Spanning subspace configurations

Abstract

A spanning configuration in the complex vector space \({{\mathbb {C}}}^k\) is a sequence \((W_1, \dots , W_r)\) of linear subspaces of \({{\mathbb {C}}}^k\) such that \(W_1 + \cdots + W_r = {{\mathbb {C}}}^k\). We present the integral cohomology of the moduli space of spanning configurations in \({{\mathbb {C}}}^k\) corresponding to a given sequence of subspace dimensions. This simultaneously generalizes the classical presentation of the cohomology of partial flag varieties and the more recent presentation of a variety of spanning line configurations defined by the author and Pawlowski. This latter variety of spanning line configurations plays the role of the flag variety for the Haglund–Remmel–Wilson Delta Conjecture of symmetric function theory.

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Acknowledgements

The author is grateful to Sara Billey, Brendan Pawlowski, Vic Reiner, and Andy Wilson for many helpful conversations. The author thanks François Bergeron for asking how to generalize \(X_{n,k}\) to higher-dimensional subspaces. The author also thanks the anonymous referees for helpful comments which improved the exposition of this paper. The author was partially supported by NSF Grants DMS-1500838 and DMS-1953781.

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Correspondence to Brendon Rhoades.

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Rhoades, B. Spanning subspace configurations. Sel. Math. New Ser. 27, 8 (2021). https://doi.org/10.1007/s00029-021-00617-6

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Keywords

  • Subspace configuration
  • Cohomology
  • Schubert calculus
  • Affine paving
  • Ordered set cover

Mathematics Subject Classification

  • 05E99
  • 14M15