Cominuscule points and Schubert varieties

Graham, William; Kreiman, Victor

Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2519-2548.

Nous définissons la notion de point cominuscule d’une variété de Schubert dans une variété de drapeaux généralisée pour un groupe semi-simple. Nous en déduisons des formules exprimant les séries de Hilbert et multiplicités des variétés de Schubert en des points cominuscules en termes de restrictions de classes de la $K$-théorie tore-équivariante et de la cohomologie en ces points, ce qui permet de généraliser des formules précédemment connues pour les variétés de drapeaux de type cominuscule. Nous pouvons ainsi calculer les séries de Hilbert et les multiplicités dans de nouveaux cas. Les formules pour les variétés de Schubert sont des cas particuliers de formules qui valent plus généralement en des points cominuscules généralisés de schémas munis d’actions de tores.

We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions.

Classification : 14M15, 05E15
Keywords: flag variety, Schubert variety, cominuscule, minuscule, equivariant, K-theory
Mot clés : variété de drapeaux, variété Schubert, cominuscule, minuscule, équivariante, K-théorie

[1] Andersen, Henning H.; Jantzen, Jens C.; Soergel, Wolfgang Representations of quantum groups at a $p$th root of unity and of semisimple groups in characteristic $p$: independence of $p$, Astérisque, 220, Société Mathématique de France, 1994 | Numdam | MR | Zbl

[2] Bergeron, Nantel; Billey, Sara C. RC-graphs and Schubert polynomials, Exp. Math., Volume 2 (1993) no. 4, pp. 257-269 | DOI | MR | Zbl

[3] Bernstein, Joseph; Gel’fand, Israil’ M.; Gel’fand, Sergeĭ I. Differential operators on the base affine space and a study of $𝔤$-modules, Lie groups and their representations (Proc. Summer School, Bolyai JLános Math. Soc., Budapest, 1971), János Bolyai Mathematical Society, 1975, pp. 21-64 | MR | Zbl

[4] Billey, Sara C. Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J., Volume 96 (1999) no. 1, pp. 205-224 | DOI | MR | Zbl

[5] Billey, Sara C.; Lakshmibai, Venkatramani Singular loci of Schubert varieties, Progress in Mathematics, 182, Birkhäuser, 2000 | DOI | MR | Zbl

[6] Boe, Brian D.; Graham, William A lookup conjecture for rational smoothness, Am. J. Math., Volume 125 (2003) no. 2, pp. 317-356 | MR | Zbl

[7] Brion, Michel Rational smoothness and fixed points of torus actions, Transform. Groups, Volume 4 (1999) no. 2-3, pp. 127-156 (Dedicated to the memory of Claude Chevalley) | DOI | MR | Zbl

[8] Brion, Michel Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties (Trends in Mathematics), Birkhäuser, 2005, pp. 33-85 | DOI | MR | Zbl

[9] Edidin, Dan; Graham, William Equivariant intersection theory, Invent. Math., Volume 131 (1998) no. 3, pp. 595-634 | DOI | MR | Zbl

[10] Edidin, Dan; Graham, William Nonabelian localization in equivariant $K$-theory and Riemann-Roch for quotients, Adv. Math., Volume 198 (2005) no. 2, pp. 547-582 | DOI | MR | Zbl

[11] Eisenbud, David Commutative algebra, Graduate Texts in Mathematics, 150, Springer, 1995 (With a view toward algebraic geometry) | DOI | MR | Zbl

[12] Fomin, Sergey; Kirillov, Anatol N. The Yang–Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math., Volume 153 (1996) no. 1-3, pp. 123-143 | DOI | MR | Zbl

[13] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2, Springer, 1984 | DOI | MR | Zbl

[14] Graham, William Equivariant $K$-theory and Schubert varieties (2002) (preprint)

[15] Graham, William; Kreiman, Victor Excited Young diagrams, equivariant $K$-theory, and Schubert varieties, Trans. Am. Math. Soc., Volume 367 (2015) no. 9, pp. 6597-6645 | DOI | MR | Zbl

