A combinatorial study on K-theoretic k-Schur functions and stable Grothendieck polynomials

[Abe80] Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. MR594432

[ACT] David Anderson, Linda Chen, and Hsian-Hua Tseng, On the quantum K-ring of the flag manifold. arXiv:1711.08414.

[BB05] Anders Bj¨orner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR2133266

[BM15] Anders S. Buch and Leonardo C. Mihalcea, Curve neighborhoods of schubert varieties, J. Di↵erential Geom. 99 (2015), no. 2, 255–283.

[Buc02] Anders Skovsted Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), no. 1, 37–78. MR1946917

[BW82] Anders Bj¨orner and Michelle Wachs, Bruhat order of Coxeter groups and shellability, Adv. in Math. 43 (1982), no. 1, 87–100. MR644668

[BW88] Anders Bj¨orner and Michelle L. Wachs, Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc. 308 (1988), no. 1, 1–37. MR946427

[Den12] Tom Denton, Canonical decompositions of ane permutations, ane codes, and split k-Schur functions, Electron. J. Combin. 19 (2012), no. 4, Paper 19, 41. MR3001656

[Deo87] Vinay V. Deodhar, A splitting criterion for the bruhat orderings on coxeter groups, Communications in Algebra 15 (1987), no. 9, 1889–1894.

[FK96] Sergey Fomin and Anatol N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), 1996, pp. 123–143. MR1394950

[GR] Darij Grinberg and Victor Reiner, Hopf algebras in combinatorics. arXiv:1409.8356.

[Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR1066460

[IIM18] Takeshi Ikeda, Shinsuke Iwao, and Toshiaki Maeno, Peterson isomorphism in K-theory and relativistic Toda lattice, International Mathematics Research Notices (2018), rny051, available at /oup/backfile/content_public/journal/ imrn/pap/10.1093_imrn_rny051/2/rny051.pdf.

[Kat] Syu Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds. arXiv:1805.01718.

[KM04] Allen Knutson and Ezra Miller, Subword complexes in coxeter groups, Advances in Mathematics 184 (2004), no. 1, 161 –176.

[Lam06] Thomas Lam, Ane Stanley symmetric functions, Amer. J. Math. 128 (2006), no. 6, 1553–1586. MR2275911

[Lam08] , Schubert polynomials for the ane Grassmannian, J. Amer. Math. Soc. 21 (2008), no. 1, 259–281.

[Len00] Cristian Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb. 4 (2000), no. 1, 67–82. MR1763950

[LLM03] Luc Lapointe, Alain Lascoux, and Jennifer Morse, Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003), no. 1, 103–146.

[LLMS10] Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono, Ane insertion and Pieri rules for the ane Grassmannian, Mem. Amer. Math. Soc. 208 (2010), no. 977, xii+82.

[LLMS18] Thomas Lam, Changzheng Li, Leonardo C. Mihalcea, and Mark Shimozono, A conjectural Peterson isomorphism in K-theory, J. Algebra 513 (2018), 326–343. MR3849889

[LLM+14] Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, and Mike Zabrocki, k-Schur functions and ane Schubert calculus, Fields Institute Monographs, vol. 33, 2014.

[LM04] L. Lapointe and J. Morse, Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures, Ann. Comb. 8 (2004), no. 2, 197–219.

[LM05] Luc Lapointe and Jennifer Morse, Tableaux on k + 1-cores, reduced words for ane permutations, and k-Schur expansions, J. Combin. Theory Ser. A 112 (2005), no. 1, 44–81.

[LM07] , A k-tableau characterization of k-Schur functions, Adv. Math. 213 (2007), no. 1, 183–204.

[LP07] Thomas Lam and Pavlo Pylyavskyy, Combinatorial Hopf algebras and K-homology of Grassmannians, Int. Math. Res. Not. IMRN 24 (2007), Art. ID rnm125, 48. MR2377012

[LS10] Thomas Lam and Mark Shimozono, Quantum cohomology of G/P and homology of ane Grassmannian, Acta Math. 204 (2010), no. 1, 49–90. MR2600433

[LS12] , From quantum Schubert polynomials to k-Schur functions via the Toda lattice, Math. Res. Lett. 19 (2012), no. 1, 81–93.

[LS82] Alain Lascoux and Marcel-Paul Sch¨utzenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 11, 629–633. MR686357

[LSS10] Thomas Lam, Anne Schilling, and Mark Shimozono, K-theory Schubert calculus of the ane Grassmannian, Compos. Math. 146 (2010), no. 4, 811–852.

[Mac95] Ian G. Macdonald, Symmetric functions and Hall polynomials, Second, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky, Oxford Science Publications.

[Mor12] Jennifer Morse, Combinatorics of the K-theory of ane Grassmannians, Adv. Math. 229 (2012), no. 5, 2950–2984.

[Shi] Mark Shimozono, private communication. 2018.

[Sta12] Richard P. Stanley, Enumerative combinatorics. Volume 1, Second, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR2868112

[Ste07] John R. Stembridge, A short derivation of the M¨obius function for the Bruhat order, J. Algebraic Combin. 25 (2007), no. 2, 141–148. MR2310418

[Swe69] Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR0252485

[Taka] Motoki Takigiku, Factorization formulas of K-k-Schur functions I. arXiv:1704.08643.

[Takb] , Factorization formulas of K-k-Schur functions II. arXiv:1704.08660.

[Wau99] Debra J. Waugh, Upper bounds in ane Weyl groups under the weak order, Order 16 (1999), no. 1, 77–87 (2000). MR1740744

[Yel] Damir Yeliussizov, Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs. arXiv:1711.09544.