No CrossRef data available.
Article contents
Positroid varieties: juggling and geometry
Published online by Cambridge University Press: 19 August 2013
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2013
References
Anderson, D., Griffeth, S. and Miller, E., Positivity and Kleiman transversality in equivariant 
$K$-theory of homogeneous spaces, J. Eur. Math. Soc. 13 (2011), 57–84.CrossRefGoogle Scholar
$K$-theory of homogeneous spaces, J. Eur. Math. Soc. 13 (2011), 57–84.CrossRefGoogle ScholarBergeron, N. and Sottile, F., Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), 373–423.CrossRefGoogle Scholar
Billey, S. and Coşkun, I., Singularities of generalized Richardson varieties, Comm. Algebra 40 (2012), 1466–1495.CrossRefGoogle Scholar
Björner, A. and Brenti, F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231 (Springer, New York, 2005).Google Scholar
Björner, A. and Wachs, M., Bruhat order of Coxeter groups and shellability, Adv. Math. 43 (1982), 87–100.Google Scholar
Bonin, J. and de Mier, A., Lattice path matroids: structural properties, European J. Combin. 27 (2006), 701–738.Google Scholar
Borovik, A., Gelfand, I. and White, N., Coxeter matroids, Progress in Mathematics, vol. 216 (Birkhäuser Boston, Inc., Boston, MA, 2003).CrossRefGoogle Scholar
Brion, M., Positivity in the Grothendieck group of complex flag varieties, J. Algebra (special volume in honor of Claudio Procesi) 258 (2002), 137–159.Google Scholar
Brown, K., Goodearl, K. and Yakimov, M., Poisson structures on affine spaces and flag varieties I. Matrix affine Poisson space, Adv. Math. 206 (2006), 567–629.Google Scholar
Brylawski, T., Appendix of matroid cryptomorphisms, in Theory of matroids, Encyclopedia of Mathematics and its Applications, vol. 26, ed. White, N. (Cambridge University Press, 1986), 298–312.CrossRefGoogle Scholar
Buch, A. S., A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37–78.Google Scholar
Buch, A. S., Kresch, A. and Tamvakis, H., Gromov–Witten invariants on Grassmannians, J. Amer. Math. Soc. 16 (2003), 901–915.Google Scholar
Buch, A. and Mihalcea, L., Quantum K-theory of Grassmannians, Duke Math. J. 156 (2011), 501–538.CrossRefGoogle Scholar
Buhler, J., Eisenbud, D., Graham, R. and Wright, C., Juggling drops and descents, Amer. Math. Monthly 101 (1994), 507–519.CrossRefGoogle Scholar
Chung, F. and Graham, R., Universal juggling cycles, in Combinatorial number theory, eds Landman, B. et al. (Walter de Gruyter, Berlin, 2007), 121–130.Google Scholar
Chung, F. and Graham, R., Primitive juggling sequences, Amer. Math. Monthly 115 (2008), 185–194.CrossRefGoogle Scholar
Dyer, M., Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993), 91–115.Google Scholar
Ehrenborg, R. and Readdy, M., Juggling and applications to $q$-analogues, Discrete Math. 157 (1996), 107–125.CrossRefGoogle Scholar
$q$-analogues, Discrete Math. 157 (1996), 107–125.CrossRefGoogle ScholarFomin, S. and Zelevinsky, A., Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335–380.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., Recognizing Schubert cells, J. Algebraic Combin. 12 (2000), 37–57.Google Scholar
Fulton, W., Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), 381–420.CrossRefGoogle Scholar
Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, in Algebraic geometry, Proceedings of Symposia in Applied Mathematics, vol. 62, part 2 (American Mathematical Society, Providence, RI, 1997), 45–96.Google Scholar
Fulton, W. and Woodward, C., On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), 641–661.CrossRefGoogle Scholar
Gel’fand, I., Goresky, M., MacPherson, R. and Serganova, V., Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math. 63 (1987), 301–316.CrossRefGoogle Scholar
Goodearl, K. and Yakimov, M., Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), 5753–5780.CrossRefGoogle Scholar
He, X. and Lam, T., Projected Richardson varieties and affine Schubert varieties, Preprint (2011), arXiv:1106.2586 [math.AG].Google Scholar
Knutson, A., The siteswap FAQ, (1993), http://www.juggling.org/help/siteswap/faq.html.Google Scholar
Knutson, A., Frobenius splitting and Möbius inversion, Preprint (2009), arXiv:0902.1930 [math.AG].Google Scholar
