zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[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
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.