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Equivariant Pieri rules for isotropic Grassmannians. (English) Zbl 1339.14030

Let \(V\) be an \(N\)-dimensional complex vector space equipped with a symmetric or skew-symmetric bilinear form \(\omega\), which can be either trivial or non-degenerate. The Grassmannians \(I G_{\omega}(m, N)\) of classical Lie type parameterize \(m\)-dimensional isotropic vector subspaces of \(V\). The cohomology ring of an isotropic Grassmannian \(X =I G_{\omega}(m, N)\), or more generally of a homogeneous variety, has an additive basis of Schubert classes represented by Schubert subvarieties \(X_{\lambda}\). One of the central problems of Schubert calculus is to find a manifestly positive formula for the structure constants of the cup product of two Schubert cohomology classes, or equivalently, for the triple intersection numbers of three Schubert subvarieties in general position. Such a positive formula, called a Littlewood-Richardson rule, has deep connections to various subjects, including geometry, combinatorics and representation theory.
An isotropic Grassmannian \(X\) can be written as a quotient of a classical complex simple Lie group \(G\) by a maximal parabolic subgroup \(P\) (with two notable exceptions of Lie type \(D_n\)). Fix a choice of maximal complex torus \(T\) and a Borel subgroup \(B\) with \(T \subset B \subset P\). The Schubert varieties \(X_{\lambda}\) are closures of \(B\)-orbits, and hence are \(T\)-stable. They give a basis \([X_{\lambda}]^T\) for the \(T\)-equivariant cohomology \({H^*}_T (X)\) as a \({H^*}_T (pt)\)-module. The structure coefficients \({N^{\nu}}_{\lambda,\mu}\) in the equivariant product, \[ [X_{\lambda}]^T \cdot [X_{\mu}]^T=\sum\limits_{\nu}{N^{\nu}}_{\lambda,\mu} [X_{\nu}]^T, \] are homogeneous polynomials which satisfy a positivity condition conjectured by D. Peterson [Lectures on quantum cohomology of \(G/B\), MIT (1996)] and proved by W. Graham [Duke Math. J. 109, No. 3, 599–614 (2001; Zbl 1069.14055)]. In particular, they are Graham-positive, meaning they are polynomials in the negative simple roots, with non-negative integer coefficients. These equivariant structure coefficients carry much more information than the triple intersection numbers of Schubert varieties, and are more challenging to study.
In the present paper, the authors give for the first time an equivariant Pieri rule for Grassmannians of Lie types \(B, C\), and \(D\), as well as a new proof of the Pieri rule in type \(A\). Such a rule concerns products with the special Schubert classes \([X_{p}]^T\) , which are related to the equivariant Chern classes of the tautological quotient bundle, and generate the \(T\) -equivariant cohomology ring. Using geometric methods, they give a manifestly positive formula for the structure coefficients \({N^{\mu}}_{\lambda,p}\) of the equivariant multiplication \([X_{\lambda}]^T \cdot [X_{p}]^T\).
Reviewer: Cenap Özel (Bolu)

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
55N91 Equivariant homology and cohomology in algebraic topology

Citations:

Zbl 1069.14055
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References:

