Abstract
We prove that twisted versions of Schubert polynomials defined by \(\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}\) and \(\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial _i)\widetilde{\mathfrak S}_w\) are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the \(\widetilde{\mathfrak S}_w\) as well as their localizations.
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References
Aluffi, P., Mihalcea, L.C.: Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds. Compos. Math. 152(12), 2603–2625 (2016)
Aluffi, P., Mihalcea, L.C., Schuermann, J., Su, C.: Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells (2017). Preprint arXiv:1709.08697
Bergeron, N., Billey, S.: RC-graphs and Schubert polynomials. Exp. Math. 2(4), 257–269 (1993)
Billey, S.C.: Kostant polynomials and the cohomology ring for \(G/B\). Duke Math. J. 96(1), 205–224 (1999)
Billey, S.C., Jockusch, W., Stanley, R.P.: Some combinatorial properties of Schubert polynomials. J. Algebraic Combin. 2(4), 345–374 (1993)
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
Buch, A.S.: Mutations of puzzles and equivariant cohomology of two-step flag varieties. Ann. Math. (2) 182(1), 173–220 (2015)
Coskun, I.: A Littlewood-Richardson rule for two-step flag varieties. Invent. Math. 176(2), 325–395 (2009)
Fomin, S., Stanley, R.P.: Schubert polynomials and the nil-Coxeter algebra. Adv. Math. 103(2), 196–207 (1994)
Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25–83 (1998)
Huh, J.: Positivity of Chern classes of Schubert cells and varieties. J. Algebraic Geom. 25(1), 177–199 (2016)
Kirillov, A.N.: Skew divided difference operators and Schubert polynomials. SIGMA Symmetry Integr. Geom. Methods Appl. 3, 14 (2007)
Kirillov, A.N.: Notes on Schubert, Grothendieck and key polynomials. SIGMA Symmetry Integr. Geom. Methods Appl. 12, 56 (2016)
Lascoux, A., Leclerc, B., Thibon, J.-Y.: Twisted action of the symmetric group on the cohomology of a flag manifold. In: Parameter spaces (Warsaw, 1994), volume 36 of Banach Center Publ., pp. 111–124. Polish Acad. Sci. Inst. Math., Warsaw (1996)
Lascoux, A., Leclerc, B., Thibon, J.-Y.: Flag varieties and the Yang–Baxter equation. Lett. Math. Phys. 40(1), 75–90 (1997)
Lascoux, A., Schützenberger, M.-P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294(13), 447–450 (1982)
Lee, S.J.: Chern class of Schubert cells in the flag manifold and related algebras. J. Algebraic Combin. 47(2), 213–231 (2018)
Liu, R.I.: Positive expressions for skew divided difference operators. J. Algebraic Combin. 42(3), 861–874 (2015)
Macdonald, I.G.: Schubert polynomials. In: Surveys in Combinatorics, 1991 (Guildford, 1991), Volume 166 of London Math. Soc. Lecture Note Ser., pp. 73–99. Cambridge University Press, Cambridge (1991)
Maulik, D., Okounkov, A.: Quantum Groups and Quantum Cohomology (2012). Preprint arXiv:1211.1287
Postnikov, A.: On a quantum version of Pieri’s formula. In: Advances in Geometry, Volume 172 of Progr. Math., pp. 371–383. Birkhäuser, Boston (1999)
Rimányi, R., Varchenko, A.: Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae. In: Schubert varieties, equivariant cohomology and characteristic classes—IMPANGA 15, EMS Ser. Congr. Rep., pp. 225–235. Eur. Math. Soc., Zürich (2018)
Sottile, F.: Pieri’s formula for flag manifolds and Schubert polynomials. Ann. Inst. Fourier (Grenoble) 46(1), 89–110 (1996)
Stryker, J.P., III.: Chern–Schwartz–MacPherson classes of graph hypersurfaces and Schubert varieties. ProQuest LLC, Ann Arbor, MI (2011). Ph. D. thesis, The Florida State University
Su, C.: Restriction formula for stable basis of the Springer resolution. Sel. Math. (N.S.) 23(1), 497–518 (2017)
Acknowledgements
The author would like to thank Alexander Yong for suggesting this line of inquiry and insightful conversations, as well as the organizers of the “Positivity in Algebraic Combinatorics” workshop at the Korea Institute for Advanced Study in June 2016.
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The author is partially supported by a National Science Foundation Grant (DMS 1700302/2204415).
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