\(WP\)-matroids and thin Schubert cells on Tits systems
\(WP\)-matroids are combinatorial structures, vastly generalizing matroids, that were introduced by \textit{I. M. Gel'fand} and \textit{V. V. Serganova} [Combinatorial geometries and torus strata on homogeneous compact manifolds, Russ. Math. Surv. 42, No. 2, 133-168 (1987); translation from Usp. Mat. Nauk 42, No. 2, 107-134 (1987; Zbl 0629.14035)]: if \(G\) is a semisimple Lie group with Weyl group \(W\), and if \(G_ P\) is a parabolic subgroup with corresponding parabolic \(P \subset W\), then the Schubert cells of \(G/G_ P\) are indexed by the set \(W/P\) of left cosets. The Schubert decomposition is, however, not invariant under \(W\). The \(W\)- invariant stratification of \(G/G_ P\) into thin Schubert cells is obtained as the common refinement of all \(W\)-images of the Schubert decomposition. \(WP\)-matroids are defined in such a way that very thin Schubert cell has a canonically associated \(WP\)-matroid. (Ordinary matroids are obtained in the special case where \(G\) is a special linear group, \(W\) is a symmetric group, and \(P\) is a maximal parabolic subgroup. Already in this case the thin Schubert cells, corresponding to realization spaces of matroids, can be arbitrarily complicated real semialgebraic sets.) The paper under review presents a new, equivalent definition of \(WP\)- matroids that is considerably simpler, and also works in situations where the Coxeter group \(W\) is infinite. The main result associates \(WP\)- matroids with the canonical retraction map \(\rho_{w,W}\) of an arbitrary thick building of type \(W\) to an appartment. It is asked under what conditions \(WP\)-matroids have a ``geometric realization,'' that is, correspond to non-empty thin Schubert cells. A further generalization to the (very general) situation of ``thin chamber systems'' (e.g., barycentric subdivisions of triangulated pseudomanifolds) is also given.
scientific article; zbMATH DE number 611213