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A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

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Abstract

We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory \(QK_{T}(G/B)\) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of \(QK_{T}(G/B)\). In type \(A_{n-1}\), we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory \(QK(SL_{n}/B)\); we also obtain very explicit information about the coefficients in the respective Chevalley formula.

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References

  1. Anderson, D., Chen, L., Tseng, H.-H.: On the finiteness of quantum \(K\)-theory of a homogeneous space. Int. Math. Res. Not. 2022(2), 1313–1349 (2022)

    Article  MathSciNet  Google Scholar 

  2. Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics. Westview Press, Oxford (1969)

  3. Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate Texts in Mathematics Vol. 231. Springer, New York (2005)

  4. Brenti, F., Fomin, S., Postnikov, A.: Mixed Bruhat operators and Yang–Baxter equations for Weyl groups. Int. Math. Res. Not. 8, 419–441 (1999)

    Article  MathSciNet  Google Scholar 

  5. Buch, A., Chaput, P.-E., Mihalcea, L., Perrin, N.: A Chevalley formula for the equivariant quantum \(K\)-theory of cominuscule varieties. Algebraic Geom. 5, 568–595 (2018)

    MathSciNet  Google Scholar 

  6. Buch, A., Chung, S., Li, C., Mihalcea, L.: Euler characteristics in the quantum \(K\)-theory of flag varieties. Selecta Math. (N.S.), 26:Article No. 29 (2020)

  7. Deodhar, V.: A splitting criterion for the Bruhat orderings on Coxeter groups. Commun. Algebra 15, 1889–1894 (1987)

    Article  MathSciNet  Google Scholar 

  8. Dyer, M.J.: Hecke algebras and shellings of Bruhat intervals. Compositio Math. 89, 91–115 (1993)

    MathSciNet  Google Scholar 

  9. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics Vol. 150. Springer, New York (1995)

  10. Fomin, S., Gelfand, S., Postnikov, A.: Quantum Schubert polynomials. J. Am. Math. Soc. 10, 565–596 (1997)

    Article  MathSciNet  Google Scholar 

  11. Finkelberg, M., Mirkovic, I.: Semi-infinite flags. I: Case of global \(\mathbb{P}^1\). Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, pp. 81–112, American Mathematical Society Translations Series 2 Vol. 194, Advances in the Mathematical Sciences Vol. 44. American Mathematical Society , Providence, RI (1999)

  12. Fulton, W., Woodward, C.: On the quantum product of Schubert classes. J. Algebraic Geom. 13, 641–661 (2004)

    Article  MathSciNet  Google Scholar 

  13. Gaussent, S., Littelmann, P.: LS-galleries, the path model and MV-cycles. Duke Math. J. 127, 35–88 (2005)

    Article  MathSciNet  Google Scholar 

  14. Gu, W., Mihalcea, L.C., Sharpe, E., Zou, H.: Quantum \(K\)-theory of Grassmannians, Wilson operators, and Schur bundles (2022). arXiv:2208.01091

  15. Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  16. Kato, S.: Loop structure on equivariant \(K\)-theory of semi-infinite flag manifolds (2018). arXiv:1805.01718

  17. Kato, S.: Frobenius splitting of Schubert varieties of semi-infinite flag manifolds. Forum Math. Pi, 9:Paper No. e5 (2021)

  18. Kato, S.: On quantum \(K\)-group of partial flag manifolds (2019). arXiv:1906.09343

  19. Kato, S.: The formal model of semi-infinite flag manifolds. ICM - International Congress of Mathematicians. Vol. III. Sections 1–4, 1600–1623. EMS Press, Berlin, (2023)

  20. Kato, S., Naito, S., Sagaki, D.: Equivariant \(K\)-theory of semi-infinite flag manifolds and the Pieri–Chevalley formula. Duke Math. J. 169, 2421–2500 (2020)

    Article  MathSciNet  Google Scholar 

  21. Kouno, T., Lenart, C., Naito, S.: New structure on the quantum alcove model with applications to representation theory and Schubert calculus. J. Comb. Algebra 7, 347–400 (2023)

    Article  MathSciNet  Google Scholar 

  22. Kouno, T., Lenart, C., Naito, S.: Generalized quantum Yang-Baxter moves and their application to Schubert calculus (extended abstract). 34th international conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2022), Sem. Lothar. Combin., 86B:Art. 13 (2022)

  23. Kouno, T., Lenart, C., Naito, S., Sagaki, D.: Quantum \(K\)-theory Chevalley formulas in the parabolic case. J. Algebra 645, 1–53 (2024)

    Article  MathSciNet  Google Scholar 

  24. Kouno, T., Naito, S., Orr, D., Sagaki, D.: Inverse \(K\)-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in \(ADE\) type. Forum Math. Sigma, 9:Paper No. e51, 25 (2021)

  25. Kouno, T., Naito, S., Sagaki, D.: Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum \(K\)-group of partial flag manifolds. J. Combin. Theory Ser. A, 192:Paper No. 105670 (2022)

  26. Lee, Y.-P.: Quantum \(K\)-theory I: Foundations. Duke Math. J. 121, 389–424 (2004)

    Article  MathSciNet  Google Scholar 

  27. Lenart, C.: From Macdonald polynomials to a charge statistic beyond type \(A\). J. Combin. Theory Ser. A 119, 683–712 (2012)

    Article  MathSciNet  Google Scholar 

  28. Lenart, C., Lubovsky, A.: A generalization of the alcove model and its applications. J. Algebraic Combin. 41, 751–783 (2015)

    Article  MathSciNet  Google Scholar 

  29. Lenart, C., Lubovsky, A.: A uniform realization of the combinatorial \(R\)-matrix. Adv. Math. 334, 151–183 (2018)

    Article  MathSciNet  Google Scholar 

  30. Lenart, C., Maeno, T.: Quantum Grothendieck Polynomials (2006). arXiv:math.CO/0608232

  31. Lenart, C., Naito, S., Orr, D., Sagaki, D.: Inverse \(K\)-Chevalley formulas for semi-infinite flag manifolds, II: arbitrary weights in ADE type. Adv. Math., 423:Paper No. 109037 (2023)

  32. Lenart, C., Naito, S., Sagaki, D.: A combinatorial Chevalley formula for semi-infinite flag manifolds and its applications (extended abstract). 33rd international conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2021), Sém. Lothar. Combin., 85B:Art. 22 (2021)

  33. Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph. Int. Math. Res. Not. 2015(7), 1848–1901 (2015)

    MathSciNet  Google Scholar 

  34. Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov-Reshetikhin crystals II: Path models and \(P=X\). Int. Math. Res. Not. 2017(14), 4259–4319 (2017)

    MathSciNet  Google Scholar 

  35. Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at \(t=0\) and Demazure characters. Transform. Groups 22, 1041–1079 (2017)

    Article  MathSciNet  Google Scholar 

  36. Lenart, C., Postnikov, A.: Affine Weyl groups in \(K\)-theory and representation theory. Int. Math. Res. Not., 2007, no. 12: Art. ID rnm038 (2007)

  37. Lenart, C., Postnikov, A.: A combinatorial model for crystals of Kac–Moody algebras. Trans. Am. Math. Soc. 360, 4349–4381 (2008)

    Article  MathSciNet  Google Scholar 

  38. Lenart, C., Schultze, A.: On combinatorial models for affine crystals. Sém. Lothar. Combin., 85B:Art. 20 (2021)

  39. Lenart, C., Shimozono, M.: Equivariant \(K\)-Chevalley rules for Kac–Moody flag manifolds. Am. J. Math. 136, 1175–1213 (2014)

    Article  MathSciNet  Google Scholar 

  40. Maeno, T., Naito, S., Sagaki, D.: A presentation of the torus-equivariant quantum \(K\)-theory ring of flag manifolds of type \(A\), I: the defining ideal (2023). arXiv:2302.09485

  41. Naito, S., Orr, D., Sagaki, D.: Pieri-Chevalley formula for anti-dominant weights in the equivariant \(K\)-theory of semi-infinite flag manifolds. Adv. Math., 387:Paper No. 107828 (2021)

  42. Naito, S., Sagaki, D.: Pieri-type multiplication formula for quantum Grothendieck polynomials (2022). arXiv:2211.01578

  43. Orr, D.: Equivariant \(K\)-theory of the semi-infinite flag manifold as a nil-DAHA module. Selecta Math. (N.S.), 29:Paper No. 45 (2023)

  44. Postnikov, A.: Quantum Bruhat graph and Schubert polynomials. Proc. Am. Math. Soc. 133, 699–709 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

C.L. was partly supported by the NSF grant DMS-1855592 and the Simons Foundation grant #584738. S.N. was partly supported by JSPS Grants-in-Aid for Scientific Research (B) 16H03920 and (C) 21K03198. D.S. was partly supported by JSPS Grants-in-Aid for Scientific Research (C) 15K04803 and 19K03415. An extended abstract of this work has appeared in the Proceedings of the 33rd international conference on Formal Power Series and Algebraic Combinatorics [32].

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Authors and Affiliations

  1. Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY, 12222, USA

    Cristian Lenart

  2. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo, 152-8551, Japan

    Satoshi Naito

  3. Department of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan

    Daisuke Sagaki

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  1. Cristian Lenart
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Lenart, C., Naito, S. & Sagaki, D. A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory. Sel. Math. New Ser. 30, 39 (2024). https://doi.org/10.1007/s00029-024-00924-8

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