Orlik-Solomon Algebras

class sage.algebras.orlik_solomon.OrlikSolomonAlgebra(R, M, ordering=None)

Bases: sage.combinat.free_module.CombinatorialFreeModule

An Orlik-Solomon algebra.

Let \(R\) be a commutative ring. Let \(M\) be a matroid with ground set \(X\). Let \(C(M)\) denote the set of circuits of \(M\). Let \(E\) denote the exterior algebra over \(R\) generated by \(\{ e_x \mid x \in X \}\). The Orlik-Solomon ideal \(J(M)\) is the ideal of \(E\) generated by

\[\partial e_S := \sum_{i=1}^t (-1)^{i-1} e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge \widehat{e}_{j_i} \wedge \cdots \wedge e_{j_t}\]

for all \(S = \left\{ j_1 < j_2 < \cdots < j_t \right\} \in C(M)\), where \(\widehat{e}_{j_i}\) means that the term \(e_{j_i}\) is being omitted. The notation \(\partial e_S\) is not a coincidence, as \(\partial e_S\) is actually the image of \(e_S := e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge e_{j_t}\) under the unique derivation \(\partial\) of \(E\) which sends all \(e_x\) to \(1\).

It is easy to see that \(\partial e_S \in J(M)\) not only for circuits \(S\), but also for any dependent set \(S\) of \(M\). Moreover, every dependent set \(S\) of \(M\) satisfies \(e_S \in J(M)\).

The Orlik-Solomon algebra \(A(M)\) is the quotient \(E / J(M)\). This is a graded finite-dimensional skew-commutative \(R\)-algebra. Fix some ordering on \(X\); then, the NBC sets of \(M\) (that is, the subsets of \(X\) containing no broken circuit of \(M\)) form a basis of \(A(M)\). (Here, a broken circuit of \(M\) is defined to be the result of removing the smallest element from a circuit of \(M\).)

In the current implementation, the basis of \(A(M)\) is indexed by the NBC sets, which are implemented as frozensets.

INPUT:

  • R – the base ring

  • M – the defining matroid

  • ordering – (optional) an ordering of the ground set

EXAMPLES:

We create the Orlik-Solomon algebra of the uniform matroid \(U(3, 4)\) and do some basic computations:

sage: M = matroids.Uniform(3, 4)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.dimension()
14
sage: G = OS.algebra_generators()
sage: M.broken_circuits()
frozenset({frozenset({1, 2, 3})})
sage: G[1] * G[2] * G[3]
OS{0, 1, 2} - OS{0, 1, 3} + OS{0, 2, 3}

REFERENCES:

algebra_generators()

Return the algebra generators of self.

These form a family indexed by the ground set \(X\) of \(M\). For each \(x \in X\), the \(x\)-th element is \(e_x\).

EXAMPLES:

sage: M = matroids.Uniform(2, 2)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.algebra_generators()
Finite family {0: OS{0}, 1: OS{1}}

sage: M = matroids.Uniform(1, 2)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.algebra_generators()
Finite family {0: OS{0}, 1: OS{0}}

sage: M = matroids.Uniform(1, 3)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.algebra_generators()
Finite family {0: OS{0}, 1: OS{0}, 2: OS{0}}
degree_on_basis(m)

Return the degree of the basis element indexed by m.

EXAMPLES:

sage: M = matroids.Wheel(3)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.degree_on_basis(frozenset([1]))
1
sage: OS.degree_on_basis(frozenset([0, 2, 3]))
3
one_basis()

Return the index of the basis element corresponding to \(1\) in self.

EXAMPLES:

sage: M = matroids.Wheel(3)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.one_basis() == frozenset([])
True
product_on_basis(a, b)

Return the product in self of the basis elements indexed by a and b.

