Abstract
For any finite poset P, we introduce a homogeneous space as a quotient of the general linear group. When P is a chain this quotient is a complete flag variety. Moreover, we provide partitions for any set in a projective space, induced by the action of incidence groups of posets. Our general framework allows to deal with several combinatorial and geometric objects, unifying and extending different structures such as Bruhat orders, parking functions and weak orders on matroids. We introduce the notion of P-flag matroid, extending flag matroids.
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Notes
More precisely, the set of bases of a matroid.
The use of the word cell in this article does not refer in general to affine spaces.
We omit parentheses when writing the elements of \([n]_<\) and \([n]^2_<\).
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Acknowledgements
The first author was partially supported by Swiss National Science Foundation Professorship Grant PP00P2_179110/1 of Prof. Emanuele Delucchi. He is grateful to the town of Zagarolo, where this paper started and finished, for the hospitality received there.
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Bolognini, D., Sentinelli, P. P-flag spaces and incidence stratifications. Sel. Math. New Ser. 27, 72 (2021). https://doi.org/10.1007/s00029-021-00685-8
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DOI: https://doi.org/10.1007/s00029-021-00685-8