×

Crystal bases and Newton-Okounkov bodies. (English) Zbl 1428.14083

Summary: Let \(G\) be a connected reductive algebraic group. We prove that the string parameterization of a crystal basis for a finite-dimensional irreducible representation of \(G\) extends to a natural valuation on the field of rational functions on the flag variety \(G/B\), which is a highest-term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary, we recover a multiplicativity property of the canonical basis due to Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations, follow recovering results by Alexeev and Brion, Caldero, and the author.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
05E10 Combinatorial aspects of representation theory
14M27 Compactifications; symmetric and spherical varieties
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] V. Alexeev and M. Brion, Toric degenerations of spherical varieties , Selecta Math. (N.S.) 10 (2004), 453-478. · Zbl 1078.14075
[2] D. Anderson, Okounkov bodies and toric degenerations , Math. Ann. 356 (2013), 1183-1202. · Zbl 1273.14104
[3] A. Berenstein and D. Kazhdan, “Geometric and unipotent crystals, II: From unipotent bicrystals to crystal bases” in Quantum Groups (Haifa, 2004) , Contemp. Math. 433 , Amer. Math. Soc., Providence, 2007, 13-88. · Zbl 1154.14035
[4] A. D. Berenstein and A. V. Zelevinsky, Tensor product multiplicities and convex polytopes in partition space , J. Geom. Phys. 5 (1988), 453-472. · Zbl 0712.17006
[5] A. D. Berenstein and A. V. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties , Invent. Math. 143 (2001), 77-128. · Zbl 1061.17006
[6] M. Brion, Groupe de Picard et nombres caractéristiques des variétiés sphériques , Duke Math. J. 58 (1989), 397-424. · Zbl 0701.14052
[7] M. Brion and S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory , Progr. Math. 231 , Birkhäuser, Boston, 2005. · Zbl 1072.14066
[8] P. Caldero, Toric degenerations of Schubert varieties , Transform. Groups 7 (2002), 51-60. · Zbl 1050.14040
[9] I. M. Gelfand and M. L. Cetlin, Finite-dimensional representations of the group of unimodular matrices , Doklady Akad. Nauk USSR (N.S.) 71 (1950), 825-828.
[10] N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties , Transform. Groups 1 (1996), 215-248. · Zbl 0909.14028
[11] J. Hong and S. Kang, Introduction to Quantum Groups and Crystal Bases , Grad. Stud. Math. 42 , Amer. Math. Soc., Providence, 2002. · Zbl 1134.17007
[12] J. E. Humphreys, Linear Algebraic Groups , Grad. Texts in Math. 21 , Springer, New York, 1975.
[13] J. C. Jantzen, Representations of Algebraic Groups , Math. Surveys Monogr. 107 , Amer. Math. Soc., Providence, 2003. · Zbl 1034.20041
[14] B. H. Kahng, S.-J. Kang, M. Kashiwara, and U. R. Suh, Dual perfect bases and dual perfect graphs , preprint, [math.RT]. arXiv:1405.1820v1 · Zbl 1383.17008
[15] M. Kashiwara, Crystalizing the \(q\)-analogue of universal enveloping algebras , Comm. Math. Phys. 133 (1990), 249-260. · Zbl 0724.17009
[16] M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula , Duke Math. J. 71 (1993), 839-858. · Zbl 0794.17008
[17] M. Kashiwara, “On crystal bases” in Representations of Groups (Banff, 1994) , CMS Conf. Proc. 16 , Amer. Math. Soc., Providence, 1995, 155-197. · Zbl 0851.17014
[18] K. Kaveh, SAGBI bases and degeneration of spherical varieties to toric varieties , Michigan Math. J. 53 (2005), 109-121. · Zbl 1076.14071
[19] K. Kaveh, Note on cohomology rings of spherical varieties and volume polynomial , J. Lie Theory 21 (2011), 263-283. · Zbl 1222.14108
[20] K. Kaveh and A. Khovanskii, Convex bodies associated to actions of reductive groups , Mosc. Math. J. 12 (2012), 369-396, 461. · Zbl 1284.14061
[21] K. Kaveh and A. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory , Ann. of Math. (2) 176 (2012), 925-978. · Zbl 1270.14022
[22] K. Kaveh and A. Khovanskii, Convex bodies and algebraic equations on affine varieties , preprint, [math.AG]. arXiv:0804.4095v1 · Zbl 1316.52011
[23] B. Ya. Kazarnovskii, Newton polyhedra and Bezout’s formula for matrix functions of finite-dimensional representations , (in Russian) Funktsional. Anal. i Prilozhen. 21 , no. 4 (1987), 73-74; English translation in Funct. Anal. Appl. 21 , no. 4 (1987), 319-321.