EXAMPLES:

sage: M = matroids.Wheel(3)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS.product_on_basis(frozenset([2]), frozenset([3,4]))
OS{0, 1, 2} - OS{0, 1, 4} + OS{0, 2, 3} + OS{0, 3, 4}

sage: G = OS.algebra_generators()
sage: prod(G)
0
sage: G[2] * G[4]
-OS{1, 2} + OS{1, 4}
sage: G[3] * G[4] * G[2]
OS{0, 1, 2} - OS{0, 1, 4} + OS{0, 2, 3} + OS{0, 3, 4}
sage: G[2] * G[3] * G[4]
OS{0, 1, 2} - OS{0, 1, 4} + OS{0, 2, 3} + OS{0, 3, 4}
sage: G[3] * G[2] * G[4]
-OS{0, 1, 2} + OS{0, 1, 4} - OS{0, 2, 3} - OS{0, 3, 4}
subset_image(S)

Return the element \(e_S\) of \(A(M)\) (== self) corresponding to a subset \(S\) of the ground set of \(M\).

INPUT:

  • S – a frozenset which is a subset of the ground set of \(M\)

EXAMPLES:

sage: M = matroids.Wheel(3)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: BC = sorted(M.broken_circuits(), key=sorted)
sage: for bc in BC: (sorted(bc), OS.subset_image(bc))
([1, 3], -OS{0, 1} + OS{0, 3})
([1, 4, 5], OS{0, 1, 4} - OS{0, 1, 5} - OS{0, 3, 4} + OS{0, 3, 5})
([2, 3, 4], OS{0, 1, 2} - OS{0, 1, 4} + OS{0, 2, 3} + OS{0, 3, 4})
([2, 3, 5], OS{0, 2, 3} + OS{0, 3, 5})
([2, 4], -OS{1, 2} + OS{1, 4})
([2, 5], -OS{0, 2} + OS{0, 5})
([4, 5], -OS{3, 4} + OS{3, 5})

sage: M4 = matroids.CompleteGraphic(4)
sage: OS = M4.orlik_solomon_algebra(QQ)
sage: OS.subset_image(frozenset({2,3,4}))
OS{0, 2, 3} + OS{0, 3, 4}

An example of a custom ordering:

sage: G = Graph([[3, 4], [4, 1], [1, 2], [2, 3], [3, 5], [5, 6], [6, 3]])
sage: M = Matroid(G)
sage: s = [(5, 6), (1, 2), (3, 5), (2, 3), (1, 4), (3, 6), (3, 4)]
sage: sorted([sorted(c) for c in M.circuits()])
[[(1, 2), (1, 4), (2, 3), (3, 4)],
 [(3, 5), (3, 6), (5, 6)]]
sage: OS = M.orlik_solomon_algebra(QQ, ordering=s)
sage: OS.subset_image(frozenset([]))
OS{}
sage: OS.subset_image(frozenset([(1,2),(3,4),(1,4),(2,3)]))
0
sage: OS.subset_image(frozenset([(2,3),(1,2),(3,4)]))
OS{(1, 2), (2, 3), (3, 4)}
sage: OS.subset_image(frozenset([(1,4),(3,4),(2,3),(3,6),(5,6)]))
-OS{(1, 2), (1, 4), (2, 3), (3, 6), (5, 6)}
 + OS{(1, 2), (1, 4), (3, 4), (3, 6), (5, 6)}
 - OS{(1, 2), (2, 3), (3, 4), (3, 6), (5, 6)}
sage: OS.subset_image(frozenset([(1,4),(3,4),(2,3),(3,6),(3,5)]))
OS{(1, 2), (1, 4), (2, 3), (3, 5), (5, 6)}
 - OS{(1, 2), (1, 4), (2, 3), (3, 6), (5, 6)}
 + OS{(1, 2), (1, 4), (3, 4), (3, 5), (5, 6)}
 + OS{(1, 2), (1, 4), (3, 4), (3, 6), (5, 6)}
 - OS{(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}
 - OS{(1, 2), (2, 3), (3, 4), (3, 6), (5, 6)}