[16] Graham, William; Kreiman, Victor The combinatorics of cominuscule points in Schubert varieties (2019) (work in progress)

[17] Graham, William; Kumar, Shrawan On positivity in $T$-equivariant $K$-theory of flag varieties, Int. Math. Res. Not., Volume 2008 (2008), rnn093 | MR | Zbl

[18] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, 1990 | DOI | MR | Zbl

[19] Ikeda, Takeshi Lectures on equivariant Schubert polynomials, Schubert calculus – Osaka 2012. Proceedings of the 5th Mathematical Society of Japan-Seasonal Institute, MSJ-SI, Osaka, Japan, July 17–27, 2012 (Advanced Studies in Pure Mathematics), Volume 71, Mathematical Society of Japan (2016), pp. 97-137 | MR | Zbl

[20] Ikeda, Takeshi; Naruse, Hiroshi Excited Young diagrams and equivariant Schubert calculus, Trans. Am. Math. Soc., Volume 361 (2009) no. 10, pp. 5193-5221 | DOI | MR | Zbl

[21] Knutson, Allen Frobenius splitting, point-counting, and degeneration (2009) (https://arxiv.org/abs/0911.4941)

[22] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. Math., Volume 161 (2005) no. 3, pp. 1245-1318 | DOI | MR | Zbl

[23] Kostant, Bertram Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math., Volume 74 (1961), pp. 329-387 | DOI | MR | Zbl

[24] Kreiman, Victor Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian (2005) (https://arxiv.org/abs/math/0512204)

[25] Kreiman, Victor Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian (2006) (https://arxiv.org/abs/math/0602245)

[26] Lakshmibai, Venkatramani Tangent spaces to Schubert varieties, Math. Res. Lett., Volume 2 (1995) no. 4, pp. 473-477 | DOI | MR | Zbl

[27] Lakshmibai, Venkatramani On tangent spaces to Schubert varieties, J. Algebra, Volume 230 (2000) no. 1, pp. 222-244 | DOI | MR | Zbl

[28] Lakshmibai, Venkatramani On tangent spaces to Schubert varieties. II, J. Algebra, Volume 224 (2000) no. 2, pp. 167-197 | DOI | MR | Zbl

[29] Lakshmibai, Venkatramani; Seshadri, Conjeeveram S. Singular locus of a Schubert variety, Bull. Am. Math. Soc., Volume 11 (1984) no. 2, pp. 363-366 | DOI | MR | Zbl

[30] Li, Li; Yong, Alexander Some degenerations of Kazhdan–Lusztig ideals and multiplicities of Schubert varieties, Adv. Math., Volume 229 (2012) no. 1, pp. 633-667 | DOI | MR | Zbl

[31] Perrin, Nicolas Small resolutions of minuscule Schubert varieties, Compos. Math., Volume 143 (2007) no. 5, pp. 1255-1312 | DOI | MR | Zbl

[32] Richardson, Roger W. Intersections of double cosets in algebraic groups, Indag. Math., New Ser., Volume 3 (1992) no. 1, pp. 69-77 | DOI | MR | Zbl

[33] Springer, Tonny A. Linear algebraic groups, Modern Birkhäuser Classics, Birkhäuser, 2009 | MR | Zbl

[34] Stembridge, John R. Minuscule elements of Weyl groups, J. Algebra, Volume 235 (2001) no. 2, pp. 722-743 | DOI | MR | Zbl

[35] Thomason, Robert W. Algebraic $K$-theory of group scheme actions, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) (Annals of Mathematics Studies), Volume 113, Princeton University Press, 1987, pp. 539-563 | MR | Zbl

[36] Willems, Matthieu $K$-théorie équivariante des tours de Bott. Application à la structure multiplicative de la $K$-théorie équivariante des variétés de drapeaux, Duke Math. J., Volume 132 (2006) no. 2, pp. 271-309 | DOI | MR | Zbl