[24] V. Kiritchenko, Gelfand-Zetlin polytopes and flag varieties , Int. Math. Res. Not. IMRN 2010 , no. 13, 2512-2531. · Zbl 1213.14089
[25] V. Kiritchenko, Divided difference operators on polytopes , to appear in Adv. Stud. Pure Math., preprint, [math.AG]. arXiv:1307.7234v1
[26] V. Kiritchenko, E. Yu. Smirnov, and V. A. Timorin, Schubert calculus and Gelfand-Tsetlin polytopes (in Russian), Uspekhi Mat. Nauk 67 (2012), no. 4 (406), 89-128; English translation in Russian Math. Surveys 67 (2012), no. 4, 685-719.
[27] S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory , Progr. Math. 204 , Birkhäuser Boston, Boston, 2002. · Zbl 1026.17030
[28] A. Küronya, V. Lozovanu, and C. Maclean, Convex bodies appearing as Okounkov bodies of divisors , Adv. Math. 229 (2012), 2622-2639. · Zbl 1253.14008
[29] R. Lazarsfeld and M. Mustata, Convex bodies associated to linear series , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 783-835. · Zbl 1182.14004
[30] P. Littelmann, Cones, crystals, and patterns , Transform. Groups 3 (1998), 145-179. · Zbl 0908.17010
[31] G. Lusztig, Canonical bases arising from quantized enveloping algebras , J. Amer. Math. Soc. 3 (1990), 447-498. · Zbl 0703.17008
[32] P. Magyar, Schubert polynomials and Bott-Samelson varieties , Comment. Math. Helv. 73 (1998), 603-636. · Zbl 0951.14036
[33] C. Manon, Toric degenerations and tropical geometry of branching algebras , preprint, [math.AG]. arXiv:1103.2484v3 · Zbl 1327.14055
[34] A. Okounkov, Brunn-Minkowski inequality for multiplicities , Invent. Math. 125 (1996), 405-411. · Zbl 0893.52004
[35] A. Okounkov, Note on the Hilbert polynomial of a spherical variety (in Russian), Funktsional. Anal. i Prilozhen. 31 (1997), no. 2, 82-85; English translation in Funct. Anal. Appl. 31 (1997), no. 2, 138-140.
[36] A. Okounkov, “Multiplicities and Newton polytopes” in Kirillov’s Seminar on Representation Theory , Amer. Math. Soc. Transl. Ser. 2 181 , Adv. Math. Sci. 35 , Amer. Math. Soc., Providence, 1998, 231-244. · Zbl 0920.20032
[37] A. Okounkov, “Why would multiplicities be log-concave?” in The Orbit Method in Geometry and Physics (Marseille, 2000) , Progr. Math. 213 , Birkhäuser Boston, Boston, 2003, 329-347. · Zbl 1063.22024
[38] A. N. Parshin, Chern classes, adèles and L-functions , J. Reine Angew. Math. 341 (1983), 174-192. · Zbl 0518.14013
[39] Z. Reichstein, SAGBI bases in rings of multiplicative invariants , Comment. Math. Helv. 78 (2003), 185-202. · Zbl 1043.13008
[40] B. Sturmfels, Gröbner Bases and Convex Polytopes , Univ. Lecture Ser. 8 , Amer. Math. Soc., Providence, 1996.
[41] B. Teissier, “Valuations, deformations, and toric geometry” in Valuation Theory and its Applications, Vol. II (Saskatoon, 1999) , Fields Inst. Commun. 33 , Amer. Math. Soc., Providence, 2003, 361-459. · Zbl 1061.14016
[42] D. A. Timashev, Homogeneous Spaces and Equivariant Embeddings , Encyclopaedia Math. Sci. 138 , Invariant Theory Algebr. Transform. Groups 8 , Springer, Heidelberg, 2011. · Zbl 1237.14057
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.