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A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions

Published online by Cambridge University Press:  20 November 2018

Chris Berg ,
Nantel Bergeron ,
Franco Saliola ,
Luis Serrano  and
Mike Zabrocki
Chris Berg
Affiliation:
Université du Québec à Montréal, Montréal, QC. e-mail: [email protected], [email protected], [email protected]
Nantel Bergeron
Affiliation:
Fields Institute, TorontoON York University, TorontoON. e-mail: [email protected], [email protected]
Franco Saliola
Affiliation:
Université du Québec à Montréal, Montréal, QC. e-mail: [email protected], [email protected], [email protected]
Luis Serrano
Affiliation:
Université du Québec à Montréal, Montréal, QC. e-mail: [email protected], [email protected], [email protected]
Mike Zabrocki
Affiliation:
Fields Institute, TorontoON York University, TorontoON. e-mail: [email protected], [email protected]
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Abstract

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We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions that expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall–Littlewood symmetric functions with similar properties to their commutative counterparts.

Keywords

05E05Hall-Littlewood polynomialsymmetric functionquasisymmetric functiontableau
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 525 - 565
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Aguiar, M., Bergeron, N., and Sottile, F., Combinatorial Hopf Algebras and generalized Dehn–Sommerville relations. Compositio Math. 142(2006), no. 1, 1–30. http://dx.doi.org/10.1112/S0010437X0500165X Google Scholar
[2] Berg, C., Bergeron, N., Saliola, F., Serrano, L. and Zabrocki, M., Indecomposable modules for the dual immaculate basis of quasi-symmetric functions. arxiv:1304.1224 Google Scholar
[3] Berg, C., Multiplicative structures of the immaculate basis of non-commutative symmetric functions. arxiv:1305.4700Google Scholar
[4] Bergeron, F., Garsia, A. M., Haiman, M., and Tesler, G., Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions. Methods Appl. Anal. 6(1999), no. 3, 363–420.Google Scholar
[5] Bergeron, N. and Zabrocki, M., q and q; t-Analogs of non-commutative symmetric functions. Discrete Math. 298(2005), no. 1–3, 79–103. http://dx.doi.org/10.1016/j.disc.2004.08.044 Google Scholar
[6] Bergeron, N., Mykytiuk, S., Sottile, F., and vanWilligenburg, S., Noncommutative Pieri operators on posets. J. Combin. Theory Ser. A 91(2000), no. 1–2, 84–110. http://dx.doi.org/10.1006/jcta.2000.3090 Google Scholar
[7] Bessenrodt, C., Luoto, K., and vanWilligenburg, S., Skew quasisymetric Schur functions and noncommutative Schur functions. Adv. Math. 226(2011), no. 5, 4492–4532. http://dx.doi.org/10.1016/j.aim.2010.12.015 Google Scholar
[8] Egge, E., Loehr, N., and Warrington, G., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix. European J. Combin. 31(2010), no. 8, 2014–2027. http://dx.doi.org/10.1016/j.ejc.2010.05.010 Google Scholar
[9] Garsia, A. M., Orthogonality of Milne's polynomials and raising operators. Discrete Math. 99(1992), no. 1–3, 247–264. http://dx.doi.org/10.1016/0012-365X(92)90375-P Google Scholar
[10] Gessel, I. M., Multipartite P-partitions and inner products of skew Schur functions. In: Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., 34, American Mathematical Society, Providence, RI, 1984, pp. 289–317.Google Scholar
[11] Gelfand, I. M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V. S., and Thibon, J.-Y., Noncommutative symmetric functions. Adv. Math. 112(1995), no. 2, 218–348. http://dx.doi.org/10.1006/aima.1995.1032 Google Scholar
[12] Gessel, I. M. and Reutenauer, C., Counting permutations with given cycle structure and descent set. J. Combin. Theory Ser. A 64(1993), no. 2, 189–215. http://dx.doi.org/10.1016/0097-3165(93)90095-P Google Scholar
[13] Haglund, J., Luoto, K., Mason, S., and vanWilligenburg, S., Quasisymmetric Schur functions. J. Combin. Theory Ser. A 118(2011), no. 2, 463–490. http://dx.doi.org/10.1016/j.jcta.2009.11.002 Google Scholar
[14] Haglund, J., Refinements of the Littlewood-Richardson rule. Trans. Amer. Math. Soc. 363(2011), no. 3, 1665–1686. http://dx.doi.org/10.1090/S0002-9947-2010-05244-4 Google Scholar
