSE);
>> axiom Ax::normalRep, Ax::canonicalRep;
>>
>> info := "A domain for rooted unordered trees";
>>
>> grafts :=
>> proc(x:dom, y:dom) : Type::ListOf(dom)
>> local i;
>> begin
>> [sort(x::append(y))] . // Base case
>> (map( (x[i])::grafts(y), // Recursively
>> z -> sort(x::subsop(i=z))) // graft back each result in turn
>> $ i=1..nops(x)) // on all childs
>> end_proc;
>> end_domain:
>> T := Trees::list(4);
-- o, o , o , o --
| | | / \ /|\ |
| | / \ | |
-- | --
Time: 100 msec
>> t1 := T[2]; t2 := subsop(Trees::list(5)[7], 0=0)
o
|
/ \
Time: 4 msec
0
/ \
| |
Time: 52 msec
>> t1::grafts(t2)
-- o , o , o , o --
| / \ | | | |
| /\ 0 /|\ / \ / \ |
| / \ 0 | | |
| | | / \ 0 0 |
| | | / \ / \ |
-- | | | | --
Time: 36 msec
>> domain preLieAlgebra(R = Dom::ExpressionField(): Cat::Ring)
>> inherits Dom::FreeModule(R, Trees);
>> category Cat::AlgebraWithBasis(R); // Non Associative...
>>
>> v := dom::term(Trees([NIL]));
>>
>> mult2Basis :=
>> proc(t1 : Trees,
>> t2 : Trees)
>> local t;
>> begin
>> dom::plus(dom::term(t) $ t in Trees::grafts(t1,t2));
>> end;
>> end_domain:
>> pr := preLieAlgebra():
>> d := pr::v
B(o)
Time: 4 msec
>> d := d * pr::v
B o
(|)
Time: 8 msec
>> d := d * pr::v
B o + B/ o \
(/ \) | | |
\ | /
Time: 4 msec
>> d := d * pr::v
3 B/ o \ + B/ o \ + B/ o \ + B o
| / \ | | | | | | | (/|\)
\ | / \ / \ / | | |
\ | /
Time: 12 msec
>> d := d * pr::v;
B o + 6 B/ o \ + 4 B/ o \ + 3 B/ o \ + B/ o \ + B/ o \ + 4 B/ o \ +
(// \\) | /|\ | | / \ | | | | | | | | | | | / \ |
\ | / | | | | / \ | | | | | | | \ /\ /
\ | / \ | / \ / \ / | | |
\ | /
B/ o \ + 3 B/ o \
| | | | / \ |
\ /|\ / \ | | /
Time: 40 msec
>> _plus(coeff(d))
24
Time: 0 msec
>> d := d * pr::v;
15 B/ o \ + 10 B/ o \ + 10 B/ o \ + 5 B/ o \ + B/ o \ + 6 B/ o \ +
| /|\ | | / \ | | / \ | | / \ | | | | | | |
\ || / \ | /\ / | | | | \ /|\ / \ // \\ / | /|\ |
\ | / \ | /
10 B/ o \ + 15 B/ o \ + 5 B/ o \ + 4 B/ o \ + 3 B/ o \ + B/ o \ +
| /|\ | | / \ | | / \ | | | | | | | | | |
\ /\ / | /\ | | | | | / \ | | | | | | |
\ | / | | | | | | | / \ | | | |
\ | / \ | / \ | / \ / \ /
B/ o \ + 5 B/ o \ + 4 B/ o \ + B/ o \ + 3 B/ o \ + 10 B/ o \ +
| | | | / \ | | | | | | | | | | | /|\ |
| | | | | | | / \ | | | | | / \ | | | |
| | | \ / \ / \ /\ / \ /|\ / \ | | / \ | /
| | |
\ | /
10 B/ o \ + B o
| // \\ | (//|\\)
\ | /
Time: 88 msec
>> _plus(coeff(d))
120
Time: 0 msec
>> d := d * pr::v;
B o + 15 B/ o \ + 45 B/ o \ + 20 B/ o \ + 20 B/ o \ +
(/// \\\) | //|\\ | | // \\ | | // \\ | | // \\ |
\ | / \ || / \ /\ / | | |
\ | /
60 B/ o \ + 45 B/ o \ + 15 B/ o \ + 15 B/ o \ + 18 B/ o \ +
| /|\ | | /|\ | | /|\ | | /|\ | | / \ |
| || | | /\ | | | | | | | | /\ |
\ | / \ | / \ / \ / | | | \ || /
\ | /
15 B/ o \ + 10 B/ o \ + 10 B/ o \ + 6 B/ o \ + 5 B/ o \ +
| | | | | | | | | | / \ | | | |
| /|\ | | / \ | | / \ | | | | | / \ |
\ || / \ | /\ / | | | | \ /|\ / \ /|\ /
\ | /
B/ o \ + 6 B/ o \ + 24 B/ o \ + 10 B/ o \ + 15 B/ o \ +
