# zbMATH — the first resource for mathematics

A conjectural Peterson isomorphism in $$K$$-theory. (English) Zbl 1423.14278
Summary: We state a precise conjectural isomorphism between localizations of the equivariant quantum $$K$$-theory ring of a flag variety and the equivariant $$K$$-homology ring of the affine Grassmannian, in particular relating their Schubert bases and structure constants. This generalizes Peterson’s isomorphism in (co)homology. We prove a formula for the Pontryagin structure constants in the $$K$$-homology ring, and we use it to check our conjecture in few situations.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 19L47 Equivariant $$K$$-theory
Full Text:
##### References:
 [1] Anderson, D.; Chen, L.; Tseng, H.-H., On the quantum K-ring of the flag manifold [2] Anderson, D.; Chen, L.; Tseng, H.-H., The quantum K-theory of a homogeneous space is finite [3] Anderson, D.; Griffeth, S.; Miller, E., Positivity and kleiman transversality in equivariant K-theory of homogeneous spaces, J. Eur. Math. Soc. (JEMS), 13, 1, 57-84, (2011) · Zbl 1213.19003 [4] Braverman, A.; Finkelberg, M., Semi-infinite Schubert varieties and quantum K-theory of flag manifolds, J. Amer. Math. Soc., 27, 1147-1168, (2014) · Zbl 1367.17011 [5] Baldwin, S.; Kumar, S., Positivity in T-equivariant K-theory of flag variaties associated to Kac-Moody groups, II, Represent. Theory, 21, 35-60, (2017) · Zbl 1390.19010 [6] Brion, M., Lectures on the geometry of flag varieties, topics in cohomological studies of algebraic varieties, Trends Math., 33-85, (2005), Birkhäuser Basel [7] Buch, A. S.; Mihalcea, L. C., Quantum K-theory of Grassmannians, Duke Math. J., 156, 3, 501-538, (2011) · Zbl 1213.14103 [8] Brion, M.; Kumar, S., Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, (2005), Birkhäuser Boston, Inc. Boston, MA · Zbl 1072.14066 [9] Buch, A. S.; Mihalcea, L. C., Curve neighborhoods of Schubert varieties, J. Differential Geom., 99, 2, 255-283, (2015) · Zbl 06423472 [10] Buch, A. S.; Chaput, P. E.; Mihalcea, L. C.; Perrin, N., Finiteness of cominuscule quantum K-theory, Ann. Sci. Éc. Norm. Supér. (4), 46, 3, 477-494, (2013) · Zbl 1282.14016 [11] Buch, A. S.; Chaput, P. E.; Mihalcea, L. C.; Perrin, N., Rational connectedness implies finiteness of quantum K-theory, Asian J. Math., 20, 1, 117-122, (2016) · Zbl 1364.14044 [12] Buch, A. S.; Chaput, P. E.; Mihalcea, L. C.; Perrin, N., A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties [13] Chaput, P. E.; Perrin, N., Rationality of some Gromov-Witten varieties and applications to quantum K theory, Commun. Contemp. Math., 13, 1, 67-90, (2011) · Zbl 1237.14058 [14] Chriss, N.; Ginzburg, V., Representation theory and complex geometry, (2010), Birkhäuser Boston, Inc. Boston, MA · Zbl 1185.22001 [15] Givental, A., On the WDVV equation in quantum K-theory, dedicated to william fulton on the occasion of his 60th birthday, Michigan Math. J., 48, 295-304, (2000) [16] Givental, A.; Kim, B., Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys., 168, 3, 609-641, (1995) · Zbl 0828.55004 [17] Givental, A.; Lee, Y.-P., Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups, Invent. Math., 151, 1, 193-219, (2003) · Zbl 1051.14063 [18] He, X.; Lam Thomas, T., Projected Richardson varieties and affine Schubert varieties, Ann. Inst. Fourier (Grenoble), 65, 6, 2385-2412, (2015) · Zbl 1350.14035 [19] Ikeda, T.; Iwao, S.; Maeno, T., Peterson isomorphism in K-theory and relativistic Toda lattice [20] Iritani, H.; Milanov, T.; Tonita, V., Reconstruction and convergence in quantum K-theory via difference equations, Int. Math. Res. Not. IMRN, 11, 2887-2937, (2015) · Zbl 1353.14064 [21] Kashiwara, M., The flag manifold of Kac-Moody Lie algebra, (Algebraic Analysis, Geometry, and Number Theory, Baltimore, MD, 1988, (1989), Johns Hopkins Univ. Press Baltimore, MD), 161-190 · Zbl 0764.17019 [22] Kashiwara, M.; Shimozono, M., Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials, Duke Math. J., 148, 3, 501-538, (2009) · Zbl 1173.19004 [23] Kato, S., Loop structure on equivariant K-theory of semi-infinite flag manifolds, preprint available on: [24] Kim, B., Quantum cohomology of flag manifolds $$G / B$$ and quantum Toda lattices, Ann. of Math. (2), 149, 1, 129-148, (1999) · Zbl 1054.14533 [25] Koroteev, P.; Pushkar, P. P.; Smirnov, A.; Zeitlin, A. M., Quantum K-theory of quiver varieties and many-body systems [26] Kostant, B.; Kumar, S., T-equivariant K-theory of generalized flag varieties, J. Differential Geom., 32, 2, 549-603, (1990) · Zbl 0731.55005 [27] Kumar, S., Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, (2002), Birkhäuser Boston, Inc. Boston, MA · Zbl 1026.17030 [28] Kumar, S., Positivity in T-equivariant K-theory of flag variaties associated to Kac-Moody groups, J. Eur. Math. Soc. (JEMS), (2018), in press [29] Kumar, S.; Schwede, K., Richardson varieties have Kawamata log-terminal singularities, Int. Math. Res. Not., 2014, 3, 842-864, (2014) [30] Lam, T.; Schilling, A.; Shimozono, M., K-theory Schubert calculus of the affine Grassmannian, Compos. Math., 146, 4, 811-852, (2010) · Zbl 1256.14056 [31] Lam, T.; Shimozono, M., Quantum cohomology of $$G / P$$ and homology of affine Grassmannian, Acta Math., 204, 1, 49-90, (2010) · Zbl 1216.14052 [32] Lee, Y.-P., Quantum K-theory. I. foundations, Duke Math. J., 121, 3, 389-424, (2004) · Zbl 1051.14064 [33] Lee, Y.-P.; Pandharipande, R., A reconstruction theorem in quantum cohomology and quantum K-theory, Amer. J. Math., 126, 6, 1367-1379, (2004) · Zbl 1080.14065 [34] Lenart, C.; Maeno, T., Quantum Grothendieck polynomials [35] Leung, N. C.; Li, C., Gromov-Witten invariants for $$G / B$$ and Pontryagin product for ωK, Trans. Amer. Math. Soc., 5, 2567-2599, (2012) · Zbl 1248.14061 [36] Li, C.; Mihalcea, L. C., K-theoretic Gromov-Witten invariants of lines in homogeneous spaces, Int. Math. Res. Not., 2014, 17, 4625-4664, (2014) · Zbl 1326.14132 [37] Mihalcea, L. C., On equivariant quantum cohomology of homogeneous spaces: Chevalley formulas and algorithms, Duke Math. J., 140, 2, 321-350, (2007) · Zbl 1135.14042 [38] Peterson, D., Quantum cohomology of $$G / P$$, Lecture Notes at MIT, (1997)
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.