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Multiplicities of Higher Lie Characters

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Finite fields and commutative rings (number-theoretic aspects) Classical measure theory

Published online by Cambridge University Press:  09 April 2009

Manfred Schocker
Manfred Schocker
Affiliation:
Mathematical Institute24–29 St Giles Oxford OX1 3LBUK e-mail: schocker@maths.ox.ac.uk
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Abstract

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The higher Lie characters of the symmetric group Sn arise from the Poincaré-Birkhoff-Witt basis of the free associative algebra. They are indexed by the partitions of n and sum up to the regular character of Sn. A combinatorial description of the multiplicities of their irreducible components is given. As a special case the Kraśkiewicz-Weyman result on the multiplicities of the classical Lie character is obtained.

Keywords

symmetric groupgeneral linear groupfree Lie algebratableaumajor index.

MSC classification

Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11T06: Polynomials 28A80: Fractals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 1 , August 2003 , pp. 9 - 21
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Bergeron, F., Bergeron, N. and Garsia, A. M., ‘Idempotents for the free Lie algebra and q-enumeration’, in: Invariant theory and tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl. 19 (Springer New York, 1990) pp. 166–190.Google Scholar
[2]Blessenohl, D. and Laue, H., ‘On the descending Loewy series of Solomon's descent algebra’, J. Algebra 180 (1996), 698–724.CrossRefGoogle Scholar
[3]Garsia, A. M. and Reutenauer, C., ‘A decomposition of Solomon's descent algebra’, Adv. in Math. 77 (1989), 189–262.CrossRefGoogle Scholar
[4]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 4th Edition (Oxford University Press, Oxford, 1960).Google Scholar
[5]James, G. and Kerber, A., The representation theory of the symmetric group (Addison-Wesley, Reading, Massachusetts, 1981).Google Scholar
[6]Jöllenbeck, A., ‘Nichtkommutative Charaktertheorie der symmetrischen Gruppen’, Bayr. Math. Schr. 56 (1999), 1–41.Google Scholar
[7]Jöllenbeck, A. and Schocker, M., ‘Cyclic characters of symmetric groups’, J. Algebraic Combin. 12 (2000), 155–161.CrossRefGoogle Scholar
[8]Klyachko, A. A., ‘Lie elements in the tensor algebra’, Siberian Math. J. 15 (1974), 914–920.CrossRefGoogle Scholar
[9]Kraśkiewicz, W. and Weyman, J., ‘Algebra of invariants and the action of a Coxeter element’, Bayr. Math. Schr. 63 (2001), 265–284.Google Scholar
[10]Leclerc, B., Scharf, T. and Thibon, J.-Y., ‘Noncommutative cyclic characters of symmetric groups’, J. Combin. Theory Ser. A (I) 75 (1996), 55–69.CrossRefGoogle Scholar
[11]Patras, F. and Reutenauer, C., ‘Higher Lie Idempotents’, J. Algebra 222 (1999), 51–64.CrossRefGoogle Scholar
[12]Reutenauer, C., Free Lie algebras, London Math. Soc. Monographs 7 (Oxford University Press, Oxford, 1993).CrossRefGoogle Scholar
[13]Schur, I., Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen (Dissertation, Berlin, 1901).Google Scholar
[14]Schur, I., ‘Über die rationalen Darstellungen der allgemeinen linearen Gruppe’, Sitzungsber. Pr. Akad. Wiss. (1927), 58–75.CrossRefGoogle Scholar
[15]Solomon, L., ‘A Mackey formula in the group ring of a Coxeter group’, J. Algebra 41 (1976), 255–268.CrossRefGoogle Scholar
[16]Thrall, R., ‘On symmetrized Kronecker powers and the structure of the free Lie ring’, Amer. J. Math. 64 (1942), 371–388.CrossRefGoogle Scholar
[17]Witt, E., ‘Treue Darstellung Liescher Ringe’, J. Reine Angew. Math. 177 (1937), 152–160.CrossRefGoogle Scholar