Descent Algebras¶

AUTHORS:

• Travis Scrimshaw (2013-07-28): Initial version
class sage.combinat.descent_algebra.DescentAlgebra(R, n)

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

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]
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()

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]
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]
class D(alg, prefix='D')

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,
B[1, 3] - B,
B[2, 2] - B,
B[3, 1] - B,
B[1, 1, 2] - B[1, 3] - B[2, 2] + B,
B[1, 2, 1] - B[1, 3] - B[3, 1] + B,
B[2, 1, 1] - B[2, 2] - B[3, 1] + B,
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]
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')

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]
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
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]
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)

The category of bases of a descent algebra.

class ElementMethods
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
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]