One-skeleton posets of Bruhat interval polytopes. (English) Zbl 1531.06001
Summary: Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these polytopes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee-Masuda, and of Lee-Masuda-Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth.
MSC:
| 06A07 | Combinatorics of partially ordered sets |
| 05E16 | Combinatorial aspects of groups and algebras |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |
References:
| [1] | Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.; Goncharov, A.; Postnikov, A.; Trnka, J., Grassmannian Geometry of Scattering Amplitudes (2016), Cambridge University Press · Zbl 1365.81004 |
| [2] | Barkley, G. T.; Gaetz, C., Combinatorial invariance for elementary intervals (2023) |
| [3] | Billey, S. C.; Fan, C. K.; Losonczy, J., The parabolic map, J. Algebra, 214, 1, 1-7 (1999) · Zbl 0922.20048 |
| [4] | Björner, A., Orderings of Coxeter groups, (Combinatorics and Algebra. Combinatorics and Algebra, Boulder, Colo., 1983. Combinatorics and Algebra. Combinatorics and Algebra, Boulder, Colo., 1983, Contemp. Math., vol. 34 (1984), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 175-195 · Zbl 0594.20029 |
| [5] | Björner, A.; Brenti, F., Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231 (2005), Springer: Springer New York · Zbl 1110.05001 |
| [6] | Björner, A.; Ekedahl, T., On the shape of Bruhat intervals, Ann. Math. (2), 170, 2, 799-817 (2009) · Zbl 1226.05268 |
| [7] | Boretsky, J.; Eur, C.; Williams, L., Polyhedral and tropical geometry of flag positroids (2022) |
| [8] | Borovik, A. V.; Gelfand, I. M.; White, N., Coxeter Matroids, Progress in Mathematics, vol. 216 (2003), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1050.52005 |
| [9] | Bousquet-Mélou, M.; Butler, S., Forest-like permutations, Ann. Comb., 11, 3-4, 335-354 (2007) · Zbl 1141.05011 |
| [10] | Bump, D.; Chetard, B., Matrix coefficients of intertwining operators and the bruhat order (2021) |
| [11] | Carrell, J. B.; Kurth, A., Normality of torus orbit closures in \(G / P\), J. Algebra, 233, 1, 122-134 (2000) · Zbl 1023.14023 |
| [12] | Carrell, J. B.; Kuttler, J., Smooth points of T-stable varieties in \(G / B\) and the Peterson map, Invent. Math., 151, 2, 353-379 (2003) · Zbl 1041.14020 |
| [13] | Fraser, C., Cyclic symmetry loci in Grasssmannians (2020) |
| [14] | Gaetz, C., Bruhat interval polytopes, 1-skeleton lattices, and smooth torus orbit closures, Sémin. Lothar. Comb., 89B (2023) |
| [15] | Gel’fand, I. M.; Serganova, V. V., Combinatorial geometries and the strata of a torus on homogeneous compact manifolds, Usp. Mat. Nauk, 42, 2(254), 107-134 (1987), 287 · Zbl 0629.14035 |
| [16] | Patricia, H., Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology (February 2018) |
| [17] | Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 2, 165-184 (1979) · Zbl 0499.20035 |
| [18] | Kazhdan, D.; Lusztig, G., Schubert varieties and Poincaré duality, (Geometry of the Laplace Operator, Proc. Sympos. Pure Math.. Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979. Geometry of the Laplace Operator, Proc. Sympos. Pure Math.. Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, Proc. Sympos. Pure Math., vol. XXXVI (1980), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 185-203 · Zbl 0461.14015 |
| [19] | Klyachko, A. A., Orbits of a maximal torus on a flag space, Funkc. Anal. Prilozh., 19, 1, 77-78 (1985) · Zbl 0581.14038 |
| [20] | Kodama, Y.; Williams, L., The full Kostant-Toda hierarchy on the positive flag variety, Commun. Math. Phys., 335, 1, 247-283 (2015) · Zbl 1319.37048 |
| [21] | Lakshmibai, V.; Sandhya, B., Criterion for smoothness of Schubert varieties in \(\operatorname{Sl}(n) / B\), Proc. Indian Acad. Sci. Math. Sci., 100, 1, 45-52 (1990) · Zbl 0714.14033 |
| [22] | Lee, E.; Masuda, M., Generic torus orbit closures in Schubert varieties, J. Comb. Theory, Ser. A, 170, Article 105143 pp. (2020) · Zbl 1475.14106 |
| [23] | Lee, E.; Masuda, M.; Park, S., Torus orbit closures in flag varieties and retractions on Weyl groups (2020) |
| [24] | Lee, E.; Masuda, M.; Park, S., Toric Bruhat interval polytopes, J. Comb. Theory, Ser. A, 179, Article 41 pp. (2021), Paper No. 105387 · Zbl 1467.52017 |
| [25] | Lee, E.; Masuda, M.; Park, S., Torus orbit closures in the flag variety (2022), arXiv preprint |
| [26] | Lee, E.; Masuda, M.; Park, S.; Song, J., Poincaré polynomials of generic torus orbit closures in Schubert varieties, (Topology, Geometry, and Dynamics. Topology, Geometry, and Dynamics, Contemp. Math., vol. 772 (2021), Amer. Math. Soc.: Amer. Math. Soc. [Providence], RI), 189-208, ©2021 · Zbl 1489.14066 |
| [27] | Postnikov, A.; Reiner, V.; Williams, L., Faces of generalized permutohedra, Doc. Math., 13, 207-273 (2008) · Zbl 1167.05005 |
| [28] | Postnikov, A., Total positivity, grassmannians, and networks (2006), arXiv preprint |
| [29] | Postnikov, A., Permutohedra, associahedra, and beyond, Int. Math. Res. Not., 6, 1026-1106 (2009) · Zbl 1162.52007 |
| [30] | Reading, N., Cambrian lattices, Adv. Math., 205, 2, 313-353 (2006) · Zbl 1106.20033 |
| [31] | Reading, N., Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., 29, 2, 736-750 (2015) · Zbl 1314.05015 |
| [32] | Reading, N., Lattice homomorphisms between weak orders, Electron. J. Comb., 26, 2, 50 (2019), Paper No. 2.23 · Zbl 1515.20203 |
| [33] | Reading, N.; Speyer, D. E., Cambrian fans, J. Eur. Math. Soc., 11, 2, 407-447 (2009) · Zbl 1213.20038 |
| [34] | Richmond, E.; Slofstra, W., Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties, Math. Ann., 366, 1-2, 31-55 (2016) · Zbl 1383.14013 |
| [35] | Tsukerman, E.; Williams, L., Bruhat interval polytopes, Adv. Math., 285, 766-810 (2015) · Zbl 1323.05010 |
| [36] | Williams, L. K., A positive Grassmannian analogue of the permutohedron, Proc. Am. Math. Soc., 144, 6, 2419-2436 (2016) · Zbl 1332.05154 |
| [37] | Woo, A.; Yong, A., When is a Schubert variety Gorenstein?, Adv. Math., 207, 1, 205-220 (2006) · Zbl 1112.14058 |
| [38] | Ziegler, G. M., Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0823.52002 |
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.
