*----* 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"): >>