Descent Algebras

AUTHORS:

  • Travis Scrimshaw (2013-07-28): Initial version

class sage.combinat.descent_algebra.DescentAlgebra(R, n)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

Solomon’s descent algebra.

The descent algebra \(\Sigma_n\) over a ring \(R\) is a subalgebra of the symmetric group algebra \(R S_n\). (The product in the latter algebra is defined by \((pq)(i) = q(p(i))\) for any two permutations \(p\) and \(q\) in \(S_n\) and every \(i \in \{ 1, 2, \ldots, n \}\). The algebra \(\Sigma_n\) inherits this product.)

There are three bases currently implemented for \(\Sigma_n\):

  • the standard basis \(D_S\) of (sums of) descent classes, indexed by subsets \(S\) of \(\{1, 2, \ldots, n-1\}\),

  • the subset basis \(B_p\), indexed by compositions \(p\) of \(n\),

  • the idempotent basis \(I_p\), indexed by compositions \(p\) of \(n\), which is used to construct the mutually orthogonal idempotents of the symmetric group algebra.

The idempotent basis is only defined when \(R\) is a \(\QQ\)-algebra.

We follow the notations and conventions in [GR1989], apart from the order of multiplication being different from the one used in that article. Schocker’s exposition [Sch2004], in turn, uses the same order of multiplication as we are, but has different notations for the bases.

INPUT:

  • R – the base ring

  • n – a nonnegative integer

REFERENCES:

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: D = DA.D(); D
Descent algebra of 4 over Rational Field in the standard basis
sage: B = DA.B(); B
Descent algebra of 4 over Rational Field in the subset basis
sage: I = DA.I(); I
Descent algebra of 4 over Rational Field in the idempotent basis
sage: basis_B = B.basis()
sage: elt = basis_B[Composition([1,2,1])] + 4*basis_B[Composition([1,3])]; elt
B[1, 2, 1] + 4*B[1, 3]
sage: D(elt)
5*D{} + 5*D{1} + D{1, 3} + D{3}
sage: I(elt)
7/6*I[1, 1, 1, 1] + 2*I[1, 1, 2] + 3*I[1, 2, 1] + 4*I[1, 3]

As syntactic sugar, one can use the notation D[i,...,l] to construct elements of the basis; note that for the empty set one must use D[[]] due to Python’s syntax:

sage: D[[]] + D[2] + 2*D[1,2]
D{} + 2*D{1, 2} + D{2}

The same syntax works for the other bases:

sage: I[1,2,1] + 3*I[4] + 2*I[3,1]
I[1, 2, 1] + 2*I[3, 1] + 3*I[4]
class B(alg, prefix='B')

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The subset basis of a descent algebra (indexed by compositions).

The subset basis \((B_S)_{S \subseteq \{1, 2, \ldots, n-1\}}\) of \(\Sigma_n\) is formed by

\[B_S = \sum_{T \subseteq S} D_T,\]

where \((D_S)_{S \subseteq \{1, 2, \ldots, n-1\}}\) is the standard basis. However it is more natural to index the subset basis by compositions of \(n\) under the bijection \(\{i_1, i_2, \ldots, i_k\} \mapsto (i_1, i_2 - i_1, i_3 - i_2, \ldots, i_k - i_{k-1}, n - i_k)\) (where \(i_1 < i_2 < \cdots < i_k\)), which is what Sage uses to index the basis.

The basis element \(B_p\) is denoted \(\Xi^p\) in [Sch2004].

By using compositions of \(n\), the product \(B_p B_q\) becomes a sum over the non-negative-integer matrices \(M\) with row sum \(p\) and column sum \(q\). The summand corresponding to \(M\) is \(B_c\), where \(c\) is the composition obtained by reading \(M\) row-by-row from left-to-right and top-to-bottom and removing all zeroes. This multiplication rule is commonly called “Solomon’s Mackey formula”.

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: B = DA.B()
sage: list(B.basis())
[B[1, 1, 1, 1], B[1, 1, 2], B[1, 2, 1], B[1, 3],
 B[2, 1, 1], B[2, 2], B[3, 1], B[4]]
one_basis()

Return the identity element which is the composition \([n]\), as per AlgebrasWithBasis.ParentMethods.one_basis.

