Indecomposable modules for the dual immaculate basis of quasi-symmetric functions

Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike

doi:10.1090/S0002-9939-2014-12298-2

Indecomposable modules for the dual immaculate basis of quasi-symmetric functions
HTML articles powered by AMS MathViewer

by Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano and Mike Zabrocki

Proc. Amer. Math. Soc. 143 (2015), 991-1000

DOI: https://doi.org/10.1090/S0002-9939-2014-12298-2

Published electronically: October 28, 2014

PDF | Request permission

Abstract:

We construct indecomposable modules for the $0$-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.

References

Marcelo Aguiar, Nantel Bergeron, and Frank Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), no. 1, 1–30. MR 2196760, DOI 10.1112/S0010437X0500165X
C. Berg, N. Bergeron, F. Saliola, L. Serrano and M. Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, arXiv:1208.5191, to appear, Canadian Journal of Mathematics, doi:10.4153/CJM-2013-013-0.
C. Berg, N. Bergeron, F. Saliola, L. Serrano and M. Zabrocki, Multiplicative structures of the immaculate basis of non-commutative symmetric functions, arXiv:1305.4700.
Gérard Duchamp, Daniel Krob, Bernard Leclerc, and Jean-Yves Thibon, Fonctions quasi-symétriques, fonctions symétriques non commutatives et algèbres de Hecke à $q=0$, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 2, 107–112 (French, with English and French summaries). MR 1373744
Ira M. Gessel, Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR 777705, DOI 10.1090/conm/034/777705
Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, and Jean-Yves Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), no. 2, 218–348. MR 1327096, DOI 10.1006/aima.1995.1032
Ira M. Gessel and Christophe Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993), no. 2, 189–215. MR 1245159, DOI 10.1016/0097-3165(93)90095-P
N. Jacobson, Basic algebra 2, (2nd ed.), Dover.
Daniel Krob and Jean-Yves Thibon, Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at $q=0$, J. Algebraic Combin. 6 (1997), no. 4, 339–376. MR 1471894, DOI 10.1023/A:1008673127310
Clauda Malvenuto and Christophe Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967–982. MR 1358493, DOI 10.1006/jabr.1995.1336
P. N. Norton, $0$-Hecke algebras, J. Austral. Math. Soc. Ser. A 27 (1979), no. 3, 337–357. MR 532754
Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028, DOI 10.1007/978-1-4757-6804-6
W. A. Stein et al. Sage Mathematics Software (Version 4.3.3), The Sage Development Team, 2010, http://www.sagemath.org.
The Sage-Combinat community. Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, http://combinat. sagemath.org, 2008.
Richard P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), no. 4, 359–372. MR 782057, DOI 10.1016/S0195-6698(84)80039-6
Jean-Yves Thibon, Lectures on noncommutative symmetric functions, Interaction of combinatorics and representation theory, MSJ Mem., vol. 11, Math. Soc. Japan, Tokyo, 2001, pp. 39–94. MR 1862149
J. Y. Thibon, Introduction to noncommutative symmetric functions, From Numbers and Languages to (Quantum) Cryptography, NATO Security through Science Series: Information and Communication Security, Volume 7.