[15] Haglund, J., Morse, J., and Zabrocki, M., A compositional shuffle conjecture specifying touch points of the Dyck path. Canad. J. Math. 64(2012), no. 4, 822–844. http://dx.doi.org/10.4153/CJM-2011-078-4 Google Scholar
[16] Hivert, F., Analogues non-commutatifs et quasi-symétriques des fonctions de Hall-Littlewood et modules de Demazure d’une algèbre quantique dégénérée. C. R. Acad. Sci. Paris 362(1998), no. 1, 1–6. http://dx.doi.org/10.1016/S0764-4442(97)82703-6 Google Scholar
[17] Hoffman, P. and Humphreys, J. F., Projective representations of symmetric groups. Q-functions and shifted tableaux. Oxford Mathematical Monographs, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992.Google Scholar
[18] Jing, N. H., Vertex operators and Hall-Littlewood symmetric functions. Adv. Math. 87(1991), no. 2, 226–248. http://dx.doi.org/10.1016/0001-8708(91)90072-F Google Scholar
[19] Jing, N. H., Vertex operators, symmetric functions and the spin group Γn. J. Algebra 138(1991), no. 2, 340–398. http://dx.doi.org/10.1016/0021-8693(91)90177-A Google Scholar
[20] Kirillov, A. N. and Noumi, M., q-difference raising operators for Macdonald polynomials and the integrality of transition coefficients. In: Algebraic methods and q-special functions (Montr´eal, QC, 1996), CRM Proc. Lecture Notes, 22, American Mathematical Society, Providence, RI, 1999, pp. 227–243.Google Scholar
[21] Krob, D. and Thibon, J.-Y., Noncommutative symmetric functions IV. Quantum linear groups and Hecke algebras at q = 0. J. Algebraic Combin. 6(1997), no. 4, 339–376. http://dx.doi.org/10.1023/A:1008673127310 Google Scholar
[22] Lapointe, L. and Vinet, L., Rodrigues formulas for the Macdonald polynomials. Adv. Math. 130(1997), no. 2, 261–279. http://dx.doi.org/10.1006/aima.1997.1662 Google Scholar
[23] Lascoux, A., Novelli, J.-C., and Thibon, J.-Y., Noncommutative symmetric functions with matrix parameters. arxiv:1110.3209v1.Google Scholar
[24] Lascoux, A. and M.-P. Schützenberger, , Sur une conjecture de H. O. Foulkes. C. R. Acad. Sci. Paris Sèr. A-B 286(1978), no. 7, A323–A324.Google Scholar
[25] Macdonald, I. G., Symmetric functions and hall polynomials. Second ed., Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.Google Scholar
[26] Malvenuto, C., Produits et coproduits des fonctions quasi-symètriques et de l’algèbre des descents. Laboratoire de combinatoire et d’informatique mathèmatique (LACIM), Univ. du Quèbec à Montrèal, Montrèal, no. 16, 1994.Google Scholar
[27] Malvenuto, C. and Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(1995), no. 3, 967–982. http://dx.doi.org/10.1006/jabr.1995.1336 Google Scholar
[28] Novelli, J.-C., Thibon, J.-Y., and Williams, L, Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux. Adv. Math. 224(2010), no. 4, 1311–1348. http://dx.doi.org/10.1016/j.aim.2010.01.006 Google Scholar
[29] Sagan, B. E., The symmetric group: representations, combinatorial algorithms, and symmetric functions. second ed., Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001.Google Scholar
[30] Stein, W. A. et al., Sage Mathematics Software (Version 4.3.3). The Sage Development Team, 2010, http://www.sagemath.org. Google Scholar
[31] The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics. http://combinat.sagemath.org, 2008.Google Scholar
[32] Shimozono, M. and Zabrocki, M., Hall-Littlewood vertex operators and generalized Kostka polynomials. Adv. Math. 158(2001), no. 1, 66–85. http://dx.doi.org/10.1006/aima.2000.1964 Google Scholar
[33] Stanley, R. P., Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999.Google Scholar
[34] Stanley, R. P., On the number of reduced decompositions of elements of Coxeter group. European J. Combin. 5(1984), no. 4, 359–372.Google Scholar
[35] Tevlin, L., Noncommutative symmetric Hall-Littlewood polynomials. In: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, pp. 915–925 Google Scholar
[36] Zabrocki, M., q-analogs of symmetric function operators. Discrete Math. 256(2002), no. 3, 831–853. http://dx.doi.org/10.1016/S0012-365X(02)00350-3 Google Scholar
[37] Zelevinsky, A. V., Representations of finite classical groups. A Hopf algebra approach. Lecture Notes in Mathematics, 869, Springer-Verlag, Berlin-New York, 1981.Google Scholar