Bialgebras with basis#
- class sage.categories.bialgebras_with_basis.BialgebrasWithBasis(base_category)#
Bases:
CategoryWithAxiom_over_base_ringThe category of bialgebras with a distinguished basis.
EXAMPLES:
sage: C = BialgebrasWithBasis(QQ); C Category of bialgebras with basis over Rational Field sage: sorted(C.super_categories(), key=str) [Category of algebras with basis over Rational Field, Category of bialgebras over Rational Field, Category of coalgebras with basis over Rational Field]
- class ElementMethods#
Bases:
object- adams_operator(n)#
Compute the \(n\)-th convolution power of the identity morphism \(\mathrm{Id}\) on
self.INPUT:
n– a nonnegative integer
OUTPUT:
the image of
selfunder the convolution power \(\mathrm{Id}^{*n}\)
Note
In the literature, this is also called a Hopf power or Sweedler power, cf. [AL2015].
See also
sage.categories.bialgebras.ElementMethods.convolution_product()Todo
Remove dependency on
modules_with_basismethods.EXAMPLES:
sage: h = SymmetricFunctions(QQ).h() # optional - sage.combinat sage: h[5].adams_operator(2) # optional - sage.combinat 2*h[3, 2] + 2*h[4, 1] + 2*h[5] sage: h[5].plethysm(2*h[1]) # optional - sage.combinat 2*h[3, 2] + 2*h[4, 1] + 2*h[5] sage: h([]).adams_operator(0) # optional - sage.combinat h[] sage: h([]).adams_operator(1) # optional - sage.combinat h[] sage: h[3,2].adams_operator(0) # optional - sage.combinat 0 sage: h[3,2].adams_operator(1) # optional - sage.combinat h[3, 2]
sage: S = NonCommutativeSymmetricFunctions(QQ).S() # optional - sage.combinat sage: S[4].adams_operator(5) # optional - sage.combinat 5*S[1, 1, 1, 1] + 10*S[1, 1, 2] + 10*S[1, 2, 1] + 10*S[1, 3] + 10*S[2, 1, 1] + 10*S[2, 2] + 10*S[3, 1] + 5*S[4]
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m() # optional - sage.combinat sage: m[[1,3],[2]].adams_operator(-2) # optional - sage.combinat 3*m{{1}, {2, 3}} + 3*m{{1, 2}, {3}} + 6*m{{1, 2, 3}} - 2*m{{1, 3}, {2}}
- convolution_product(*maps)#
Return the image of
selfunder the convolution product (map) of the maps.Let \(A\) and \(B\) be bialgebras over a commutative ring \(R\). Given maps \(f_i : A \to B\) for \(1 \leq i < n\), define the convolution product
\[(f_1 * f_2 * \cdots * f_n) := \mu^{(n-1)} \circ (f_1 \otimes f_2 \otimes \cdots \otimes f_n) \circ \Delta^{(n-1)},\]where \(\Delta^{(k)} := \bigl(\Delta \otimes \mathrm{Id}^{\otimes(k-1)}\bigr) \circ \Delta^{(k-1)}\), with \(\Delta^{(1)} = \Delta\) (the ordinary coproduct in \(A\)) and \(\Delta^{(0)} = \mathrm{Id}\); and with \(\mu^{(k)} := \mu \circ \bigl(\mu^{(k-1)} \otimes \mathrm{Id})\) and \(\mu^{(1)} = \mu\) (the ordinary product in \(B\)). See [Swe1969].
(In the literature, one finds, e.g., \(\Delta^{(2)}\) for what we denote above as \(\Delta^{(1)}\). See [KMN2012].)
INPUT:
maps– any number \(n \geq 0\) of linear maps \(f_1, f_2, \ldots, f_n\) onself.parent(); or a singlelistortupleof such maps
OUTPUT:
the convolution product of
mapsapplied toself
AUTHORS:
Amy Pang - 12 June 2015 - Sage Days 65
Todo
Remove dependency on
modules_with_basismethods.EXAMPLES:
We compute convolution products of the identity and antipode maps on Schur functions:
sage: Id = lambda x: x sage: Antipode = lambda x: x.antipode() sage: s = SymmetricFunctions(QQ).schur() # optional - sage.combinat sage: s[3].convolution_product(Id, Id) # optional - sage.combinat 2*s[2, 1] + 4*s[3] sage: s[3,2].convolution_product(Id) == s[3,2] # optional - sage.combinat True
The method accepts multiple arguments, or a single argument consisting of a list of maps:
sage: s[3,2].convolution_product(Id, Id) # optional - sage.combinat 2*s[2, 1, 1, 1] + 6*s[2, 2, 1] + 6*s[3, 1, 1] + 12*s[3, 2] + 6*s[4, 1] + 2*s[5] sage: s[3,2].convolution_product([Id, Id]) # optional - sage.combinat 2*s[2, 1, 1, 1] + 6*s[2, 2, 1] + 6*s[3, 1, 1] + 12*s[3, 2] + 6*s[4, 1] + 2*s[5]
We test the defining property of the antipode morphism; namely, that the antipode is the inverse of the identity map in the convolution algebra whose identity element is the composition of the counit and unit:
sage: (s[3,2].convolution_product() # optional - sage.combinat ....: == s[3,2].convolution_product(Antipode, Id) ....: == s[3,2].convolution_product(Id, Antipode)) True
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi() # optional - sage.combinat sage: Psi[2,1].convolution_product(Id, Id, Id) # optional - sage.combinat 3*Psi[1, 2] + 6*Psi[2, 1] sage: (Psi[5,1] - Psi[1,5]).convolution_product(Id, Id, Id) # optional - sage.combinat -3*Psi[1, 5] + 3*Psi[5, 1]
sage: G = SymmetricGroup(3) # optional - sage.combinat sage: QG = GroupAlgebra(G, QQ) # optional - sage.combinat sage: x = QG.sum_of_terms([(p, p.length()) # optional - sage.combinat ....: for p in Permutations(3)]); x [1, 3, 2] + [2, 1, 3] + 2*[2, 3, 1] + 2*[3, 1, 2] + 3*[3, 2, 1] sage: x.convolution_product(Id, Id) # optional - sage.combinat 5*[1, 2, 3] + 2*[2, 3, 1] + 2*[3, 1, 2] sage: x.convolution_product(Id, Id, Id) # optional - sage.combinat 4*[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + 3*[3, 2, 1] sage: x.convolution_product([Id] * 6) # optional - sage.combinat 9*[1, 2, 3]
- class ParentMethods#
Bases:
object- convolution_product(*maps)#
Return the convolution product (a map) of the given maps.
