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)}
class sage.algebras.orlik_solomon.OrlikSolomonInvariantAlgebra(R, M, G, action_on_groundset=None, *args, **kwargs)

Bases: sage.modules.with_basis.invariant.FiniteDimensionalInvariantModule

The invariant algebra of the Orlik-Solomon algebra from the action on \(A(M)\) induced from the action_on_groundset.

INPUT:

  • R – the ring of coefficients

  • M – a matroid

  • G – a semigroup

  • action_on_groundset – (optional) a function defining the action of G on the elements of the groundset of M; default is g(x)

EXAMPLES:

Lets start with the action of \(S_3\) on the rank \(2\) braid matroid:

sage: M = matroids.CompleteGraphic(3)
sage: M.groundset()
frozenset({0, 1, 2})
sage: G = SymmetricGroup(3)

Calling elements g of G on an element \(i\) of \(\{1, 2, 3\}\) defines the action we want, but since the groundset is \(\{0, 1, 2\}\) we first add \(1\) and then subtract \(1\):

sage: def on_groundset(g, x):
....:     return g(x+1) - 1

Now that we have defined an action we can create the invariant, and get its basis:

sage: OSG = M.orlik_solomon_algebra(QQ, invariant=(G, on_groundset))
sage: OSG.basis()
Finite family {0: B[0], 1: B[1]}
sage: [OSG.lift(b) for b in OSG.basis()]
[OS{}, OS{0} + OS{1} + OS{2}]

Since it is invariant, the action of any g in G is trivial:

sage: x = OSG.an_element(); x
2*B[0] + 2*B[1]
sage: g = G.an_element(); g
(2,3)
sage: g * x
2*B[0] + 2*B[1]

sage: x = OSG.random_element()
sage: g = G.random_element()
sage: g * x == x
True

The underlying ambient module is the Orlik-Solomon algebra, which is accessible via ambient():

sage: M.orlik_solomon_algebra(QQ) is OSG.ambient()
True

There is not much structure here, so lets look at a bigger example. Here we will look at the rank \(3\) braid matroid, and to make things easier, we’ll start the indexing at \(1\) so that the \(S_6\) action on the groundset is simply calling \(g\):

sage: M = matroids.CompleteGraphic(4); M.groundset()
frozenset({0, 1, 2, 3, 4, 5})
sage: new_bases = [frozenset(i+1 for i in j) for j in M.bases()]
sage: M = Matroid(bases=new_bases); M.groundset()
frozenset({1, 2, 3, 4, 5, 6})
sage: G = SymmetricGroup(6)
sage: OSG = M.orlik_solomon_algebra(QQ, invariant=G)
sage: OSG.basis()
Finite family {0: B[0], 1: B[1]}
sage: [OSG.lift(b) for b in OSG.basis()]
[OS{}, OS{1} + OS{2} + OS{3} + OS{4} + OS{5} + OS{6}]
sage: (OSG.basis()[1])^2
0
sage: 5 * OSG.basis()[1]
5*B[1]

Next, we look at the same matroid but with an \(S_3 \times S_3\) action (here realized as a Young subgroup of \(S_6\)):

sage: H = G.young_subgroup([3, 3])
sage: OSH = M.orlik_solomon_algebra(QQ, invariant=H)
sage: OSH.basis()
Finite family {0: B[0], 1: B[1], 2: B[2]}
sage: [OSH.lift(b) for b in OSH.basis()]
[OS{}, OS{4} + OS{5} + OS{6}, OS{1} + OS{2} + OS{3}]

We implement an \(S_4\) action on the vertices:

sage: M = matroids.CompleteGraphic(4)
sage: G = SymmetricGroup(4)
sage: edge_map = {i: M.groundset_to_edges([i])[0][:2]
....:             for i in M.groundset()}
sage: inv_map = {v: k for k, v in edge_map.items()}
sage: def vert_action(g, x):
....:     a, b = edge_map[x]
....:     return inv_map[tuple(sorted([g(a+1)-1, g(b+1)-1]))]
sage: OSG = M.orlik_solomon_algebra(QQ, invariant=(G, vert_action))
sage: B = OSG.basis()
sage: [OSG.lift(b) for b in B]
[OS{}, OS{0} + OS{1} + OS{2} + OS{3} + OS{4} + OS{5}]

We use this to describe the Young subgroup \(S_2 \times S_2\) action:

sage: H = G.young_subgroup([2,2])
sage: OSH = M.orlik_solomon_algebra(QQ, invariant=(H, vert_action))
sage: B = OSH.basis()
sage: [OSH.lift(b) for b in B]
[OS{}, OS{5}, OS{1} + OS{2} + OS{3} + OS{4}, OS{0},
 -1/2*OS{1, 2} + OS{1, 5} - 1/2*OS{3, 4} + OS{3, 5},
 OS{0, 5}, OS{0, 1} + OS{0, 2} + OS{0, 3} + OS{0, 4},
 -1/2*OS{0, 1, 2} + OS{0, 1, 5} - 1/2*OS{0, 3, 4} + OS{0, 3, 5}]

We demonstrate the algebra structure:

sage: matrix([[b*bp for b in B] for bp in B])
[   B[0]    B[1]    B[2]    B[3]    B[4]    B[5]    B[6]    B[7]]
[   B[1]       0  2*B[4]    B[5]       0       0  2*B[7]       0]
[   B[2] -2*B[4]       0    B[6]       0 -2*B[7]       0       0]
[   B[3]   -B[5]   -B[6]       0    B[7]       0       0       0]
[   B[4]       0       0    B[7]       0       0       0       0]
[   B[5]       0 -2*B[7]       0       0       0       0       0]
[   B[6]  2*B[7]       0       0       0       0       0       0]
[   B[7]       0       0       0       0       0       0       0]

Note

The algebra structure only exists when the action on the groundset yeilds an equivariant matroid, in the sense that \(g \cdot I \in \mathcal{I}\) for every \(g \in G\) and for every \(I \in \mathcal{I}\).