Shuffle algebras¶
AUTHORS:
- Frédéric Chapoton (2013-03): Initial version
- Matthieu Deneufchatel (2013-07): Implemented dual PBW basis
-
sage.algebras.shuffle_algebra.
DualPBWBasis
¶ The basis dual to the Poincaré-Birkhoff-Witt basis of the free algebra.
We recursively define the dual PBW basis as the basis of the shuffle algebra given by
\[\begin{split}S_w = \begin{cases} w & |w| = 1, \\ x S_u & w = xu \text{ and } w \in \mathrm{Lyn}(X), \\ \displaystyle \frac{S_{\ell_{i_1}}^{\ast \alpha_1} \ast \cdots \ast S_{\ell_{i_k}}^{\ast \alpha_k}}{\alpha_1! \cdots \alpha_k!} & w = \ell_{i_1}^{\alpha_1} \cdots \ell_{i_k}^{\alpha_k} \text{ with } \ell_1 > \cdots > \ell_k \in \mathrm{Lyn}(X). \end{cases}\end{split}\]where \(S \ast T\) denotes the shuffle product of \(S\) and \(T\) and \(\mathrm{Lyn}(X)\) is the set of Lyndon words in the alphabet \(X\).
The definition may be found in Theorem 5.3 of [Reu1993].
INPUT:
R
– ringnames
– names of the generators (string or an alphabet)
EXAMPLES:
sage: S = ShuffleAlgebra(QQ, 'ab').dual_pbw_basis() sage: S The dual Poincare-Birkhoff-Witt basis of Shuffle Algebra on 2 generators ['a', 'b'] over Rational Field sage: S.one() S[word: ] sage: S.one_basis() word: sage: T = ShuffleAlgebra(QQ, 'abcd').dual_pbw_basis(); T The dual Poincare-Birkhoff-Witt basis of Shuffle Algebra on 4 generators ['a', 'b', 'c', 'd'] over Rational Field sage: T.algebra_generators() (S[word: a], S[word: b], S[word: c], S[word: d])
-
sage.algebras.shuffle_algebra.
ShuffleAlgebra
¶ 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
andshuffle()
.REFERENCES:
INPUT:
R
– ringnames
– generator names (string or an alphabet)
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]