# Orlik-Solomon Algebras¶

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

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)}