[1] Andersen, HH; Jantzen, JC; Soergel, W, Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque, 220, 297-302, (1994) · Zbl 0802.17009
[2] Anderson, D.: Introduction to equivariant cohomology in algebraic geometry. In: Contributions to Algebraic Geometry, pp. 71-92. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2012) · Zbl 1262.14019
[3] Arabia, A, Cohomologie T-équivariante de la variété de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. France, 117, 129-165, (1989) · Zbl 0706.57024
[4] Bergeron, N; Sottile, F, A Pieri-type formula for isotropic flag manifolds, Trans. Am. Math. Soc., 354, 2659-2705, (2002) · Zbl 0992.14014
[5] Billey, S, Kostant polynomials and the cohomology ring for \(G/B\), Duke Math. J., 96, 205-224, (1999) · Zbl 0980.22018
[6] Billey, S; Haiman, M, Schubert polynomials for the classical groups, J. Am. Math. Soc., 8, 443-482, (1995) · Zbl 0832.05098
[7] Bourbaki, B.: Lie groups and Lie algebras: Chapters 4-6. In: Elements of Mathematics (Berlin). Springer, Berlin (2002) · Zbl 0983.17001
[8] Brion, M, Equivariant Chow groups for torus actions, Transform. Groups, 2, 225-267, (1997) · Zbl 0916.14003
[9] Buch, AS, Mutations of puzzles and equivariant cohomology of two-step flag varieties, Ann. Math. (2), 182, 173-220, (2015) · Zbl 1354.14072
[10] Buch, A.S., Kresch, A., Tamvakis, H.: A Giambelli formula for even orthogonal Grassmannians. J. Reine Angew. Math. (to appear). arXiv:1109.6669 [math.AG] · Zbl 1342.14108
[11] Buch, A.S.: Equivariant quantum calculator, a free software package for Maple. http://math.rutgers.edu/ asbuch/equivcalc/. Accessed on 6 July 2015 · Zbl 1205.05244
[12] Buch, AS; Kresch, A; Tamvakis, H, Quantum Pieri rules for isotropic Grassmannians, Invent. Math., 178, 345-405, (2009) · Zbl 1193.14071
[13] Buch, AS; Mihalcea, LC, Quantum K-theory of Grassmannians, Duke Math. J., 156, 501-538, (2011) · Zbl 1213.14103
[14] Buch, AS; Ravikumar, V, Pieri rules for the K-theory of cominuscule Grassmannians, J. Reine Angew. Math., 668, 109-132, (2012) · Zbl 1298.14059
[15] Buch, AS; Rimanyi, R, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 339, 1-4, (2004) · Zbl 1051.14062
[16] Fulton, W.: Equivariant cohomology in algebraic geometry. In: Eilenberg lectures. Columbia University, Springer, Berlin (2007) (Notes by D. Anderson) · Zbl 1361.14029
[17] Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997) · Zbl 0878.14034
[18] Fun, A.: Raising operators and the Littlewood-Richardson polynomials. arXiv:1203.4729 [math.CO] · Zbl 0972.05053
[19] Gatto, L; Santiago, T, Equivariant Schubert calculus, Ark. Mat., 48, 41-55, (2010) · Zbl 1188.14033
[20] Graham, W, Positivity in equivariant Schubert calculus, Duke Math. J., 109, 599-614, (2001) · Zbl 1069.14055
[21] Huang, Y., Li, C.: On equivariant quantum Schubert calculus for \(G/P\). J. Algebra (to appear). arXiv:1506.00872 [math.AG] · Zbl 1188.14033
[22] Ikeda, T, Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian, Adv. Math., 215, 1-23, (2007) · Zbl 1126.14060
[23] Ikeda, T; Mihalcea, L; Naruse, H, Double Schubert polynomials for the classical groups, Adv. Math., 226, 840-886, (2011) · Zbl 1291.05222
[24] Ikeda, T; Matsumura, T, Pfaffian sum formula for the symplectic Grassmannians, Math. Zeit., 280, 269-306, (2015) · Zbl 1361.14029
[25] Ikeda, T; Naruse, H, Excited Young diagrams and equivariant Schubert calculus, Trans. Am. Math. Soc., 361, 5193-5221, (2009) · Zbl 1229.05287
[26] Knutson, A; Tao, T, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., 119, 221-260, (2003) · Zbl 1064.14063
[27] Kostant, B; Kumar, S, The nil Hecke ring and the cohomology of \(G/P\) for a Kac-Moody group \(G\), Adv. Math., 62, 187-237, (1986) · Zbl 0641.17008
[28] Kreiman, V, Equivariant Littlewood-Richardson skew tableaux, Trans. Am. Math. Soc., 362, 2589-2617, (2010) · Zbl 1205.05244
[29] Kumar, S.: Kac-Moody groups, their flag varieties and representation theory. In: Progress in Mathematics, vol. 204. Birhäuser, Boston (2002) · Zbl 1026.17030
[30] Lakshmibai, V., Raghavan, K.N., Sankaran, P.: Equivariant Giambelli and determinantal restriction formulas for the Grassmannian. Pure Appl. Math. Q. 2(3) (2006) (Special Issue: In honor of Robert D. MacPherson. Part 1, 699-717) · Zbl 1105.14065
[31] Laksov, D, Schubert calculus and equivariant cohomology of Grassmannians, Adv. Math., 217, 1869-1888, (2008) · Zbl 1136.14042
[32] Lam, T; Shimozono, M, Equivariant Pieri rule for the homology of the affine Grassmannian, J. Algebr. Combin., 36, 623-648, (2012) · Zbl 1267.14071
[33] Lam, T; Shimozono, M, Quantum double Schubert polynomials represent Schubert classes, Proc. Am. Math. Soc., 142, 835-850, (2014) · Zbl 1299.14044
[34] Leung, NC; Li, C, Quantum Pieri rules for tautological subbundles, Adv. Math., 248, 279-307, (2013) · Zbl 1291.14083
[35] Mihalcea, LC, Equivariant quantum cohomology of homogeneous spaces, Duke Math. J., 140, 321-350, (2007) · Zbl 1135.14042
[36] Mihalcea, LC, Giambelli formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Am. Math. Soc., 360, 2285-2301, (2008) · Zbl 1136.14046
[37] Molev, A.I., Sagan, B.E.: A Pieri rule for generalized factorial Schur functions. In: Proceedings of the 9-th Conference on Factorial Power Series and Algebraic Combinatorics, Vienna, vol. 3, pp. 517-523 (1997) · Zbl 1229.05287
[38] Molev, AI, Littlewood-Richardson polynomials, J. Algebra, 321, 3450-3468, (2009) · Zbl 1169.05050
[39] Molev, AI; Sagan, BE, A Littlewood-Richardson rule for factorial Schur functions, Trans. Am. Math. Soc., 351, 4429-4443, (1999) · Zbl 0972.05053
[40] Peterson D.: Lectures on quantum cohomology of G/P, Unpublished (1997)
[41] Ravikumar, V.: Triple intersection formulas for isotropic Grassmannians. Algebra Num. Theory (to appear, preprint). arXiv:1403.1741 [math.AG] · Zbl 1342.14109
[42] Robinson, S, A Pieri-type formula for \(H_T^*(SL_n(\mathbb{C})/B)\), J. Algebra, 249, 38-58, (2002) · Zbl 1061.14060
[43] Sottile, F, Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloq. Math., 82, 49-63, (1999) · Zbl 0977.14023
[44] Tamvakis, H., Wilson, E.: Double theta polynomials and equivariant Giambelli formulas (preprint). arXiv:1410.8329 [math.AG] · Zbl 1371.14058
[45] Tamvakis, H.: Giambelli and degeneracy locus formulas for classical \(G/P\) spaces (preprint). arXiv:1305.3543 [math.AG] · Zbl 1379.14026
[46] Thomas, H., Yong, A.: Equivariant Schubert calculus and jeu de taquin. Ann. l’Inst. Four. (to appear, preprint). arXiv:1207.3209 [math.CO] · Zbl 0977.14023
[47] Wilson, E.V.: Equivariant Giambelli Formulae for Grassmannians, Ph.D. Thesis, University of Maryland, College Park (2010)
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.