Newton-Okounkov polytopes of flag varieties for classical groups. (English) Zbl 1436.52015
Uniformly geometric valuations on the corresponding complete flag varieties are defined for classical groups \(\mathrm{SL}_n(\mathbb{C})\), \(\mathrm{SO}_n(\mathbb{C})\) and \(\mathrm{Sp}_{2n}(\mathbb{C})\). The valuation in every type is combinatorially related to the Gelfand-Zetlin pattern in the same type and comes from a natural coordinate system on the open Schubert cell. In types A and C, FFLV are exhibited as Newton-Okounkov polytopes with respect to this valuation. The result is new in type C. Low-dimensional examples are computed in types B and D. Further directions of research are opened, as the assumption that the valuation and the corresponding Newton-Okounkov polytopes might give a tool for describing FFLV bases in types B and D.
Reviewer: Gabriela Cristescu (Arad)
MSC:
| 52B45 | Dissections and valuations (Hilbert’s third problem, etc.) |
| 05E18 | Group actions on combinatorial structures |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Keywords:
Newton-Okounkov convex body; flag variety; FFLV polytope; uniformly geometric valuations; Gelfand-Zetlin patternReferences:
| [1] | Ardila, F.; Bliem, Th; Salazar, D., Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes, J. Comb. Theory Ser. A, 118, 2454-2462 (2011) · Zbl 1234.52009 |
| [2] | Backhaus, T.; Kus, D., The PBW filtration and convex polytopes in type \[B B\], J. Pure Appl. Algebra, 223, 245-276 (2019) · Zbl 1464.17011 |
| [3] | Belyaev, A.; Avramenko, S.; Agakishiev, G.; Pechenov, V.; Rikhvitsky, V., On the initial approximation of charged particle tracks in detectors with linear sensing elements, Nucl. Instrum. Methods Phy. Res., 938, 1-4 (2019) · doi:10.1016/j.nima.2019.05.082 |
| [4] | Berenstein, AD; Zelevinsky, AV, Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys., 5, 453-472 (1989) · Zbl 0712.17006 |
| [5] | Bernstein, DN, The number of roots of a system of equations, Funct. Anal. Appl., 9, 183-185 (1975) · Zbl 0328.32001 |
| [6] | Brion, M.: Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, 33-85. Trends Math. Birkhäuser, Basel (2005) · Zbl 1487.14105 |
| [7] | Brion, M., Groupe de Picard et nombres caracteristiques des varietes spheriques, Duke Math J., 58, 397-424 (1989) · Zbl 0701.14052 |
| [8] | De Concini, C., Procesi, C.: Complete symmetric varieties II Intersection theory, Advanced Studies in Pure Mathematics 6 (1985), Algebraic groups and related topics, 481-513 · Zbl 0596.14041 |
| [9] | Feigin, E.; Fourier, Gh; Littelmann, P., PBW filtration and bases for irreducible modules in type \[A_n\] An, Transform. Groups, 165, 71-89 (2011) · Zbl 1237.17011 |
| [10] | Feigin, E., Fourier, Gh, Littelmann, P.: PBW filtration and bases for for symplectic Lie algebras. IMRN (24):5760-5784 (2011) · Zbl 1233.17007 |
| [11] | Feigin, E.; Fourier, Gh; Littelmann, P., Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations, Transform. Groups, 22, 321-352 (2017) · Zbl 1461.14068 |
| [12] | Fujita, N., Oya, H.: A comparison of Newton-Okounkov polytopes of Schubert varieties. J. London Math. Soc. 2, (2017). https://doi.org/10.1112/jlms.12059 · Zbl 1427.17024 |
| [13] | Fulton, W., Harris, J.: Representation theory: a first course. Springer, New York (2004) |
| [14] | Gornitskii, A. A.: Essential Signatures and Canonical Bases for Irreducible Representations of \[D_4\] D4, preprint arXiv:1507.07498 [math.RT] · Zbl 1377.17010 |
| [15] | Kaveh, K., Crystal basis and Newton-Okounkov bodies, Duke Math. J., 164, 2461-2506 (2015) · Zbl 1428.14083 |
| [16] | Kaveh, K., Khovanskii, A.: Newton convex bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. (2) 176(2), 925-978 (2012) · Zbl 1270.14022 |
| [17] | Kaveh, K.; Khovanskii, AG, Convex bodies associated to actions of reductive groups, Moscow Math. J., 12, 369-396 (2012) · Zbl 1284.14061 |
| [18] | Kazarnovskii, B.Ya.: Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations. Funct. Anal. Appl. 21(4), 319-321 (1987) · Zbl 0662.22014 |
| [19] | Kiritchenko, V., Geometric mitosis, Math. Res. Lett., 23, 1069-1096 (2016) · Zbl 1356.14040 |
| [20] | Kiritchenko, V., Newton-Okounkov polytopes of flag varieties, Transform. Groups, 22, 387-402 (2017) · Zbl 1396.14047 |
| [21] | Khovanskii, AG, Newton polyhedra, and the genus of complete intersections, Funct. Anal. Appl., 12, 38-46 (1978) · Zbl 0406.14035 |
| [22] | Kushnirenko, AG, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32, 1-31 (1976) · Zbl 0328.32007 |
| [23] | Lazarsfeld, R.; Mustata, M., Convex Bodies Associated to Linear Series, Annales Scientifiques de l’ENS, 42, 783-835 (2009) · Zbl 1182.14004 |
| [24] | Makhlin, I.: FFLV-type monomial bases for type \[BB\], preprint arXiv:1610.07984 [math.RT] · Zbl 1472.17033 |
| [25] | Molev, A.I.: Gelfand-Tsetlin bases for classical Lie algebras, Handbook of Algebra (M. Hazewinkel, Ed.), 4, Elsevier, Amsterdam, 2006, 109-170 · Zbl 1211.17009 |
| [26] | Okounkov, A., Note on the Hilbert polynomial of a spherical variety, Funct. Anal. Appl., 31, 138-140 (1997) · Zbl 0928.14032 |
| [27] | Okounkov, A., Multiplicities and Newton polytopes, Kirillov’s seminar on representation theory, Am. Math. Soc. Transl. Ser., 2, 231-244 (1998) · Zbl 0920.20032 |
| [28] | Zhelobenko, DP, An analogue of the Gel’fand-Tsetlin basis for symplectic Lie algebras, Russ. Math. Surveys, 42, 247-248 (1987) · Zbl 0659.17006 |
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