A Möbius identity arising from modularity in a matroid bilinear form. (English) Zbl 0966.05014
Summary: The matrix for the bilinear form of the flag space of a matroid has (with respect to an appropriate basis) a tensor product structure when the matroid has a modular flat \(K\). When determinants are taken, an identity is obtained for the rho function (a certain product of the Möbius and beta functions) summed over flats with a fixed intersection with \(K\). When the identity is interpreted for Dowling lattices and finite projective spaces, identities with similar combinatorial proofs are obtained for binomial and Gaussian coefficients, respectively.
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
| 06C10 | Semimodular lattices, geometric lattices |
| 05A30 | \(q\)-calculus and related topics |
Keywords:
bilinear form; flag space; matroid; Möbius and beta functions; Dowling lattices; projective spaces; Gaussian coefficientsReferences:
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