Möbius Algebras

sage.combinat.posets.moebius_algebra.BasisAbstract

Abstract base class for a basis.

sage.combinat.posets.moebius_algebra.MoebiusAlgebra

The Möbius algebra of a lattice.

Let \(L\) be a lattice. The Möbius algebra \(M_L\) was originally constructed by Solomon [Solomon67] and has a natural basis \(\{ E_x \mid x \in L \}\) with multiplication given by \(E_x \cdot E_y = E_{x \vee y}\). Moreover this has a basis given by orthogonal idempotents \(\{ I_x \mid x \in L \}\) (so \(I_x I_y = \delta_{xy} I_x\) where \(\delta\) is the Kronecker delta) related to the natural basis by

\[I_x = \sum_{x \leq y} \mu_L(x, y) E_y,\]

where \(\mu_L\) is the Möbius function of \(L\).

Note

We use the join \(\vee\) for our multiplication, whereas [Greene73] and [Etienne98] define the Möbius algebra using the meet \(\wedge\). This is done for compatibility with QuantumMoebiusAlgebra.

REFERENCES:

[Solomon67]Louis Solomon. The Burnside Algebra of a Finite Group. Journal of Combinatorial Theory, 2, 1967. doi:10.1016/S0021-9800(67)80064-4.
[Greene73]Curtis Greene. On the Möbius algebra of a partially ordered set. Advances in Mathematics, 10, 1973. doi:10.1016/0001-8708(73)90106-0.
[Etienne98]Gwihen Etienne. On the Möbius algebra of geometric lattices. European Journal of Combinatorics, 19, 1998. doi:10.1006/eujc.1998.0227.
sage.combinat.posets.moebius_algebra.MoebiusAlgebraBases

The category of bases of a Möbius algebra.

INPUT:

  • base – a Möbius algebra
sage.combinat.posets.moebius_algebra.QuantumMoebiusAlgebra

The quantum Möbius algebra of a lattice.

Let \(L\) be a lattice, and we define the quantum Möbius algebra \(M_L(q)\) as the algebra with basis \(\{ E_x \mid x \in L \}\) with multiplication given by

\[E_x E_y = \sum_{z \geq a \geq x \vee y} \mu_L(a, z) q^{\operatorname{crk} a} E_z,\]

where \(\mu_L\) is the Möbius function of \(L\) and \(\operatorname{crk}\) is the corank function (i.e., \(\operatorname{crk} a = \operatorname{rank} L - \operatorname{rank}\) a). At \(q = 1\), this reduces to the multiplication formula originally given by Solomon.