A Solomon descent theory for the wreath products 𝐺≀𝔖_{𝔫}

Baumann, Pierre; Hohlweg, Christophe

doi:10.1090/S0002-9947-07-04237-7

A Solomon descent theory for the wreath products $G\wr \mathfrak S_n$

Authors: Pierre Baumann and Christophe Hohlweg
Journal: Trans. Amer. Math. Soc. 360 (2008), 1475-1538
MSC (2000): Primary 16S99; Secondary 05E05, 05E10, 16S34, 16W30, 20B30, 20E22
DOI: https://doi.org/10.1090/S0002-9947-07-04237-7
Published electronically: October 22, 2007
MathSciNet review: 2357703
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Abstract: We propose an analogue of Solomon’s descent theory for the case of a wreath product $G\wr \mathfrak S_n$, where $G$ is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht’s theory for the representations of wreath products, Okada’s extension to wreath products of the Robinson-Schensted correspondence, and Poirier’s quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.

Ron M. Adin, Francesco Brenti, and Yuval Roichman, Descent representations and multivariate statistics, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3051–3082. MR 2135735, DOI https://doi.org/10.1090/S0002-9947-04-03494-4
Marcelo Aguiar, Nantel Bergeron, and Kathryn Nyman, The peak algebra and the descent algebras of types $B$ and $D$, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2781–2824. MR 2052597, DOI https://doi.org/10.1090/S0002-9947-04-03541-X
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 https://doi.org/10.1112/S0010437X0500165X
M. D. Atkinson, A new proof of a theorem of Solomon, Bull. London Math. Soc. 18 (1986), no. 4, 351–354. MR 838800, DOI https://doi.org/10.1112/blms/18.4.351
J.-C. Aval, F. Bergeron, and N. Bergeron, Diagonal Temperley-Lieb invariants and harmonics, Sém. Lothar. Combin. 54A (2005/07), Art. B54Aq, 19. MR 2264937

E. Bagno and R. Biagioli, Colored-descent representations of complex reflection groups $G(r,p,n)$, preprint arXiv:math.CO/0503238.

F. Bergeron, N. Bergeron, R. B. Howlett, and D. E. Taylor, A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin. 1 (1992), no. 1, 23–44. MR 1162640, DOI https://doi.org/10.1023/A%3A1022481230120

Dieter Blessenohl, Christophe Hohlweg, and Manfred Schocker, A symmetry of the descent algebra of a finite Coxeter group, Adv. Math. 193 (2005), no. 2, 416–437. MR 2137290, DOI https://doi.org/10.1016/j.aim.2004.05.007

D. Blessenohl and M. Schocker, Noncommutative character theory of the symmetric group. River Edge: World Scientific, 2005.

Cédric Bonnafé and Christophe Hohlweg, Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 1, 131–181 (English, with English and French summaries). With an appendix by Pierre Baumann and Hohlweg. MR 2228684

Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548

Gérard Duchamp, Florent Hivert, and Jean-Yves Thibon, Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput. 12 (2002), no. 5, 671–717. MR 1935570, DOI https://doi.org/10.1142/S0218196702001139

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 https://doi.org/10.1006/aima.1995.1032

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 https://doi.org/10.1090/conm/034/777705

Florent Hivert, Jean-Christophe Novelli, and Jean-Yves Thibon, Yang-Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki-Koike-Shoji algebras, Adv. Math. 205 (2006), no. 2, 504–548. MR 2258265, DOI https://doi.org/10.1016/j.aim.2005.07.016

Armin Jöllenbeck, Nichtkommutative Charaktertheorie der symmetrischen Gruppen, Bayreuth. Math. Schr. 56 (1999), 1–41 (German, with English summary). MR 1717091

A. Jöllenbeck and C. Reutenauer, Eine Symmetrieeigenschaft von Solomons Algebra und der höheren Lie-Charaktere, Abh. Math. Sem. Univ. Hamburg 71 (2001), 105–111 (German, with English summary). MR 1872717, DOI https://doi.org/10.1007/BF02941465

R. Kilmoyer, Some irreducible complex representations of a finte group with a $BN$-pair, Ph.D. dissertation, M.I.T., Cambridge, 1969.

Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727. MR 272654

Jean-Louis Loday and María O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), no. 2, 293–309. MR 1654173, DOI https://doi.org/10.1006/aima.1998.1759

I. G. Macdonald, Polynomial functors and wreath products, J. Pure Appl. Algebra 18 (1980), no. 2, 173–204. MR 585222, DOI https://doi.org/10.1016/0022-4049%2880%2990128-0

I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144

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 https://doi.org/10.1006/jabr.1995.1336

R. Mantaci and C. Reutenauer, A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products, Comm. Algebra 23 (1995), no. 1, 27–56. MR 1311773, DOI https://doi.org/10.1080/00927879508825205

Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637

Warren D. Nichols, Bialgebras of type one, Comm. Algebra 6 (1978), no. 15, 1521–1552. MR 506406, DOI https://doi.org/10.1080/00927877808822306

J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level, preprint arXiv:math.CO/0405597.

Soichi Okada, Wreath products by the symmetric groups and product posets of Young’s lattices, J. Combin. Theory Ser. A 55 (1990), no. 1, 14–32. MR 1070012, DOI https://doi.org/10.1016/0097-3165%2890%2990044-W

Stéphane Poirier, Cycle type and descent set in wreath products, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995), 1998, pp. 315–343. MR 1603753, DOI https://doi.org/10.1016/S0012-365X%2897%2900123-4

Stéphane Poirier and Christophe Reutenauer, Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec 19 (1995), no. 1, 79–90 (French, with English and French summaries). MR 1334836

Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799

Marc Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998), no. 2, 399–416. MR 1632802, DOI https://doi.org/10.1007/s002220050249

Louis Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), no. 2, 255–264. MR 444756, DOI https://doi.org/10.1016/0021-8693%2876%2990182-4

John R. Stembridge, Enriched $P$-partitions, Trans. Amer. Math. Soc. 349 (1997), no. 2, 763–788. MR 1389788, DOI https://doi.org/10.1090/S0002-9947-97-01804-7

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

Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099

Andrey V. Zelevinsky, Representations of finite classical groups, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR 643482