Let \(A\) and \(B\) be bialgebras over a commutative ring \(R\). Given maps \(f_i : A \to B\) for \(1 \leq i < n\), define the convolution product
\[(f_1 * f_2 * \cdots * f_n) := \mu^{(n-1)} \circ (f_1 \otimes f_2 \otimes \cdots \otimes f_n) \circ \Delta^{(n-1)},\]where \(\Delta^{(k)} := \bigl(\Delta \otimes \mathrm{Id}^{\otimes(k-1)}\bigr) \circ \Delta^{(k-1)}\), with \(\Delta^{(1)} = \Delta\) (the ordinary coproduct in \(A\)) and \(\Delta^{(0)} = \mathrm{Id}\); and with \(\mu^{(k)} := \mu \circ \bigl(\mu^{(k-1)} \otimes \mathrm{Id})\) and \(\mu^{(1)} = \mu\) (the ordinary product in \(B\)). See [Swe1969].
(In the literature, one finds, e.g., \(\Delta^{(2)}\) for what we denote above as \(\Delta^{(1)}\). See [KMN2012].)
INPUT:
maps– any number \(n \geq 0\) of linear maps \(f_1, f_2, \ldots, f_n\) onself; or a singlelistortupleof such maps
OUTPUT:
the new map \(f_1 * f_2 * \cdots * f_2\) representing their convolution product
See also
sage.categories.bialgebras.ElementMethods.convolution_product()AUTHORS:
Aaron Lauve - 12 June 2015 - Sage Days 65
Todo
Remove dependency on
modules_with_basismethods.EXAMPLES:
We construct some maps: the identity, the antipode and projection onto the homogeneous component of degree 2:
sage: Id = lambda x: x sage: Antipode = lambda x: x.antipode() sage: Proj2 = lambda x: x.parent().sum_of_terms([(m, c) for (m, c) in x if m.size() == 2])
Compute the convolution product of the identity with itself and with the projection
Proj2on the Hopf algebra of non-commutative symmetric functions:sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon() # optional - sage.combinat sage: T = R.convolution_product([Id, Id]) # optional - sage.combinat sage: [T(R(comp)) for comp in Compositions(3)] # optional - sage.combinat [4*R[1, 1, 1] + R[1, 2] + R[2, 1], 2*R[1, 1, 1] + 4*R[1, 2] + 2*R[2, 1] + 2*R[3], 2*R[1, 1, 1] + 2*R[1, 2] + 4*R[2, 1] + 2*R[3], R[1, 2] + R[2, 1] + 4*R[3]] sage: T = R.convolution_product(Proj2, Id) # optional - sage.combinat sage: [T(R([i])) for i in range(1, 5)] # optional - sage.combinat [0, R[2], R[2, 1] + R[3], R[2, 2] + R[4]]
Compute the convolution product of no maps on the Hopf algebra of symmetric functions in non-commuting variables. This is the composition of the counit with the unit:
sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m() # optional - sage.combinat sage: T = m.convolution_product() # optional - sage.combinat sage: [T(m(lam)) # optional - sage.combinat ....: for lam in SetPartitions(0).list() + SetPartitions(2).list()] [m{}, 0, 0]
Compute the convolution product of the projection
Proj2with the identity on the Hopf algebra of symmetric functions in non-commuting variables:sage: T = m.convolution_product(Proj2, Id) # optional - sage.combinat sage: [T(m(lam)) for lam in SetPartitions(3)] # optional - sage.combinat [0, m{{1, 2}, {3}} + m{{1, 2, 3}}, m{{1, 2}, {3}} + m{{1, 2, 3}}, m{{1, 2}, {3}} + m{{1, 2, 3}}, 3*m{{1}, {2}, {3}} + 3*m{{1}, {2, 3}} + 3*m{{1, 3}, {2}}]
Compute the convolution product of the antipode with itself and the identity map on group algebra of the symmetric group:
sage: G = SymmetricGroup(3) # optional - sage.groups sage.combinat sage: QG = GroupAlgebra(G, QQ) # optional - sage.groups sage.combinat sage: x = QG.sum_of_terms( # optional - sage.groups sage.combinat ....: [(p, p.number_of_peaks() + p.number_of_inversions()) ....: for p in Permutations(3)] ....: ); x 2*[1, 3, 2] + [2, 1, 3] + 3*[2, 3, 1] + 2*[3, 1, 2] + 3*[3, 2, 1] sage: T = QG.convolution_product(Antipode, Antipode, Id) # optional - sage.groups sage.combinat sage: T(x) # optional - sage.groups sage.combinat 2*[1, 3, 2] + [2, 1, 3] + 2*[2, 3, 1] + 3*[3, 1, 2] + 3*[3, 2, 1]