Product structure and regularity theorem for totally nonnegative flag varieties. (English) Zbl 07870143
Summary: The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) \(J\)-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the \(J\)-totally nonnegative flag variety has a cellular decomposition into totally positive \(J\)-Richardson varieties. Moreover, each totally positive \(J\)-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive \(J\)-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the \(J\)-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614-620, 2019). We also show that the link of the totally nonnegative part of \(U^-\) for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57-114, 2014) for reductive groups.
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 20G44 | Kac-Moody groups |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
References:
| [1] | Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.; Goncharov, A.; Postnikov, A.; Trnka, J., Grassmannian Geometry of Scattering Amplitudes, 2016, Cambridge: Cambridge University Press, Cambridge · Zbl 1365.81004 |
| [2] | Bao, H.; He, X., Flag manifolds over semifields, Algebra Number Theory, 15, 8, 2037-2069, 2021 · Zbl 1490.14084 |
| [3] | Bao, H.; He, X., A Birkhoff-Bruhat atlas for partial flag varieties, Indag. Math., 32, 1152-1173, 2021 · Zbl 1471.14102 |
| [4] | Billig, Y.; Dyer, M. J., Decompositions of Bruhat type for Kac-Moody groups, Nova J. Algebra Geom., 3, 1, 11-39, 1994 · Zbl 0874.22020 |
| [5] | Bjorner, A., Posets, regular CW complexes and Bruhat order, Eur. J. Comb., 5, 7-16, 1984 · Zbl 0538.06001 |
| [6] | Bochnak, J.; Coste, M.; Roy, M.-F., Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), 1998, Berlin: Springer, Berlin · Zbl 0633.14016 |
| [7] | Dyer, M. J., Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders, Kazhdan-Lusztig Theory and Related Topics, 141-165, 1992, Providence: Am. Math. Soc., Providence · Zbl 0833.20048 |
| [8] | Fomin, S.; Shapiro, M., Stratified spaces formed by totally positive varieties, Mich. Math. J., 48, 253-270, 2000 · Zbl 1009.20056 |
| [9] | Fomin, S.; Zelevinksy, A., Cluster algebras. I. Foundations, J. Am. Math. Soc., 15, 2, 497-529, 2002 · Zbl 1021.16017 |
| [10] | Freedman, M. H., The topology of four-dimensional manifolds, J. Differ. Geom., 17, 357-453, 1982 · Zbl 0528.57011 |
| [11] | Galashin, P.; Karp, S.; Lam, T., The totally nonnegative part of \(G/P\) is a ball, Adv. Math., 351, 614-620, 2019 · Zbl 1440.14224 |
| [12] | Galashin, P.; Karp, S.; Lam, T., Regularity theorem for totally nonnegative flag varieties, J. Am. Math. Soc., 35, 2, 513-579, 2021 · Zbl 1493.14074 |
| [13] | Galashin, P.; Karp, S.; Lam, T., The totally nonnegative Grassmannian is a ball, Adv. Math., 397, 2022 · Zbl 1482.05355 |
| [14] | He, X., A subalgebra of 0-Hecke algebra, J. Algebra, 322, 4030-4039, 2009 · Zbl 1186.20005 |
| [15] | Hersh, P., Regular cell complexes in total positivity, Invent. Math., 197, 1, 57-114, 2014 · Zbl 1339.20041 |
| [16] | Kac, V. G., Infinite-Dimensional Lie Algebras, 1990, Cambridge: Cambridge University Press, Cambridge · Zbl 0716.17022 |
| [17] | Kazhdan, D.; Lusztig, G., Representation of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184, 1979 · Zbl 0499.20035 |
| [18] | Knutson, A.; Woo, A.; Yong, A., Singularities of Richardson varieties, Math. Res. Lett., 20, 2, 391-400, 2013 · Zbl 1298.14053 |
| [19] | Knutson, A.; Lam, T.; Speyer, D. E., Projections of Richardson varieties, J. Reine Angew. Math., 687, 133-157, 2014 · Zbl 1345.14047 |
