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A $$k$$-tableau characterization of $$k$$-Schur functions. (English) Zbl 1118.05096
Summary: We study $$k$$-Schur functions characterized by $$k$$-tableaux, proving combinatorial properties such as a $$k$$-Pieri rule and a $$k$$-conjugation. This new approach relies on developing the theory of $$k$$-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of $$k$$-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.

##### MSC:
 05E05 Symmetric functions and generalizations 05E10 Combinatorial aspects of representation theory 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 17B65 Infinite-dimensional Lie (super)algebras
##### Keywords:
Schur functions; tableaux; Gromov-Witten invariants
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##### References:
 [1] S. Agnihotri, Quantum cohomology and the Verlinde algebra, PhD thesis, Oxford University, 1995 [2] Bender, E.; Knuth, D., Enumeration of plane partitions, J. combin. theory ser. A, 13, 40-54, (1972) · Zbl 0246.05010 [3] Bégin, L.; Kirillov, A.; Mathieu, P.; Walton, M.A., Berenstein-Zelevinsky triangles, elementary couplings and fusion rules, Lett. math. phys., 28, 257-268, (1993) · Zbl 0811.17032 [4] Bégin, L.; Mathieu, P.; Walton, M.A., $$\hat{\mathit{su}}(3)_k$$ fusion coefficients, Modern phys. lett. A, 7, 35, 3255-3265, (1995) · Zbl 1021.81530 [5] Buch, A.; Kresch, A.; Tamvakis, H., Gromov-Witten invariants on Grassmannians, J. amer. math. soc., 16, 4, 901-915, (2003) · Zbl 1063.53090 [6] Garsia, A.M.; Haiman, M., A graded representation module for Macdonald’s polynomials, Proc. natl. acad. sci. USA, 90, 3607-3610, (1993) · Zbl 0831.05062 [7] Garsia, A.M.; Procesi, C., On certain graded $$S_n$$-modules and the q-kostka polynomials, Adv. math., 87, 82-138, (1992) · Zbl 0797.20012 [8] Goodman, F.; Wenzl, H., Littlewood-Richardson coefficients for Hecke algebras at roots of unity, Adv. math., 82, 244-265, (1990) · Zbl 0714.20004 [9] Haglund, J., A combinatorial model for the Macdonald polynomials, Proc. natl. acad. sci. USA, 101, 16127-16131, (2004) · Zbl 1064.05147 [10] Haglund, J.; Haiman, M.; Loehr, N., A combinatorial formula for Macdonald polynomials, J. amer. math. soc., 18, 3, 735-761, (2005) · Zbl 1061.05101 [11] Haiman, M., Hilbert schemes, polygraphs, and the Macdonald positivity conjecture, J. amer. math. soc., 14, 941-1006, (2001) · Zbl 1009.14001 [12] James, G.D.; Kerber, A., The representation theory of the symmetric group, Encyclopedia math. appl., vol. 16, (1981), Addison-Wesley Reading, MA [13] Kirrilov, A.N.; Mathieu, P.; Sénéchal, D.; Walton, M.A., Can fusion coefficients be calculated from the depth rule?, Nuclear phys., 391, 3, 651-674, (1993) [14] Knutson, A.; Tao, T.; Woodward, C., The honeycomb model of $$\mathit{GL}(n)$$ tensor products II: puzzles determine facets of the Littlewood-Richardson cone, J. amer. math. soc., 17, 1, 19-48, (2004) · Zbl 1043.05111 [15] Lam, T., Affine Stanley symmetric functions, arXiv: · Zbl 1107.05095 [16] Lam, T., Schubert polynomials for the affine Grassmannian, arXiv: · Zbl 1149.05045 [17] Lapointe, L.; Morse, J., Schur function analogs and a filtration for the symmetric function space, J. combin. theory ser. A, 101, 2, 191-224, (2003) · Zbl 1018.05101 [18] Lapointe, L.; Morse, J., Schur function identities, their t-analogs, and k-Schur irreducibility, Adv. math., 180, 222-247, (2003) · Zbl 1031.05129 [19] Lapointe, L.; Morse, J., Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures, Ann. comb., 8, 197-219, (2004) · Zbl 1063.06001 [20] Lapointe, L.; Morse, J., Tableaux on $$k + 1$$-cores, reduced words for affine permutations, and k-Schur expansions, J. combin. theory ser. A, 112, 1, 44-81, (2005) · Zbl 1120.05093 [21] Lapointe, L.; Morse, J., Quantum cohomology and the k-Schur basis, Trans. Amer. Math. Soc., in press, arXiv: · Zbl 1132.05060 [22] Lapointe, L.; Lascoux, A.; Morse, J., Tableau atoms and a new Macdonald positivity conjecture, Duke math. J., 116, 1, 103-146, (2003) · Zbl 1020.05069 [23] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Clarendon Press Oxford · Zbl 0487.20007 [24] Schilling, A.; Shimozono, M., Bosonic formula for level restricted paths, (), 305-325 · Zbl 1058.17500 [25] Schilling, A.; Shimozono, M., Fermionic formulas for level-restricted generalized kostka polynomials and coset branching functions, Comm. math. phys., 220, 105-164, (2001) · Zbl 0992.05073 [26] Tsuchiya, A.; Ueno, K.; Yamada, Y., Conformal field theory on universal family of stable curves with gauge symmetries, (), 459-566 [27] Tudose, G., A special case of $$\mathit{sl}(n)$$-fusion coefficients, arXiv: [28] Witten, E., The Verlinde algebra and the cohomology of the Grassmanian, (), 357-422 · Zbl 0863.53054
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