*----* 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:./index.html (local mirror)
-- Interface:
packages::Combinat::dotCategories, packages::Combinat::help,
packages::Combinat::viewDot, packages::Combinat::viewDotTeX,
packages::Combinat::viewTeX
>> TEXTWIDTH:=128:
>> 1+1
2
DOM_INT 0 ms
>> export(combinat):
>> trees::list(5)
-- o , o , o , o , o , o , o , o , o , o , o , o , o , o --
| // \\ /|\ /|\ / \ / \ /|\ / \ / \ / \ | | | | | |
| | | /\ | | | | /\ | /|\ / \ / \ | | |
| | | | | / \ | |
-- | --
DOM_LIST 364 ms
>> trees::count(5)
14
DOM_INT 0 ms
>> trees::random(50)
o
|
/ / \\
/ \| /\
/ / |\\ |
// \ \
| | |
/ \ | /|\
/ \ |
|
/ \
/|\ |
|/ \
| / \
/|\
|
combinat::trees 92 ms
>> 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
DOM_EXPR 112 ms
>> C1 := crystals::kirillovReshetikhin(2,2,["A",2,1]):
>> C2 := crystals::kirillovReshetikhin(1,1,["A",2,1]):
>> C1::list()
-- +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ --
| | 2 | 2 | | 2 | 3 | | 2 | 3 | | 3 | 3 | | 3 | 3 | | 3 | 3 | |
| +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ |
| | 1 | 1 |, | 1 | 1 |, | 1 | 2 |, | 1 | 1 |, | 1 | 2 |, | 2 | 2 | |
-- +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ --
DOM_LIST 164 ms
>> operators::setTensorSymbol("#"):
>> C := C1 # C2:
>> viewTeX(C::TeXClass())
>> C1::isomorphisms(C1)
[proc g(x) ... end]
DOM_LIST 108 ms
>> C1::isomorphisms(C1)
[proc g(x) ... end]
DOM_LIST 92 ms
>> C1::isomorphisms(C2)
[]
DOM_LIST 12 ms
>> viewTeX(packages::Combinat::categoryTeX())
>> 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 7092 2007-11-03 06:38:41Z 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)
KD(3, Q(II, epsilon))
DOM_DOMAIN 460 ms
>> KD3 := KD(3)
KD(3, Q(II, epsilon))
DOM_DOMAIN 32 ms
>> [aa,bb] := KD3::group::algebraGenerators::list()
[B(a), B(b)]
DOM_LIST 8 ms
>> bb^2
B(1)
(KD(3, Q(II, epsilon)))::G 12 ms
>> aa^2, aa^6, bb*aa
2 5
B(a ), B(1), B(a b)
(KD(3, Q(II, epsilon)))::G 4 ms
>> (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)
(KD(3, Q(II, epsilon)))::G 32 ms
>> 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 |
+- -+
(KD(3, Q(II, epsilon)))::M 52 ms
>> KD3::group(KD3::M(aa + 2*bb))
2 B(b) + B(a)
(KD(3, Q(II, epsilon)))::G 268 ms
>> coproduct(aa^4), coproduct(bb)
/ II \ 2 / II \ 2 4
3/16 B(a) # B(a) + | - -- - 1/16 | B(a) # B(a b) + -1/16 B(a) # B(a ) + | - -- - 1/16 | B(a) # B(a b) + 3/16 B(a) # B(a ) +
\ 8 / \ 8 /
/ II \ 4 5 / II \ 5 / II \
| -- - 1/16 | B(a) # B(a b) + -1/16 B(a) # B(a ) + | -- - 1/16 | B(a) # B(a b) + | -- - 1/16 | B(a b) # B(a) +
\ 8 / \ 8 / \ 8 /
/ II \ 2 2 4
1/16 B(a b) # B(a b) + | - -- - 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 5 / II \ 5 2
| - -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) + | -- - 1/16 | B(a b) # B(a b) + -1/16 B(a ) # B(a) +
\ 8 / \ 8 /
/ II \ 2 2 2 / II \ 2 2 2 4
| -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) + | -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) +
\ 8 / \ 8 /
/ II \ 2 4 2 5 / II \ 2 5 / II \ 2
| - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) + | - -- - 1/16 | B(a ) # B(a b) + | -- - 1/16 | B(a b) # B(a) +
\ 8 / \ 8 / \ 8 /
2 / II \ 2 2 2 2 2 4
1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) + 1/16 B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
