Shuffle algebras

AUTHORS:

  • Frédéric Chapoton (2013-03)
class sage.algebras.shuffle_algebra.ShuffleAlgebra(R, names)

Bases: sage.combinat.free_module.CombinatorialFreeModule

The shuffle algebra on some generators over a base ring.

Shuffle algebras are commutative and associative algebras, with a basis indexed by words. The product of two words w_1 \cdot w_2 is given by the sum over the shuffle product of w_1 and w_2.

See also

For more on shuffle products, see shuffle_product and shuffle().

REFERENCES:

INPUT:

  • R – ring
  • names – generator names (string)

EXAMPLES:

sage: F = ShuffleAlgebra(QQ, 'xyz'); F
Shuffle Algebra on 3 generators ['x', 'y', 'z'] over Rational Field

sage: mul(F.gens())
B[word: xyz] + B[word: xzy] + B[word: yxz] + B[word: yzx] + B[word: zxy] + B[word: zyx]

sage: mul([ F.gen(i) for i in range(2) ]) + mul([ F.gen(i+1) for i in range(2) ])
B[word: xy] + B[word: yx] + B[word: yz] + B[word: zy]

sage: S = ShuffleAlgebra(ZZ, 'abcabc'); S
Shuffle Algebra on 3 generators ['a', 'b', 'c'] over Integer Ring
sage: S.base_ring()
Integer Ring

sage: G = ShuffleAlgebra(S, 'mn'); G
Shuffle Algebra on 2 generators ['m', 'n'] over Shuffle Algebra on 3 generators ['a', 'b', 'c'] over Integer Ring
sage: G.base_ring()
Shuffle Algebra on 3 generators ['a', 'b', 'c'] over Integer Ring

Shuffle algebras commute with their base ring:

sage: K = ShuffleAlgebra(QQ,'ab')
sage: a,b = K.gens()
sage: K.is_commutative()
True
sage: L = ShuffleAlgebra(K,'cd')
sage: c,d = L.gens()
sage: L.is_commutative()
True
sage: s = a*b^2 * c^3; s
(12*B[word:abb]+12*B[word:bab]+12*B[word:bba])*B[word: ccc]
sage: parent(s)
Shuffle Algebra on 2 generators ['c', 'd'] over Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field
sage: c^3 * a * b^2
(12*B[word:abb]+12*B[word:bab]+12*B[word:bba])*B[word: ccc]

Shuffle algebras are commutative:

sage: c^3 * b * a * b == c * a * c * b^2 * c
True

We can also manipulate elements in the basis and coerce elements from our base field:

sage: F = ShuffleAlgebra(QQ, 'abc')
sage: B = F.basis()
sage: B[Word('bb')] * B[Word('ca')]  
B[word: bbca] + B[word: bcab] + B[word: bcba] + B[word: cabb] + B[word: cbab] + B[word: cbba]
sage: 1 - B[Word('bb')] * B[Word('ca')] / 2
B[word: ] - 1/2*B[word: bbca] - 1/2*B[word: bcab] - 1/2*B[word: bcba] - 1/2*B[word: cabb] - 1/2*B[word: cbab] - 1/2*B[word: cbba]
algebra_generators()

Return the generators of this algebra.

EXAMPLES:

sage: A = ShuffleAlgebra(ZZ,'fgh'); A
Shuffle Algebra on 3 generators ['f', 'g', 'h'] over Integer Ring
sage: A.algebra_generators()
Family (B[word: f], B[word: g], B[word: h])
gen(i)

The i-th generator of the algebra.

INPUT:

  • i – an integer

EXAMPLES:

sage: F = ShuffleAlgebra(ZZ,'xyz')
sage: F.gen(0)
B[word: x]

sage: F.gen(4)
Traceback (most recent call last):
...
IndexError: argument i (= 4) must be between 0 and 2
gens()

Return the generators of this algebra.

EXAMPLES:

sage: A = ShuffleAlgebra(ZZ,'fgh'); A
Shuffle Algebra on 3 generators ['f', 'g', 'h'] over Integer Ring
sage: A.algebra_generators()
Family (B[word: f], B[word: g], B[word: h])
is_commutative()

Return True as the shuffle algebra is commutative.

EXAMPLES:

sage: R = ShuffleAlgebra(QQ,'x')
sage: R.is_commutative()
True
sage: R = ShuffleAlgebra(QQ,'xy')
sage: R.is_commutative()
True
one_basis()

Return the empty word, which index of 1 of this algebra, as per AlgebrasWithBasis.ParentMethods.one_basis().

EXAMPLES:

sage: A = ShuffleAlgebra(QQ,'a')
sage: A.one_basis()
word:
sage: A.one()
B[word: ]
product_on_basis(w1, w2)

Return the product of basis elements w1 and w2, as per AlgebrasWithBasis.ParentMethods.product_on_basis().

INPUT:

  • w1, w2 – Basis elements

EXAMPLES:

sage: A = ShuffleAlgebra(QQ,'abc')
sage: W = A.basis().keys()
sage: A.product_on_basis(W("acb"), W("cba"))
B[word: acbacb] + B[word: acbcab] + 2*B[word: acbcba] + 2*B[word: accbab] + 4*B[word: accbba] + B[word: cabacb] + B[word: cabcab] + B[word: cabcba] + B[word: cacbab] + 2*B[word: cacbba] + 2*B[word: cbaacb] + B[word: cbacab] + B[word: cbacba]

sage: (a,b,c) = A.algebra_generators()
sage: a * (1-b)^2 * c
2*B[word: abbc] - 2*B[word: abc] + 2*B[word: abcb] + B[word: ac] - 2*B[word: acb] + 2*B[word: acbb] + 2*B[word: babc] - 2*B[word: bac] + 2*B[word: bacb] + 2*B[word: bbac] + 2*B[word: bbca] - 2*B[word: bca] + 2*B[word: bcab] + 2*B[word: bcba] + B[word: ca] - 2*B[word: cab] + 2*B[word: cabb] - 2*B[word: cba] + 2*B[word: cbab] + 2*B[word: cbba]
variable_names()

Return the names of the variables.

EXAMPLES:

sage: R = ShuffleAlgebra(QQ,'xy')
sage: R.variable_names()
{'x', 'y'}

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