| | | | | | | / \ | | | | | | |
| | | | | | | /\ | | /|\ | | / \ |
\ // \\ / | /|\ | \ /\ / \ /\ / | /\ |
\ | / \ | /
15 B/ o \ + 18 B/ o \ + 6 B/ o \ + 5 B/ o \ + 4 B/ o \ +
| / \ | | / \ | | / \ | | | | | | |
| | | | | | | | | | | / \ | | | |
\ / \ / | / \ | | | | | | | | / \ |
\ | / | | | | | | | | |
\ | / \ | / \ | /
3 B/ o \ + B/ o \ + B/ o \ + 6 B/ o \ + 5 B/ o \ + 4 B/ o \ +
| | | | | | | | | | / \ | | | | | | |
| | | | | | | | | | | | | / \ | | | |
| | | | | | | | | | | | | | | | / \ |
| / \ | | | | | | | \ / \ / \ / \ / \ /\ /
\ | / \ / \ / | | |
\ | /
B/ o \ + 3 B/ o \ + 24 B/ o \ + 10 B/ o \ + 15 B/ o \ +
| | | | | | | / \ | | | | | / \ |
| | | | | | | /\ | | /|\ | | | | |
| | | | / \ | | | | | | | | | |
\ /|\ / \ | | / \ | / \ | / \ | /
45 B/ o \ + 36 B/ o \ + 60 B/ o \ + 15 B/ o \ + 10 B/ o \ +
| / \ | | / \ | | /|\ | | /| \ | | | |
| | /\ | | /|\ | \ |/\ / \ /|\ / | // \\ |
\ | / \ | / \ | /
6 B/ o \ + B/ o \ + 15 B/ o \ + 20 B/ o \ + 10 B/ o \ +
| / \ | | | | | / \ | | / \ | | / \ |
\ //\\ / \ //|\\ / \ |/|\ / | | /\ | | | | |
\ | / \ | | /
10 B/ o \ + 15 B/ o \
| / \ | | /|\ |
\ /\ /\ / \ ||| /
Time: 188 msec
>> _plus(coeff(d))
720
Time: 0 msec
>> // Identit� Pr�lie :
>> bool( (x*y)*z - x*(y*z) = (x*z)*y - x*(z*y) )
>> $ x in pr::basis(3) $ y in pr::basis(3) $ z in pr::basis(3);
TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE
Time: 120 msec
>> read("experimental/2005-09-08-David.mu"):
//////////////////////////////////////////////////////////////////////
Loading worksheet: Twisted Kac algebras
Cf. p. 715 of '2-cocycles and twisting of Kac algebras'
Version: $Id: 2005-09-08-David.mu 7495 2008-04-30 20:01:44Z nthiery $
To update to the latest version, go to the MuPAD-Combinat directory and type:
svn update
Content:
G := DihedralGroup(4)
SkewTensorProduct(A, B) -- Skew tensor product of A and B (A being the dual of B)
coidealDual([ p ]) -- Basis of the dual of the left coideal generated by p
TwistedDihedralGroupAlgebra:
KD4 := TwistedDihedralGroupAlgebra(4):
KD4 := KD(4): -- shortcut
KD4::G = KD4::group -- KD4 expressed on group elements
KD4::G([3,1]) -- a^3 b
KD4::M = KD4::matrix -- KD4 expressed as block diagonal matrices
KD4::G::tensorSquare -- the tensor product KD4::G # KD4::G
KD4::M::tensorSquare --
KD4::coeffRing -- the coefficient field
KD4::coeffRing::primitiveUnitRoot(4)-- the complex value I
KD4::M(x), KD4::G(x) -- conversions between bases
KD4::e(1), KD4::e(2,2,1) -- matrix units
KD4::p(2,2,j), KD4::r(2,2,j) -- some projectors of the j-th block
KD4::p1, KD4::p2, KD4::q1, KD4::q1 -- some projectors
KD4::G::Omega -- Omega in the group basis
KD4::M::tensorSquare( KD4::G::Omega )-- Omega in the matrix basis
KD4::M::coproductAsMatrix(e(1)) -- the coproduct of e(1) as a matrix
KD4::automorphismReverseOddBlocks -- some (potential) automorphisms
KD4::automorphismTransposeEvenBlocks--
KD4::automorphismTransposeOddBlocks -- (not an automorphism for KD4!)