\ 8 /
/ II \ 2 4 2 5 / II \ 2 5 4
| -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) + | - -- - 1/16 | B(a b) # B(a b) + 3/16 B(a ) # B(a) +
\ 8 / \ 8 /
4 4 2 4 2 4 4 4 4
1/16 B(a ) # B(a b) + 3/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + 7/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) +
4 5 4 5 / II \ 4 / II \ 4
-1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + | - -- - 1/16 | B(a b) # B(a) + | -- - 1/16 | B(a b) # B(a b) +
\ 8 / \ 8 /
/ II \ 4 2 / II \ 4 2 4 4 4 4
| -- - 1/16 | B(a b) # B(a ) + | - -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) + 1/16 B(a b) # B(a b) +
\ 8 / \ 8 /
4 5 4 5 5 5 5 2
1/16 B(a b) # B(a ) + 1/16 B(a b) # B(a b) + -1/16 B(a ) # B(a) + 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
5 2 5 4 5 4 5 5 5 5
1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) +
/ II \ 5 / II \ 5 / II \ 5 2
| - -- - 1/16 | B(a b) # B(a) + | - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 / \ 8 /
/ II \ 5 2 5 4 5 4 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) + 1/16 B(a b) # B(a ) +
\ 8 /
5 5
1/16 B(a b) # B(a b), B(b) # B(b)
(KD(3, Q(II, epsilon)))::G # (KD(3, Q(II, epsilon)))::G 488 ms
>> coproduct(aa)
2 2 4 4
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) + 3/16 B(a) # B(a ) + 1/16 B(a) # B(a b) +
5 5 2
3/16 B(a) # B(a ) + 1/16 B(a) # B(a b) + 1/16 B(a b) # B(a) + 1/16 B(a b) # B(a b) + 1/16 B(a b) # B(a ) +
2 / II \ 4 / II \ 4 / II \ 5
1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) + | -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 / \ 8 /
/ II \ 5 2 2 2 2 2 2
| - -- - 1/16 | B(a b) # B(a b) + -1/16 B(a ) # B(a) + 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) +
\ 8 /
2 4 2 4 2 5 2 5 2
-1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a b) + 1/16 B(a b) # B(a) +
2 2 2 2 2 / II \ 2 4
1/16 B(a b) # B(a b) + 1/16 B(a b) # B(a ) + 1/16 B(a b) # B(a b) + | - -- - 1/16 | B(a b) # B(a ) +
\ 8 /
/ II \ 2 4 / II \ 2 5 / II \ 2 5 4
| - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) + | -- - 1/16 | B(a b) # B(a b) + 3/16 B(a ) # B(a) +
\ 8 / \ 8 / \ 8 /
/ II \ 4 4 2 / II \ 4 2 4 4
| -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) + | -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) +
\ 8 / \ 8 /
/ II \ 4 4 4 5 / II \ 4 5 4
| - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) + | - -- - 1/16 | B(a ) # B(a b) + 1/16 B(a b) # B(a) +
\ 8 / \ 8 /
/ II \ 4 4 2 / II \ 4 2 / II \ 4 4
| - -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) + | -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 8 / \ 8 /
4 4 / II \ 4 5 4 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) + 3/16 B(a ) # B(a) +
\ 8 /
/ II \ 5 5 2 / II \ 5 2 5 4
| - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) + | - -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
\ 8 / \ 8 /
/ II \ 5 4 5 5 / II \ 5 5 5
| -- - 1/16 | B(a ) # B(a b) + 3/16 B(a ) # B(a ) + | -- - 1/16 | B(a ) # B(a b) + 1/16 B(a b) # B(a) +
\ 8 / \ 8 /
/ II \ 5 5 2 / II \ 5 2 / II \ 5 4
| -- - 1/16 | B(a b) # B(a b) + 1/16 B(a b) # B(a ) + | - -- - 1/16 | B(a b) # B(a b) + | -- - 1/16 | B(a b) # B(a ) +
\ 8 / \ 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 /
(KD(3, Q(II, epsilon)))::G # (KD(3, Q(II, epsilon)))::G 252 ms