Knutson, A., Lam, T. and Speyer, D., Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG].Google Scholar
Knutson, A. and Tao, T., Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), 221–260.CrossRefGoogle Scholar
Kostant, B. and Kumar, S., The nil Hecke ring and cohomology of $G/ P$ for a Kac-Moody group $G$, Adv. in Math. 62 (1986), 187–237.Google Scholar
$G/ P$ for a Kac-Moody group 
$G$, Adv. in Math. 62 (1986), 187–237.Google ScholarLam, T., Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc. 21 (2008), 259–281.CrossRefGoogle Scholar
Lam, T., Lapointe, L., Morse, J. and Shimozono, M., Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc. 208 (2010).Google Scholar
Lam, T., Schilling, A. and Shimozono, M., $K$-theory Schubert calculus of the affine Grassmannian, Compositio Math. 146 (2010), 811–852.CrossRefGoogle Scholar
$K$-theory Schubert calculus of the affine Grassmannian, Compositio Math. 146 (2010), 811–852.CrossRefGoogle ScholarLascoux, A. and Schützenberger, M., Symmetry and flag manifolds, in Invariant theory, Montecatini, 1982, Lecture Notes in Mathematics, vol. 996 (Springer, 1983), 118–144.Google Scholar
Launois, S., Lenagan, T. and Rigal, L., Prime ideals in the quantum Grassmannian, Selecta Math. 13 (2008), 697–725.Google Scholar
Lusztig, G., Total positivity for partial flag varieties, Represent. Theory 2 (1998), 70–78.Google Scholar
Magyar, P., Notes on Schubert classes of a loop group, Preprint (2007), arXiv:0705.3826 [math.RT].Google Scholar
Marsh, R. and Rietsch, K., Parametrizations of flag varieties, Represent. Theory 8 (2004), 212–242.CrossRefGoogle Scholar
Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005).Google Scholar
Mnëv, N. E., The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in Topology and geometry—Rohlin seminar, Lecture Notes in Mathematics, vol. 1346 (Springer, Berlin, 1988), 527–543.Google Scholar
Oh, S., Positroids and Schubert matroids, J. Combin. Theory Ser. A 118 (2011), 2426–2435.Google Scholar
Postnikov, A., Affine approach to quantum Schubert calculus, Duke Math. J. 128 (2005), 473–509.CrossRefGoogle Scholar
Postnikov, A., Total positivity, Grassmannians, and networks, Preprint (2005). http://www-math.mit.edu/~apost/papers/tpgrass.pdf.Google Scholar
Pressley, A. and Segal, G., Loop groups, Oxford Mathematical Monographs (Oxford University Press, 1986).Google Scholar
Ramanathan, A., Equations defining Schubert varieties and Frobenius splitting of diagonals, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 61–90.CrossRefGoogle Scholar
Rietsch, K., Closure relations for totally nonnegative cells in $G/ P$, Math. Res. Lett. 13 (2006), 775–786.Google Scholar
$G/ P$, Math. Res. Lett. 13 (2006), 775–786.Google ScholarRietsch, K. and Williams, L., The totally nonnegative part of $G/ P$ is a CW complex, Transform. Groups 13 (2008), 839–853.Google Scholar
$G/ P$ is a CW complex, Transform. Groups 13 (2008), 839–853.Google ScholarSnider, M., Affine patches on positroid varieties and affine pipe dreams, PhD dissertation, Cornell University (2010).Google Scholar
Stanley, R., On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), 359–372.Google Scholar
Stanley, R., Enumerative combinatorics Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999).Google Scholar
Vámos, P., The missing axiom of matroid theory is lost forever, J. Lond. Math. Soc. (2) 18 (1978), 403–408.Google Scholar
Verma, D.-N., Möbius inversion for the Bruhat order on a Weyl group, Ann. Sci. Éc. Norm. Super. 4 (1971), 393–398.Google Scholar
Williams, L., Shelling totally nonnegative flag varieties, J. Reine Angew. Math. (Crelle’s Journal) 609 (2007), 1–22.Google Scholar
Yakimov, M., Cyclicity of Lusztig’s stratification of Grassmannians and Poisson geometry, in Noncommutative structures in mathematics and physics, eds Caenepeel, S. et al. (Royal Flemish Academy of Belgium for Sciences and Arts, 2010), 258–262.Google Scholar
Yakimov, M., A classification of $H$-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), 1249–1261.Google Scholar
$H$-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), 1249–1261.Google Scholar
You have
Access
- 107
- Cited by