EXAMPLES:

sage: DescentAlgebra(QQ, 4).B().one_basis()
[4]
sage: DescentAlgebra(QQ, 0).B().one_basis()
[]

sage: all( U * DescentAlgebra(QQ, 3).B().one() == U
....:      for U in DescentAlgebra(QQ, 3).B().basis() )
True
product_on_basis(p, q)

Return \(B_p B_q\), where \(p\) and \(q\) are compositions of \(n\).

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: B = DA.B()
sage: p = Composition([1,2,1])
sage: q = Composition([3,1])
sage: B.product_on_basis(p, q)
B[1, 1, 1, 1] + 2*B[1, 2, 1]
to_D_basis(p)

Return \(B_p\) as a linear combination of \(D\)-basis elements.

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: B = DA.B()
sage: D = DA.D()
sage: list(map(D, B.basis()))  # indirect doctest
[D{} + D{1} + D{1, 2} + D{1, 2, 3}
  + D{1, 3} + D{2} + D{2, 3} + D{3},
 D{} + D{1} + D{1, 2} + D{2},
 D{} + D{1} + D{1, 3} + D{3},
 D{} + D{1},
 D{} + D{2} + D{2, 3} + D{3},
 D{} + D{2},
 D{} + D{3},
 D{}]
to_I_basis(p)

Return \(B_p\) as a linear combination of \(I\)-basis elements.

This is done using the formula

\[B_p = \sum_{q \leq p} \frac{1}{\mathbf{k}!(q,p)} I_q,\]

where \(\leq\) is the refinement order and \(\mathbf{k}!(q,p)\) is defined as follows: When \(q \leq p\), we can write \(q\) as a concatenation \(q_{(1)} q_{(2)} \cdots q_{(k)}\) with each \(q_{(i)}\) being a composition of the \(i\)-th entry of \(p\), and then we set \(\mathbf{k}!(q,p)\) to be \(l(q_{(1)})! l(q_{(2)})! \cdots l(q_{(k)})!\), where \(l(r)\) denotes the number of parts of any composition \(r\).

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: B = DA.B()
sage: I = DA.I()
sage: list(map(I, B.basis()))  # indirect doctest
[I[1, 1, 1, 1],
 1/2*I[1, 1, 1, 1] + I[1, 1, 2],
 1/2*I[1, 1, 1, 1] + I[1, 2, 1],
 1/6*I[1, 1, 1, 1] + 1/2*I[1, 1, 2] + 1/2*I[1, 2, 1] + I[1, 3],
 1/2*I[1, 1, 1, 1] + I[2, 1, 1],
 1/4*I[1, 1, 1, 1] + 1/2*I[1, 1, 2] + 1/2*I[2, 1, 1] + I[2, 2],
 1/6*I[1, 1, 1, 1] + 1/2*I[1, 2, 1] + 1/2*I[2, 1, 1] + I[3, 1],
 1/24*I[1, 1, 1, 1] + 1/6*I[1, 1, 2] + 1/6*I[1, 2, 1]
  + 1/2*I[1, 3] + 1/6*I[2, 1, 1] + 1/2*I[2, 2] + 1/2*I[3, 1] + I[4]]
to_nsym(p)

Return \(B_p\) as an element in \(NSym\), the non-commutative symmetric functions.

This maps \(B_p\) to \(S_p\) where \(S\) denotes the Complete basis of \(NSym\).

EXAMPLES:

sage: B = DescentAlgebra(QQ, 4).B()
sage: S = NonCommutativeSymmetricFunctions(QQ).Complete()
sage: list(map(S, B.basis()))  # indirect doctest
[S[1, 1, 1, 1],
 S[1, 1, 2],
 S[1, 2, 1],
 S[1, 3],
 S[2, 1, 1],
 S[2, 2],
 S[3, 1],
 S[4]]
class D(alg, prefix='D')

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The standard basis of a descent algebra.

This basis is indexed by \(S \subseteq \{1, 2, \ldots, n-1\}\), and the basis vector indexed by \(S\) is the sum of all permutations, taken in the symmetric group algebra \(R S_n\), whose descent set is \(S\). We denote this basis vector by \(D_S\).

Occasionally this basis appears in literature but indexed by compositions of \(n\) rather than subsets of \(\{1, 2, \ldots, n-1\}\). The equivalence between these two indexings is owed to the bijection from the power set of \(\{1, 2, \ldots, n-1\}\) to the set of all compositions of \(n\) which sends every subset \(\{i_1, i_2, \ldots, i_k\}\) of \(\{1, 2, \ldots, n-1\}\) (with \(i_1 < i_2 < \cdots < i_k\)) to the composition \((i_1, i_2-i_1, \ldots, i_k-i_{k-1}, n-i_k)\).

