Derangements and Tensor Powers of Adjoint Modules for \mathfrak s\mathfrak l n \mathfrak{s}\mathfrak{l}_n
Georgia Benkart1
and Stephen Doty2
1University of Wisconsin Department of Mathematics Madison Wisconsin 53706 USA
2Loyola University Chicago Department of Mathematical and Computer Sciences Chicago Illinois 60626 USA
2Loyola University Chicago Department of Mathematical and Computer Sciences Chicago Illinois 60626 USA
DOI: 10.1023/A:1020830430464
Abstract
We obtain the decomposition of the tensor space \mathfrak s\mathfrak l n Ä k \mathfrak{s}\mathfrak{l}_n^{ \otimes k} as a module for \mathfrak s\mathfrak l n \mathfrak{s}\mathfrak{l}_n , find an explicit formula for the multiplicities of its irreducible summands, and (when n C \mathcal{C} = \text End \mathfrak s\mathfrak l n {\text{End}}_{\mathfrak{s}\mathfrak{l}_n } ( \mathfrak s\mathfrak l n Ä k \mathfrak{s}\mathfrak{l}_n^{ \otimes k} ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of C \mathcal{C} is given by the number of derangements of a set of 2 k elements.
Pages: 31–42
Keywords: derangements; centralizer algebras; walled Brauer algebras; tensor powers; adjoint representation
Full Text: PDF
References
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2. G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer, “Tensor product representations of general linear groups and their connections with Brauer algebras,” J. Algebra 166 (1994), 529-567.
3. R.A. Brualdi, Introductory Combinatorics, 3rd ed., Prentice Hall, Englewood Cliffs, N.J., 1999.
4. C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Vol. XI, Pure and Applied Math, Interscience Publ. John Wiley, New York, 1962.
5. J.A. Green, Polynomial Representations of GLn, Lecture Notes in Math., Vol. 830, Springer-Verlag, Heidelberg, 1980.
6. P. Hanlon, “On the construction of the maximal vectors in the tensor algebra of gln,” Combinatorics and Algebra (Boulder, Colo., 1983) Contemp. Math., Vol. 34, Amer. Math. Soc., Providence R.I., 1984, pp. 73-80.
7. P. Hanlon, “On the decomposition of the tensor algebra of the classical Lie algebras,” Adv. in Math. 56 (1985), 238-282.