// To get shorter notations:
export(KD4, Alias, e, p1, p2, q1, q2):
alias(view = KD4::M::coproductAsMatrix):
// Then you can do:
e(2,2,1) ...
view(e(1))
TwistedQuaternionGroupAlgebra(N)
KQ4 := TwistedDihedralGroupAlgebra(4):
KQ4 := KD(4): -- shortcut
Same usage as for KD(N)
algebraClosure([a,b,c])
coidealClosure([a,b,c])
coidealAndAlgebraClosure([a,b,c])
echelonForm([a,b,c], Reduced)
Isomorphism KD(2N) <-> KQ(2N)
The most natural isomorphism, in the G basis:
KQ4::G(KD4::G([1,0])): -- The image of a of KD4 in KQ4
KD4::G(KQ4::G([0,1])): -- The image of b of KQ4 in KD4
The 8 possible isomorphisms in the M basis:
phi := isomorphismKDMKQM(4, 3, TRUE)-- isomorphism KD(4)::M -> KQ(4)::M
KD4::M::isHopfAlgebraMorphism(f);
inv := KD4::M::inverseOfModuleMorphism(phi);
KQ4::M::isHopfAlgebraMorphism(inv);
A sample computation:
M := KQ(4):
Fbasis := coidealAndAlgebraClosure([M::e(1) + M::e(2)]):
F := Dom::SubFreeModule(Fbasis, [Cat::FiniteDimensionalHopfAlgebraWithBasis(M::coeffRing)]):
Fdual := Dom::DualOfFreeModule(F):
G := Fdual::intrisicGroup():
G::list(); // C'est le groupe dihedral D4
//////////////////////////////////////////////////////////////////////
>> KD3 := KD(3):
>> KD3::categories
[Cat::HopfAlgebraWithSeveralBases(Q(II, epsilon)),
TwistedDihedralOrQuaternionGroupAlgebra(3),
Cat::AlgebraWithSeveralBases(Q(II, epsilon)), Cat::Algebra(Q(II, epsilon)),
Cat::ModuleWithSeveralBases(Q(II, epsilon)), Cat::Ring,
Cat::Module(Q(II, epsilon)), Cat::DomainWithSeveralRepresentations, Cat::Rng,
Cat::SemiRing, Cat::LeftModule(KD(3, Q(II, epsilon))),
Cat::LeftModule(Q(II, epsilon)), Cat::RightModule(Q(II, epsilon)),
Cat::UseOverloading, Cat::FacadeDomain, Cat::SemiRng, Cat::Monoid,
Cat::AbelianGroup, Cat::SemiGroup, Cat::CancellationAbelianMonoid,
Cat::AbelianMonoid, Cat::AbelianSemiGroup, Cat::Object, Cat::BaseCategory]
Time: 4 msec
>> [aa,bb] := KD3::group::algebraGenerators::list()
[B(a), B(b)]
Time: 4 msec
>> bb^2
B(1)
Time: 12 msec
>> aa^2, aa^6, bb*aa
2 5
B(a ), B(1), B(a b)
Time: 4 msec
>> (1 - aa^3)*(bb + aa^3) + 1/2*bb*aa^3
3 3
-1 B(1) + B(b) + B(a ) + -1/2 B(a b)
Time: 24 msec
>> KD3::M(aa + 2*bb)
+- -+
| 3, 0, 0, 0, 0, 0, 0, 0 |
| |
| 0, -1, 0, 0, 0, 0, 0, 0 |
| |
| 0, 0, -3, 0, 0, 0, 0, 0 |
| |
| 0, 0, 0, 1, 0, 0, 0, 0 |
| |
| 0, 0, 0, 0, epsilon, 2, 0, 0 |
| |
| 0, 0, 0, 0, 2, 1 - epsilon, 0, 0 |
| |
| 0, 0, 0, 0, 0, 0, epsilon - 1, 2 |
| |
| 0, 0, 0, 0, 0, 0, 2, -epsilon |
+- -+
Time: 48 msec
>> coproduct(aa^3), coproduct(bb)
3 3
B(a ) # B(a ), B(b) # B(b)
Time: 496 msec
>> coproduct(aa)
2 2
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) +
4 4 5
3/16 B(a) # B(a ) + 1/16 B(a) # B(a b) + 3/16 B(a) # B(a ) +
5
1/16 B(a) # B(a b) + 1/16 B(a b) # B(a) + 1/16 B(a b) # B(a b) +
2 2 / II \ 4
1/16 B(a b) # B(a ) + 1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
\ 8 /
/ II \ 4 / II \ 5
| -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 /
/ II \ 5 2 2
| - -- - 1/16 | B(a b) # B(a b) + -1/16 B(a ) # B(a) + 1/16 B(a ) # B(a b) +
\ 8 /
2 2 2 2 2 4
-1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
2 4 2 5 2 5
1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) +
2 2 2 2
1/16 B(a b) # B(a) + 1/16 B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
2 2 / II \ 2 4
1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
\ 8 /
/ II \ 2 4 / II \ 2 5
| - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 /
/ II \ 2 5 4
| -- - 1/16 | B(a b) # B(a b) + 3/16 B(a ) # B(a) +
\ 8 /