The basis element corresponding to a composition \(p\) (or to the subset of \(\{1, 2, \ldots, n-1\}\)) is denoted \(\Delta^p\) in [Sch2004].

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: D = DA.D()
sage: list(D.basis())
[D{}, D{1}, D{2}, D{3}, D{1, 2}, D{1, 3}, D{2, 3}, D{1, 2, 3}]

sage: DA = DescentAlgebra(QQ, 0)
sage: D = DA.D()
sage: list(D.basis())
[D{}]
one_basis()

Return the identity element, as per AlgebrasWithBasis.ParentMethods.one_basis.

EXAMPLES:

sage: DescentAlgebra(QQ, 4).D().one_basis()
()
sage: DescentAlgebra(QQ, 0).D().one_basis()
()

sage: all( U * DescentAlgebra(QQ, 3).D().one() == U
....:      for U in DescentAlgebra(QQ, 3).D().basis() )
True
product_on_basis(S, T)

Return \(D_S D_T\), where \(S\) and \(T\) are subsets of \([n-1]\).

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: D = DA.D()
sage: D.product_on_basis((1, 3), (2,))
D{} + D{1} + D{1, 2} + 2*D{1, 2, 3} + D{1, 3} + D{2} + D{2, 3} + D{3}
to_B_basis(S)

Return \(D_S\) as a linear combination of \(B_p\)-basis elements.

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: D = DA.D()
sage: B = DA.B()
sage: list(map(B, D.basis()))  # indirect doctest
[B[4],
 B[1, 3] - B[4],
 B[2, 2] - B[4],
 B[3, 1] - B[4],
 B[1, 1, 2] - B[1, 3] - B[2, 2] + B[4],
 B[1, 2, 1] - B[1, 3] - B[3, 1] + B[4],
 B[2, 1, 1] - B[2, 2] - B[3, 1] + B[4],
 B[1, 1, 1, 1] - B[1, 1, 2] - B[1, 2, 1] + B[1, 3]
  - B[2, 1, 1] + B[2, 2] + B[3, 1] - B[4]]
to_symmetric_group_algebra_on_basis(S)

Return \(D_S\) as a linear combination of basis elements in the symmetric group algebra.

EXAMPLES:

sage: D = DescentAlgebra(QQ, 4).D()
sage: [D.to_symmetric_group_algebra_on_basis(tuple(b))
....:  for b in Subsets(3)]
[[1, 2, 3, 4],
 [2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3],
 [1, 3, 2, 4] + [1, 4, 2, 3] + [2, 3, 1, 4]
  + [2, 4, 1, 3] + [3, 4, 1, 2],
 [1, 2, 4, 3] + [1, 3, 4, 2] + [2, 3, 4, 1],
 [3, 2, 1, 4] + [4, 2, 1, 3] + [4, 3, 1, 2],
 [2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1]
  + [4, 1, 3, 2] + [4, 2, 3, 1],
 [1, 4, 3, 2] + [2, 4, 3, 1] + [3, 4, 2, 1],
 [4, 3, 2, 1]]
class I(alg, prefix='I')

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The idempotent basis of a descent algebra.

The idempotent basis \((I_p)_{p \models n}\) is a basis for \(\Sigma_n\) whenever the ground ring is a \(\QQ\)-algebra. One way to compute it is using the formula (Theorem 3.3 in [GR1989])

\[I_p = \sum_{q \leq p} \frac{(-1)^{l(q)-l(p)}}{\mathbf{k}(q,p)} B_q,\]

where \(\leq\) is the refinement order and \(l(r)\) denotes the number of parts of any composition \(r\), and where \(\mathbf{k}(q,p)\) is defined as follows: When \(q \leq p\), we can write \(q\) as a concatenation \(q_{(1)} q_{(2)} \cdots q_{(k)}\) with each \(q_{(i)}\) being a composition of the \(i\)-th entry of \(p\), and then we set \(\mathbf{k}(q,p)\) to be the product \(l(q_{(1)}) l(q_{(2)}) \cdots l(q_{(k)})\).

