The (1 -\mathbb E) (1-\mathbb{E})-transform in combinatorial Hopf algebras
Florent Hivert
, Jean-Gabriel Luque
, Jean-Christophe Novelli
and Jean-Yves Thibon
DOI: 10.1007/s10801-010-0245-5
Abstract
We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1 -\mathbb E) (1-\mathbb{E})-transform, where \mathbb E \mathbb{E} is the “exponential alphabet,” whose elementary symmetric functions are e n=\frac1 n! e_{n}=\frac{1}{n!}. In the case of noncommutative symmetric functions, we recover Schocker's idempotents for derangement numbers (Schocker, Discrete Math. 269:239-248, 2003). From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon-Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.
Pages: 277–312
Keywords: keywords combinatorial Hopf algebras; symmetric functions; derangements
Full Text: PDF
References
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2. Bidigare, P., Hanlon, P., Rockmore, D.: A combinatorial description of the spectrum of the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99, 135-174 (1999)
3. Bergeron, N., Hivert, F., Thibon, J.-Y.: The peak algebra and the Hecke-Clifford algebras at q =
0. J. Comb. Theory A 117, 1-19 (2004)
4. Blessenohl, D., Laue, H.: The module structure of Solomon's descent algebra. J. Aust. Math. Soc. 72(3), 317-333 (2002)
5. Désarménien, J.: Une autre interprétation du nombre de dérangements. Sémin. Lothar. Comb. 8, 11- 16 (1984)
6. Désarménien, J., Wachs, M.: Descentes sur les dérangements et mots circulaires. Sémin. Lothar. Comb. 19, 13-21 (1988)
7. Désarménien, J., Wachs, M.: Descent classes of permutations with a given number of fixed points. J. Comb. Theory Ser. A 64, 311-328 (1993)
8. Deutsch, E., Shapiro, L.: A survey of the Fine numbers. Discrete Math. 241, 241-265 (2001)
9. Duchamp, G., Hivert, F., Thibon, J.-Y.: Noncommutative symmetric functions VI: free quasisymmetric functions and related algebras. Int. J. Algebra Comput. 12, 671-717 (2002)
10. Duchamp, G., Hivert, F., Novelli, J.-C., Thibon, J.-Y.: Noncommutative symmetric functions VII: free quasi-symmetric functions revisited,
11. Duchamp, G., Klyachko, A., Krob, D., Thibon, J.-Y.: Noncommutative symmetric functions III: De- formations of Cauchy and convolution algebras. Discrete Math. Theor. Comput. Sci. 1, 159-216 (1997)
12. Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V.S., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218-348 (1995)
13. Hivert, F., Novelli, J.-C., Thibon, J.-Y.: The algebra of binary search trees. Theor. Comput. Sci. 339, 129-165 (2005)
14. Lothaire, N.: Combinatorics on Words. Cambridge University Press, Cambridge (1997) J Algebr Comb (2011) 33: 277-312
15. Garsia, A.M., Reutenauer, C.: A decomposition of Solomon's descent algebra. Adv. Math. 77, 189- 262 (1989)
16. Garsia, A.M., Wallach, N.: r -QSym is free over Sym. J. Comb. Theory A 114, 704-732 (2007)
17. Gessel, I., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Comb. Theory A 64, 189-215 (1993)
18. Hivert, F.: Combinatoire des fonctions quasi-symétriques. Thèse de Doctorat, Marne-La-Vallée (1999)
19. Krob, D., Leclerc, B., Thibon, J.-Y.: Noncommutative symmetric functions II: Transformations of alphabets. Int. J. Algebra Comput. 7, 181-264 (1997)
20. Lascoux, A.: Symmetric Functions and Combinatorial Operators on Polynomials. CBMS Regional Conference Series in Mathematics, vol.
99. American Math. Soc., Providence (2003), xii+268 pp.
21. Lascoux, A., Schützenberger, M.P.: Formulaire Raisonné de Fonctions Symétriques. Publ. Math. Univ. Paris 7, Paris (1985)
22. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
23. Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967-982 (1995)
24. Novelli, J.-C., Saliola, F., Thibon, J.-Y.: Representation theory of the higher order peak algebras. Preprint [math.CO]
25. Novelli, J.-C., Thibon, J.-Y.: A Hopf algebra of parking functions. In: FPSAC'04, Vancouver (2004)
26. Novelli, J.-C., Thibon, J.-Y.: Parking functions and descent algebras. Ann. Comb. 11, 59-68 (2007)
27. Novelli, J.-C., Thibon, J.-Y.: Polynomial realizations of some trialgebras. In: FPSAC'06. Also preprint
28. Novelli, J.-C., Thibon, J.-Y.: Hopf algebras and dendriform structures arising from parking functions. Fundam. Math. 193, 189-241 (2007)
29. Novelli, J.-C., Thibon, J.-Y.: Noncommutative symmetric functions and Lagrange inversion. Adv.
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