| [20] | Kumar, S., Kac-Moody Groups, Their Flag Varieties and Representation Theory, 2002, Boston: Birkhäuser, Boston · Zbl 1026.17030 |
| [21] | Kumar, S., Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups (with an appendix by M. Kashiwara), J. Eur. Math. Soc., 19, 2469-2519, 2017 · Zbl 1372.19005 |
| [22] | Lam, T., Totally nonnegative Grassmannian and Grassmann polytopes, Current Developments in Mathematics 2014, 51-152, 2016, Somerville: International Press, Somerville · Zbl 1506.14103 |
| [23] | Lusztig, G., Total positivity in reductive groups, Lie Theory and Geometry, 531-568, 1994, Boston: Birkhäuser, Boston · Zbl 0845.20034 |
| [24] | Lusztig, G., Total positivity in partial flag manifolds, Represent. Theory, 2, 70-78, 1998 · Zbl 0895.14014 |
| [25] | Lusztig, G., Introduction to Quantum Groups, Modern Birkhäuser Classics, Reprint of Theition, 2010, Boston: Birkhäuser, Boston |
| [26] | Lusztig, G., Introduction to Total Positivity, Positivity in Lie Theory: Open Problems, 133-146, 2011, Berlin: de Gruyter, Berlin · Zbl 0929.20035 |
| [27] | Lusztig, G., Total positivity in reductive groups, II, Bull. Inst. Math. Acad. Sin. (N.S.), 14, 403-460, 2019 · Zbl 1522.20199 |
| [28] | Lusztig, G., Positive structures in Lie theory, ICCM Not., 8, 1, 50-54, 2020 · Zbl 1481.22012 |
| [29] | Lusztig, G., The flag manifold over the semifield ℤ, Bull. Inst. Math. Acad. Sin. (N.S.), 15, 1, 63-92, 2020 · Zbl 1481.20181 |
| [30] | Lusztig, G., Partial flag manifolds over a semifield, Represent. Theory, 24, 397-402, 2020 · Zbl 1530.20165 |
| [31] | Lusztig, G., Total positivity in Springer fibres, Q. J. Math., 72, 1-2, 31-49, 2021 · Zbl 1475.20080 |
| [32] | Marquis, T., An Introduction to Kac-Moody Groups over Fields, 2018, Zurich: Eur. Math. Soc., Zurich · Zbl 1405.20003 |
| [33] | Marsh, B. R.; Rietsch, K., Parametrizations of flag varieties, Represent. Theory, 8, 212-242, 2004 · Zbl 1053.14057 |
| [34] | Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002). arXiv:math/0211159 · Zbl 1130.53001 |
| [35] | Postnikov, A.: Total positivity, Grassmannians, and networks (2007). http://math.mit.edu/ apost/papers/tpgrass.pdf |
| [36] | Postnikov, A.; Speyer, D.; Williams, L., Matching polytopes, toric geometry, and the totally non-negative Grassmannian, J. Algebraic Comb., 30, 2, 173-191, 2009 · Zbl 1264.20045 |
| [37] | Rietsch, K., Intersections of Bruhat cells in real flag varieties, Int. Math. Res. Not., 13, 623-640, 1997 · Zbl 0889.20022 |
| [38] | Rietsch, K., An algebraic cell decomposition of the nonnegative part of a flag variety, J. Algebra, 213, 1, 144-154, 1999 · Zbl 0920.20041 |
| [39] | Rietsch, K., Closure relations for totally nonnegative cells in \(G/P\), Math. Res. Lett., 13, 5-6, 775-786, 2006 · Zbl 1107.14040 |
| [40] | Rietsch, K.; Williams, L., The totally nonnegative part of \(G/P\) is a CW complex, Transform. Groups, 13, 839-853, 2008 · Zbl 1191.14067 |
| [41] | Rietsch, K.; Williams, L., Discrete Morse theory for totally non-negative flag varieties, Adv. Math., 223, 6, 1855-1884, 2010 · Zbl 1206.57044 |
| [42] | Smale, S., Generalized Poincaré’s conjecture in dimensions greater than four, Ann. Math., 74, 391-406, 1961 · Zbl 0099.39202 |
| [43] | Tits, J., Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra, 105, 2, 542-573, 1987 · Zbl 0626.22013 |
| [44] | Wiliams, L. K., Shelling totally nonnegative flag varieties, J. Reine Angew. Math., 609, 1-21, 2007 · Zbl 1132.14045 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