Let \(\lambda(p)\) denote the partition obtained from a composition \(p\) by sorting. This basis is called the idempotent basis since for any \(q\) such that \(\lambda(p) = \lambda(q)\), we have:

\[I_p I_q = s(\lambda) I_p\]

where \(\lambda\) denotes \(\lambda(p) = \lambda(q)\), and where \(s(\lambda)\) is the stabilizer of \(\lambda\) in \(S_n\). (This is part of Theorem 4.2 in [GR1989].)

It is also straightforward to compute the idempotents \(E_{\lambda}\) for the symmetric group algebra by the formula (Theorem 3.2 in [GR1989]):

\[E_{\lambda} = \frac{1}{k!} \sum_{\lambda(p) = \lambda} I_p.\]

Note

The basis elements are not orthogonal idempotents.

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: I = DA.I()
sage: list(I.basis())
[I[1, 1, 1, 1], I[1, 1, 2], I[1, 2, 1], I[1, 3], I[2, 1, 1], I[2, 2], I[3, 1], I[4]]
idempotent(la)

Return the idempotent corresponding to the partition la of \(n\).

EXAMPLES:

sage: I = DescentAlgebra(QQ, 4).I()
sage: E = I.idempotent([3,1]); E
1/2*I[1, 3] + 1/2*I[3, 1]
sage: E*E == E
True
sage: E2 = I.idempotent([2,1,1]); E2
1/6*I[1, 1, 2] + 1/6*I[1, 2, 1] + 1/6*I[2, 1, 1]
sage: E2*E2 == E2
True
sage: E*E2 == I.zero()
True
one()

Return the identity element, which is \(B_{[n]}\), in the \(I\) basis.

EXAMPLES:

sage: DescentAlgebra(QQ, 4).I().one()
1/24*I[1, 1, 1, 1] + 1/6*I[1, 1, 2] + 1/6*I[1, 2, 1]
 + 1/2*I[1, 3] + 1/6*I[2, 1, 1] + 1/2*I[2, 2]
 + 1/2*I[3, 1] + I[4]
sage: DescentAlgebra(QQ, 0).I().one()
I[]
one_basis()

The element \(1\) is not (generally) a basis vector in the \(I\) basis, thus this returns a TypeError.

EXAMPLES:

sage: DescentAlgebra(QQ, 4).I().one_basis()
Traceback (most recent call last):
...
TypeError: 1 is not a basis element in the I basis.
product_on_basis(p, q)

Return \(I_p I_q\), where \(p\) and \(q\) are compositions of \(n\).

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: I = DA.I()
sage: p = Composition([1,2,1])
sage: q = Composition([3,1])
sage: I.product_on_basis(p, q)
0
sage: I.product_on_basis(p, p)
2*I[1, 2, 1]
to_B_basis(p)

Return \(I_p\) as a linear combination of \(B\)-basis elements.

This is computed using the formula (Theorem 3.3 in [GR1989])

\[I_p = \sum_{q \leq p} \frac{(-1)^{l(q)-l(p)}}{\mathbf{k}(q,p)} B_q,\]

where \(\leq\) is the refinement order and \(l(r)\) denotes the number of parts of any composition \(r\), and where \(\mathbf{k}(q,p)\) is defined as follows: When \(q \leq p\), we can write \(q\) as a concatenation \(q_{(1)} q_{(2)} \cdots q_{(k)}\) with each \(q_{(i)}\) being a composition of the \(i\)-th entry of \(p\), and then we set \(\mathbf{k}(q,p)\) to be \(l(q_{(1)}) l(q_{(2)}) \cdots l(q_{(k)})\).

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: B = DA.B()
sage: I = DA.I()
sage: list(map(B, I.basis()))  # indirect doctest
[B[1, 1, 1, 1],
 -1/2*B[1, 1, 1, 1] + B[1, 1, 2],
 -1/2*B[1, 1, 1, 1] + B[1, 2, 1],
 1/3*B[1, 1, 1, 1] - 1/2*B[1, 1, 2] - 1/2*B[1, 2, 1] + B[1, 3],
 -1/2*B[1, 1, 1, 1] + B[2, 1, 1],
 1/4*B[1, 1, 1, 1] - 1/2*B[1, 1, 2] - 1/2*B[2, 1, 1] + B[2, 2],
 1/3*B[1, 1, 1, 1] - 1/2*B[1, 2, 1] - 1/2*B[2, 1, 1] + B[3, 1],
 -1/4*B[1, 1, 1, 1] + 1/3*B[1, 1, 2] + 1/3*B[1, 2, 1]
  - 1/2*B[1, 3] + 1/3*B[2, 1, 1] - 1/2*B[2, 2]
  - 1/2*B[3, 1] + B[4]]
a_realization()

Return a particular realization of self (the \(B\)-basis).