/ II \ 4 4 2
| -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
\ 8 /
/ II \ 4 2 4 4
| -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) +
\ 8 /
/ II \ 4 4 4 5
| - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
\ 8 /
/ II \ 4 5 4
| - -- - 1/16 | B(a ) # B(a b) + 1/16 B(a b) # B(a) +
\ 8 /
/ II \ 4 4 2
| - -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
\ 8 /
/ II \ 4 2 / II \ 4 4
| -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 /
4 4 / II \ 4 5
1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
\ 8 /
4 5 5 / II \ 5
1/16 B(a b) # B(a b) + 3/16 B(a ) # B(a) + | - -- - 1/16 | B(a ) # B(a b) +
\ 8 /
5 2 / II \ 5 2 5 4
-1/16 B(a ) # B(a ) + | - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
\ 8 /
/ II \ 5 4 5 5
| -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) +
\ 8 /
/ II \ 5 5 5
| -- - 1/16 | B(a ) # B(a b) + 1/16 B(a b) # B(a) +
\ 8 /
/ II \ 5 5 2
| -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
\ 8 /
/ II \ 5 2 / II \ 5 4
| - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 /
5 4 / II \ 5 5 5 5
1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) + 1/16 B(a b) # B(a b)
\ 8 /
Time: 260 msec
>> K := KD3::G: // un simple raccourci
>> checkAntipode := K::mu @ ( K::id # K::antipode ) @ K::coproduct:
>> checkAntipode(x) $ x in K::basis::list()
B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1)
Time: 2020 msec
>> e := KD3::e:
>> K2basis := coidealAndAlgebraClosure([ e(1)+e(2) ])
-- +- -+ +- -+
| | 1, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | | |
| | 0, 1, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | | |
| | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 1, 0, 0, 0, 0, 0 |
| | | | |
| | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 1, 0, 0, 0, 0 |
| | |, | |,
| | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | | |
| | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | | |
| | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | | |
| | 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
-- +- -+ +- -+
+- -+ +- -+
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| |, | |,
| 0, 0, 0, 0, 0, -1, 0, 0 | | 0, 0, 0, 0, 1, 0, 0, 0 |
| | | |
| 0, 0, 0, 0, 1, 0, 0, 0 | | 0, 0, 0, 0, 0, 1, 0, 0 |
| | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
| | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 |
+- -+ +- -+
+- -+ +- -+ --
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| | | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| | | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| | | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| |, | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| | | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| | | | |
| 0, 0, 0, 0, 0, 0, 1, 0 | | 0, 0, 0, 0, 0, 0, 0, 0 | |
| | | | |
| 0, 0, 0, 0, 0, 0, 0, 0 | | 0, 0, 0, 0, 0, 0, 0, 1 | |
+- -+ +- -+ --
Time: 13601 msec
>> K2 := Dom::SubFreeModule(K2basis,
>> [Cat::FiniteDimensionalHopfAlgebraWithBasis(KD3::coeffRing)]):
>> K2::isCommutative(), K2::isCocommutative()
TRUE, FALSE
Time: 56 msec
>> K2dual := K2::Dual():
>> K2dual::groupLikeElements()
_ _ _ _ _ _
[B([3, 3]), B([1, 1]), B([8, 8]), B([7, 7]), -II B([6, 5]) + B([5, 5]),
_ _
II B([6, 5]) + B([5, 5])]
Time: 1236 msec
>> G := K2dual::intrinsicGroup():
>> G::list()
[[], [1], [1, 1], [2], [1, 2], [1, 1, 2]]
Time: 0 msec
>> K2dual::isSemiSimple()
TRUE
Time: 56 msec
>> K2dual::simpleModulesDimensions()
[2, 1, 1]
Time: 704 msec