>> KD3::G::antipode
proc moduleMorphism(f)(x) ... end
DOM_PROC 0 ms
>> KD3::G::moduleMorphismMatrix(KD3::G::antipode)
+- -+
| 1, 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, 1, 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, 1, 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, 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, 1, 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, 1 |
+- -+
Dom::Matrix(Q(II, epsilon)) 116 ms
>> KD3::G::kernelOfModuleMorphism(KD3::G::antipode)
[]
DOM_LIST 44 ms
>> checkAntipode := KD3::G::mu @ ( KD3::G::id # KD3::G::antipode ) @ KD3::G::coproduct:
>> checkAntipode(x) $ x in KD3::G::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)
(KD(3, Q(II, epsilon)))::G 1 s 812 ms
>> e := KD3::e:
>> e(1)+e(2)
+- -+
| 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 |
+- -+
(KD(3, Q(II, epsilon)))::M 28 ms
>> K2basis := coidealAndAlgebraClosure([ e(1)+e(2) ])
-- +- -+ +- -+ +- -+ +- -+
| | 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, 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, 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, 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, -1, 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, 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, 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 | |
+- -+ +- -+ --
DOM_LIST 11 s 953 ms
>> expose(KD3::G::algebraClosure)
proc(generators : Type::ListOf((TwistedDihedralGroupAlgebra(3, Q(II, epsilon)))::G)) : Type::ListOf((TwistedDihedralGroupAlgebr\
a(3, Q(II, epsilon)))::G)
name (TwistedDihedralGroupAlgebra(3, Q(II, epsilon)))::G::algebraClosure;
begin
userinfo(3, "Computing the (non unital!) algebra closure");
dom::moduleClosure(generators, [proc(x : dom) : Type::SequenceOf(dom)
local generator;
begin
x*generator $ generator in generators
end_proc])
end_proc
stdlib::Exposed 4 ms
>> K2 := Dom::SubFreeModule(K2basis,
>> [Cat::FiniteDimensionalHopfAlgebraWithBasis(KD3::coeffRing)]):
>> K2::isCommutative(), K2::isCocommutative()
TRUE, FALSE
DOM_BOOL 108 ms
>> 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])]
DOM_LIST 948 ms
>> G := K2dual::intrisicGroup():
>> G::list()
[[], [1], [1, 1], [2], [1, 2], [1, 1, 2]]
DOM_LIST 4 ms
>> K2dual::isSemiSimple()
TRUE
DOM_BOOL 60 ms
>> K2dual::simpleModulesDimensions()
[2, 1, 1]
DOM_LIST 620 ms
>> reset():
+---+
| 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:./index.html (local mirror)
-- Interface:
packages::Combinat::dotCategories, packages::Combinat::help,
packages::Combinat::viewDot, packages::Combinat::viewDotTeX,
packages::Combinat::viewTeX
>> operators::setTensorSymbol("#"):
>> export(operators, coproduct):
>> S := examples::SymmetricFunctions():
>> S::Name := hold(S):
>> operators::setTensorSymbol("#"):
>> export(operators, coproduct):
>> S := examples::SymmetricFunctions():
>> S::Name := hold(S):
>> viewTeX((operators::overloaded::conversionGraph())::TeX()):
>> Mcd := S::Macdonald(q,t):
>> QSym := examples::QuasiSymmetricFunctions():
>> QSym::Name := hold(QSym): QSym::fixBasesNames():
>> NCSF := examples::NonCommutativeSymmetricFunctions():
>> NCSF::Name := hold(NCSF): NCSF::fixBasesNames():
>> viewTeX((operators::overloaded::conversionGraph())::TeX(), "conversion.pdf"):
>>