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Journal of Algebraic Combinatorics

, Volume 46, Issue 1, pp 51–75 | Cite as

Matroidal Schur algebras

  • Tom Braden
  • Carl MautnerEmail author
Article

Abstract

Motivated by a geometric description of the Schur algebra due to the second author, we define for any matroid M and principal ideal domain k, a quasi-hereditary algebra R(M) defined over k which we call a matroidal Schur algebra. We show that the Ringel dual of R(M) is the matroidal Schur algebra \(R(M^*)\) associated with the dual matroid \(M^*\). When k is a field of characteristic zero, the algebra R(M) can be seen as a categorification of a formula due to Kook–Reiner–Stanton and related work of Denham. More generally, the behavior of R(M) in positive characteristic is closely related to determinant formulas of Schechtman–Varchenko and Brylawski–Varchenko.

Keywords

Matroid Quasi-hereditary algebra Ringel duality 

Mathematics Subject Classification

52B40 16G99 

Notes

Acknowledgements

We are grateful to Ben Webster and Geordie Williamson for pointing out to us the connection to the papers [8, 19], which they had found while doing related computations, and to the referee for his or her comments. We would also like to thank the MPIM in Bonn for excellent working conditions. The material in this article is also partly based upon work supported by the NSF under Grant No. 0932078000, while the second author was in residence at MSRI in Berkeley, California, during the Fall 2014 semester.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA

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