An unpublished theorem of Manfred Schocker and the Patras-Reutenauer algebra
Dieter Blessenohl
Christian-Albrechts-Universität Kiel Mathematisches Seminar Ludewig-Meyn-Str. 4 24098 Kiel Germany
DOI: 10.1007/s10801-007-0118-8
Abstract
Patras, Reutenauer (J. Algebr. Comb. 16:301-314, [ 2002]) describe a subalgebra \mathfrak A \mathfrak{A} of the Malvenuto-Reutenauer algebra \?. Their paper contains several characteristic properties of this subalgebra. In an unpublished manuscript Manfred Schocker states without proof a theorem, providing two further characterizations of the Patras-Reutenauer algebra. In this paper we establish a slightly generalized version of Schocker's theorem, and give some applications. Finally we describe a derivation of the convolution algebra \mathfrak A \mathfrak{A} , which is a homomorphism for the inner product.
Pages: 25–42
Keywords: keywords symmetric group algebras; reciprocity laws; Lie idempotents; Solomon's descent algebra
Full Text: PDF
References
1. Bergeron, F., Garsia, A.M., Reutenauer, C.: Homomorphisms between Solomon's descent algebras. J. Algebra 150, 503-519 (1992)
2. Bialynicki-Birula, I., Mielnik, B., Plebański, J.: Explicit solution of the continuous Baker-Campbell- Hausdorff problem. Ann. Phys. 51, 187-200 (1969)
3. Blessenohl, D., Schocker, M.: Noncommutative Character Theory of the Symmetric Group. Imperial College Press (2005)
4. Foissy, L.: Bidendriform bialgebras, trees, and free quasi-symmetric functions. arXiv:math 0505207v1[math.RA]
5. Gelfand, L.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112(2), 218-348 (1995)
6. Magnus, W.: Über Gruppen und zugeordnete Liesche Ringe. J. Reine Angew. Math. 182, 142-149 (1940)
7. Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967-982 (1995) J Algebr Comb (2008) 28: 25-42
8. Mielnik, B., Plebański, J.: Combinatorial approach to Baker-Campbell-Hausdorff exponents. Ann. Inst. H. Poincaré Sect. A (N.S.) 12, 215-254 (1970)
9. Patras, F., Reutenauer, C.: Lie representations and an algebra containing Solomon's. J. Algebr. Comb. 16, 301-314 (2002)
10. Reutenauer, C.: Theorem of Poincaré-Birkhoff-Witt, logarithm and representations of the symmetric group whose order are the Stirling numbers. In: Labelle, G., Leroux, P. (eds.) Combinatoire Énumérative. Lecture Notes in Mathematics, vol. 1234, pp. 267-284. Springer, Berlin (1985)
11. Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs, vol.
7. Oxford University Press, Oxford (1993). New series
12. Solomon, L.: On the Poincaré-Birkhoff-Witt theorem. J. Comb. Theory (A) 4, 363-375 (1968)
13. Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41, 255-268 (1967)
14. Sweedler, M.: Hopf Algebras. Benjamin, New York (1969).
2. Bialynicki-Birula, I., Mielnik, B., Plebański, J.: Explicit solution of the continuous Baker-Campbell- Hausdorff problem. Ann. Phys. 51, 187-200 (1969)
3. Blessenohl, D., Schocker, M.: Noncommutative Character Theory of the Symmetric Group. Imperial College Press (2005)
4. Foissy, L.: Bidendriform bialgebras, trees, and free quasi-symmetric functions. arXiv:math 0505207v1[math.RA]
5. Gelfand, L.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112(2), 218-348 (1995)
6. Magnus, W.: Über Gruppen und zugeordnete Liesche Ringe. J. Reine Angew. Math. 182, 142-149 (1940)
7. Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967-982 (1995) J Algebr Comb (2008) 28: 25-42
8. Mielnik, B., Plebański, J.: Combinatorial approach to Baker-Campbell-Hausdorff exponents. Ann. Inst. H. Poincaré Sect. A (N.S.) 12, 215-254 (1970)
9. Patras, F., Reutenauer, C.: Lie representations and an algebra containing Solomon's. J. Algebr. Comb. 16, 301-314 (2002)
10. Reutenauer, C.: Theorem of Poincaré-Birkhoff-Witt, logarithm and representations of the symmetric group whose order are the Stirling numbers. In: Labelle, G., Leroux, P. (eds.) Combinatoire Énumérative. Lecture Notes in Mathematics, vol. 1234, pp. 267-284. Springer, Berlin (1985)
11. Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs, vol.
7. Oxford University Press, Oxford (1993). New series
12. Solomon, L.: On the Poincaré-Birkhoff-Witt theorem. J. Comb. Theory (A) 4, 363-375 (1968)
13. Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41, 255-268 (1967)
14. Sweedler, M.: Hopf Algebras. Benjamin, New York (1969).