EXAMPLES:

sage: DA = DescentAlgebra(QQ, 4)
sage: DA.a_realization()
Descent algebra of 4 over Rational Field in the subset basis
idempotent

alias of DescentAlgebra.I

standard

alias of DescentAlgebra.D

subset

alias of DescentAlgebra.B

class sage.combinat.descent_algebra.DescentAlgebraBases(base)

Bases: sage.categories.realizations.Category_realization_of_parent

The category of bases of a descent algebra.

class ElementMethods

Bases: object

to_symmetric_group_algebra()

Return self in the symmetric group algebra.

EXAMPLES:

sage: B = DescentAlgebra(QQ, 4).B()
sage: B[1,3].to_symmetric_group_algebra()
[1, 2, 3, 4] + [2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3]
sage: I = DescentAlgebra(QQ, 4).I()
sage: elt = I(B[1,3])
sage: elt.to_symmetric_group_algebra()
[1, 2, 3, 4] + [2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3]
class ParentMethods

Bases: object

is_commutative()

Return whether this descent algebra is commutative.

EXAMPLES:

sage: B = DescentAlgebra(QQ, 4).B()
sage: B.is_commutative()
False
sage: B = DescentAlgebra(QQ, 1).B()
sage: B.is_commutative()
True
is_field(proof=True)

Return whether this descent algebra is a field.

EXAMPLES:

sage: B = DescentAlgebra(QQ, 4).B()
sage: B.is_field()
False
sage: B = DescentAlgebra(QQ, 1).B()
sage: B.is_field()
True
to_symmetric_group_algebra()

Morphism from self to the symmetric group algebra.

EXAMPLES:

sage: D = DescentAlgebra(QQ, 4).D()
sage: D.to_symmetric_group_algebra(D[1,3])
[2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1] + [4, 1, 3, 2] + [4, 2, 3, 1]
sage: B = DescentAlgebra(QQ, 4).B()
sage: B.to_symmetric_group_algebra(B[1,2,1])
[1, 2, 3, 4] + [1, 2, 4, 3] + [1, 3, 4, 2] + [2, 1, 3, 4]
 + [2, 1, 4, 3] + [2, 3, 4, 1] + [3, 1, 2, 4] + [3, 1, 4, 2]
 + [3, 2, 4, 1] + [4, 1, 2, 3] + [4, 1, 3, 2] + [4, 2, 3, 1]
to_symmetric_group_algebra_on_basis(S)

Return the basis element index by S as a linear combination of basis elements in the symmetric group algebra.

EXAMPLES:

sage: B = DescentAlgebra(QQ, 3).B()
sage: [B.to_symmetric_group_algebra_on_basis(c)
....:  for c in Compositions(3)]
[[1, 2, 3] + [1, 3, 2] + [2, 1, 3]
  + [2, 3, 1] + [3, 1, 2] + [3, 2, 1],
 [1, 2, 3] + [2, 1, 3] + [3, 1, 2],
 [1, 2, 3] + [1, 3, 2] + [2, 3, 1],
 [1, 2, 3]]
sage: I = DescentAlgebra(QQ, 3).I()
sage: [I.to_symmetric_group_algebra_on_basis(c)
....:  for c in Compositions(3)]
[[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1]
  + [3, 1, 2] + [3, 2, 1],
 1/2*[1, 2, 3] - 1/2*[1, 3, 2] + 1/2*[2, 1, 3]
  - 1/2*[2, 3, 1] + 1/2*[3, 1, 2] - 1/2*[3, 2, 1],
 1/2*[1, 2, 3] + 1/2*[1, 3, 2] - 1/2*[2, 1, 3]
  + 1/2*[2, 3, 1] - 1/2*[3, 1, 2] - 1/2*[3, 2, 1],
 1/3*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3]
  - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] + 1/3*[3, 2, 1]]
super_categories()

The super categories of self.

EXAMPLES:

sage: from sage.combinat.descent_algebra import DescentAlgebraBases
sage: DA = DescentAlgebra(QQ, 4)
sage: bases = DescentAlgebraBases(DA)
sage: bases.super_categories()
[Category of finite dimensional algebras with basis over Rational Field,
 Category of realizations of Descent algebra of 4 over Rational Field]