arXiv math/01081062001-08math.RTmath.CO

Derangements and tensor powers of adjoint modules for sl_n

Georgia Benkart, