Geodesic
Borel-type presentation of the torus-equivariant quantum K-ring of flag manifolds of type C
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e198

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We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring. 
Kouno, Takafumi; Naito, Satoshi. Borel-type presentation of the torus-equivariant quantum K-ring of flag manifolds of type C. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e198. doi: 10.1017/fms.2025.10145
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[1] Anderson, D., Chen, L. and Tseng, H.-H., ‘On the quantum -ring of the flag manifold’, . Google Scholar | arXiv

[2] Brenti, F., Fomin, S. and Postnikov, A., ‘Mixed Bruhat operators and Yang-Baxter equations for Weyl groups’, Int. Math. Res. Not. 1999(8) (1999), 419–441. doi:10.1155/S1073792899000215. Google Scholar | DOI

[3] Finkelberg, M. and Mirković, I., ‘Semi-infinite flags. I. Case of global curve ’, in Differential Topology, Infinite-dimensional Lie Algebras, and Applications, pp. 81–112, vol. 194 of Amer. Math. Soc. Transl. Ser. 2 (American Mathematical Society, Providence, RI, 1999). doi:10.1090/trans2/194/05. Google Scholar

[4] Givental, A., ‘On the WDVV equation in quantum -theory’, Michigan Math. J. 48 (2000), 295–304. doi:10.1307/mmj/1030132720. Google Scholar | DOI

[5] Givental, A. and Kim, B., ‘Quantum cohomology of flag manifolds and Toda lattices’, Commun. Math. Phys. 168(3) (1995), 609–641. doi:10.1007/BF02101846. Google Scholar | DOI

[6] Gu, W., Mihalcea, L., Sharpe, E. and Zou, H., ‘Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles’, Forum Math. Sigma 13 (2025), Paper no. e140. doi:10.1017/fms.2025.10088. Google Scholar | DOI

[7] Gu, W., Mihalcea, L., Sharpe, E. and Zou, H., ‘Quantum K theory of symplectic Grassmannians’, J. Geom. Phys. 177 (2022), Paper no. 104548, 38 pp. doi:10.1016/j.geomphys.2022.104548. Google Scholar | DOI

[8] Gu, W., Mihalcea, L.C., Sharpe, E., Xu, W., Zhang, H. and Zou, H., ‘Quantum K Whitney relations for partial flag varieties’, . Google Scholar | arXiv

[9] Kato, S., ‘Loop structure on equivariant -theory of semi-infinite flag manifolds’, Ann. Math. (2) 202(3) (2025), 1001–1075. doi:10.4007/annals.2025.202.3.2. Google Scholar | DOI

[10] Kato, S., Naito, S. and Sagaki, D., ‘Equivariant -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula’, Duke Math. J. 169(13) (2020), 2421–2500. doi:10.1215/00127094-2020-0015. Google Scholar | DOI

[11] Kim, B., ‘Quantum cohomology of flag manifolds and quantum Toda lattices’, Ann. Math. (2) 149(1) (1999), 129–148. doi:10.2307/121021. Google Scholar | DOI

[12] Kouno, T., Lenart, C. and Naito, S., ‘New structure on the quantum alcove model with applications to representation theory and Schubert calculus’, J. Comb. Algebra 7(3–4) (2023), 347–400. doi:10.4171/jca/77. Google Scholar | DOI

[13] Kouno, T., Naito, S. and Orr, D., ‘Identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules over quantum affine algebras of type ’, Algebr. Represent. Theory 27(1) (2024), 429–460. doi:10.1007/s10468-023-10221-1. Google Scholar | DOI

[14] Kouno, T., Naito, S., Orr, D. and Sagaki, D., ‘Inverse -Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type’, Forum Math. Sigma 9 (2021), Paper no. e51, 25 pp. doi:10.1017/fms.2021.45. Google Scholar | DOI

[15] Lee, Y.-P., ‘Quantum -theory, I: Foundations’, Duke Math. J. 121(3) (2004), 389–424. doi:10.1215/S0012-7094-04-12131-1. Google Scholar | DOI

[16] Lenart, C., ‘From Macdonald polynomials to a charge statistic beyond type ’, J. Combin. Theory Ser. A 119(3) (2012), 683–712. doi:10.1016/j.jcta.2011.11.013. Google Scholar | DOI

[17] Lenart, C. and Lubovsky, A., ‘A generalization of the alcove model and its applications’, J. Algebr. Comb. 41(3) (2015), 751–783. doi:10.1007/s10801-014-0552-3. Google Scholar | DOI

[18] Lenart, C. and Maeno, T., ‘Quantum Grothendieck polynomials’, . Google Scholar | arXiv

[19] Lenart, C., Naito, S., Orr, D. and Sagaki, D., ‘Inverse -Chevalley formulas for semi-infinite flag manifolds, II: arbitrary weights in ADE type’, Adv. Math. 423 (2023), Paper no. 109037, 63 pp. doi:10.1016/j.aim.2023.109037. Google Scholar | DOI

[20] Lenart, C., Naito, S. and Sagaki, D., ‘A general Chevalley formula for semi-infinite flag manifolds and quantum -theory’, Selecta Math. (N.S.) 30(3) (2024), Paper no. 39, 44 pp. doi:10.1007/s00029-024-00924-8. Google Scholar | DOI

[21] Maeno, T., Naito, S. and Sagaki, D., ‘A presentation of the torus-equivariant quantum -theory ring of flag manifolds of type , Part I: the defining ideal’, J. Lond. Math. Soc. (2) 111(3) (2025), Paper no. e70095, 43 pp. doi:10.1112/jlms.70095. Google Scholar | DOI

[22] Maeno, T., Naito, S. and Sagaki, D., ‘A presentation of the torus-equivariant quantum -theory ring of flag manifolds of type , Part II: quantum double Grothendieck polynomials’, Forum Math. Sigma 13(2025), Paper no. e19, 26 pp. doi:10.1017/fms.2024.147. Google Scholar | DOI

[23] Orr, D., ‘Equivariant -theory of the semi-infinite flag manifold as a nil-DAHA module’, Selecta Math. (N.S.) 29(3) (2023), Paper no. 45, 26 pp. doi:10.1007/s00029-023-00848-9. Google Scholar | DOI

[24] Pittie, H. and Ram, A., ‘A Pieri-Chevalley formula in the -theory of a -bundle’, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 102–107. doi:10.1090/S1079-6762-99-00067-0. Google Scholar | DOI

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