×

zbMATH — the first resource for mathematics

Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials. (English) Zbl 1278.05241
Summary: We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials \(P_{{\lambda}/{\mu}}(t)\) and Hivert’s quasisymmetric Hall-Littlewood polynomials \(G_{\gamma}(t)\). More specifically, we provide: 0.5 cm
1.
the \(G\)-expansions of the Hall-Littlewood polynomials \(P_\lambda(t)\), the monomial quasisymmetric polynomials \(M_{\alpha}\), the quasisymmetric Schur polynomials \(S_{\alpha}\), and the peak quasisymmetric functions \(K_{\alpha}\);
2.
an expansion of \(P_{{\lambda}/{\mu}}(t)\) in terms of the \(F_{\alpha}\)’s.
The \(F\)-expansion of \(P_{{\lambda}/{\mu}}(t)\) is facilitated by introducing starred tableaux.

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Berg, C.; Bergeron, N.; Saliola, F.; Serrano, L.; Zabrocki, M., A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Canad. J. Math., (2013), in press
[2] Bergeron, N.; Zabrocki, M., q and \(q, t\)-analogs of non-commutative symmetric functions, Discrete Math., 298, 79-103, (2005) · Zbl 1070.05080
[3] Butler, L., Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc., 112, (1994) · Zbl 0813.05067
[4] Carbonara, J. O., A combinatorial interpretation of the inverse t-kostka matrix, Discrete Math., 193, 117-145, (1998), Selected papers in honor of Adriano Garsia (Taormina, 1994) · Zbl 1061.05504
[5] Désarménien, J.; Leclerc, B.; Thibon, J.-Y., Hall-Littlewood functions and kostka-foulkes polynomials in representation theory, Sém. Lothar. Combin., 32, (1994), Art. B32c, approx. 38 pp · Zbl 0855.05100
[6] Eğecioğlu, O.; Remmel, J., A combinatorial interpretation of the inverse kostka matrix, Linear Multilinear Algebra, 26, 59-84, (1990) · Zbl 0735.05013
[7] Egge, E.; Loehr, N.; Warrington, G., From quasisymmetric expansions to Schur expansions via a modified inverse kostka matrix, European J. Combin., 31, 2014-2027, (2010) · Zbl 1219.05195
[8] Garsia, A., Orthogonality of milneʼs polynomials and raising operators, Discrete Math., 99, 247-264, (1992) · Zbl 0763.05100
[9] Gelfand, I. M.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V. S.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. Math., 112, 218-348, (1995) · Zbl 0831.05063
[10] Gessel, I. M., Multipartite P-partitions and inner products of skew Schur functions, (Combinatorics and Algebra, Boulder, CO, 1983, Contemp. Math., vol. 34, (1984), Amer. Math. Soc. Providence, RI), 289-317
[11] Haglund, J., A combinatorial model for the Macdonald polynomials, Proc. Natl. Acad. Sci. USA, 101, 16 127-16 131, (2004), (electronic) · Zbl 1064.05147
[12] Haglund, J.; Haiman, M.; Loehr, N., A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc., 102, 2690-2696, (2005) · Zbl 1208.05149
[13] Haglund, J.; Luoto, K.; Mason, S.; van Willigenburg, S., Quasisymmetric Schur functions, J. Combin. Theory Ser. A, 118, 463-490, (2011) · Zbl 1229.05270
[14] Haiman, M. D., On mixed insertion, symmetry, and shifted Young tableaux, J. Combin. Theory Ser. A, 50, 196-225, (1989) · Zbl 0697.05005
[15] Hivert, F., Hecke algebras, difference operators, and quasi-symmetric functions, Adv. Math., 155, 181-238, (2000) · Zbl 0990.05129
[16] Kirillov, A., New combinatorial formula for modified Hall-Littlewood polynomials (in q-series from a contemporary perspective), Contemp. Math., 254, 283-333, (2000) · Zbl 0956.05101
[17] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Ribbon tableaux, Hall-Littlewood functions and unipotent varieties, Sém. Lothar. Combin., 34, (1995), Art. B34g, approx. 23 pp. (electronic) · Zbl 0855.05099
[18] Lascoux, A.; Novelli, J.-C.; Thibon, J.-Y., Noncommutative symmetric functions with matrix parameters, J. Algebraic Combin., 37, 621-642, (2013) · Zbl 1268.05230
[19] Lascoux, A.; Schützenberger, M.-P., Sur une conjecture de H. O. foulkes, C. R. Acad. Sci. Paris Sér. A-B, 286A, 323-324, (1978) · Zbl 0374.20010
[20] Littlewood, D. E., On certain symmetric functions, Proc. Lond. Math. Soc. (3), 11, 485-498, (1961) · Zbl 0099.25102
[21] Loehr, N.; Serrano, L.; Warrington, G., Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials, (2013) · Zbl 1278.05241
[22] Loehr, N.; Warrington, G., Nested quantum Dyck paths and \(\operatorname{\nabla}(s_\lambda)\), Intl. Math. Research Notices, 2008, (2008), article ID rnm157, 29 pp · Zbl 1159.33002
[23] Loehr, N.; Warrington, G., Quasisymmetric expansions of Schur-function plethysms, Proc. Amer. Math. Soc., 140, 1159-1171, (2012) · Zbl 1239.05192
[24] Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Math. Monogr., (1995), Oxford University Press New York, with contributions by A. Zelevinsky, Oxford Science Publications · Zbl 0899.05068
[25] Novelli, J.-C.; Thibon, J.-Y.; Williams, L. K., Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux, Adv. Math., 224, 1311-1348, (2010) · Zbl 1236.05202
[26] Sagan, B. E., Shifted tableaux, Schur Q-functions, and a conjecture of R.P. Stanley, J. Combin. Theory Ser. A, 45, 62-103, (1987) · Zbl 0661.05010
[27] Stembridge, J. R., Shifted tableaux and the projective representations of symmetric groups, Adv. Math., 74, 87-134, (1989) · Zbl 0677.20012
[28] Stembridge, J. R., Enriched P-partitions, Trans. Amer. Math. Soc., 349, 763-788, (1997) · Zbl 0863.06005
[29] Tevlin, L., Noncommutative analogs of monomial symmetric functions, Cauchy identity, and Hall scalar product, (2007)
[30] Tevlin, L., Noncommutative symmetric Hall-Littlewood polynomials, (23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2011, Discrete Math. Theor. Comput. Sci. Proc., AO, (2011), Assoc. Discrete Math. Theor. Comput. Sci. Nancy), 915-925 · Zbl 1355.05262
[31] Warnaar, S. O.; Zudilin, W., Dedekindʼs η-function and Rogers-Ramanujan identities, Bull. Lond. Math. Soc., 44, 1-11, (2012) · Zbl 1234.05040
[32] Warrington, G. S., Sage worksheet for Hall-Littlewood transition matrices, (2013), available online at:
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.