Derangements and tensor powers of adjoint modules for $\frak{sl}_n$

Benkart, Georgia ; Doty, Stephen

Journal of Algebraic Combinatorics, Tome 16 (2002) no. 1, pp. 31-42.

Résumé

Summary: We obtain the decomposition of the tensor space $\mathfrak s\mathfrak l _{ n} ^{ \"A k}$ mathfraksmathfrakl_n^ otimesk as a module for $\mathfrak s\mathfrak l _{ n}$ mathfraksmathfrakl_n , find an explicit formula for the multiplicities of its irreducible summands, and (when $n C$ mathcalC = $End _mathfraksmathfrakl _ n $ textEnd_mathfraksmathfrakl_n ( $\mathfrak s\mathfrak l _{ n} ^{ \"A k}$ mathfraksmathfrakl_n^ $\otimes k$ ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of $C$ mathcalC is given by the number of derangements of a set of $2 k$ elements.

Détail

Keywords: derangements, centralizer algebras, walled Brauer algebras, tensor powers, adjoint representation