Positroids and Schubert matroids. (English) Zbl 1231.05061

Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. In this paper author proves his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a nice combinatorial description of positroids that is easily computable.


05B35 Combinatorial aspects of matroids and geometric lattices
14M15 Grassmannians, Schubert varieties, flag manifolds
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[1] Bonin, J.; de Mier, A.; Noy, M., Lattice path matroids: enumerative aspects and Tutte polynomials, J. Combin. Theory Ser. A, 104, 63-94 (2003) · Zbl 1031.05031
[2] Fulton, W., Young Tableaux. With Applications to Representation Theory and Geometry (1997), Cambridge University Press: Cambridge University Press New York · Zbl 0878.14034
[3] Knutson, A.; Lam, T.; Speyer, D., Positroid varieties I: juggling and geometry · Zbl 1330.14086
[4] Lusztig, G., Introduction to total positivity, (Hilgert, J.; Lawson, J. D.; Neeb, K. H.; Vinberg, E. B., Positivity in Lie Theory: Open Problems (1998), de Gruyter: de Gruyter Berlin), 133-145 · Zbl 0929.20035
[5] Postnikov, A., Total positivity, Grassmannians, and networks
[6] K. Rietsch, Total positivity and real flag varieties, PhD dissertation, MIT, 1998.; K. Rietsch, Total positivity and real flag varieties, PhD dissertation, MIT, 1998. · Zbl 1059.14068
[7] Talaska, K., Combinatorial formulas for Le-coordinates on a totally nonnegative Grassmannian, J. Combin. Theory Ser. A, 118, 1, 58-66 (2011) · Zbl 1232.05047
[8] Yakimov, M., Cyclicity of Lusztigʼs stratification of grassmannians and Poisson geometry, (Caenepeel, S.; Fuchs, J.; Gutt, S.; Schweigert, Ch.; Stolin, A.; van Oystaeyen, F., Noncommutative Structures in Mathematics and Physics (2010), Royal Flemish Academy of Belgium for Sciences and Arts), 258-262
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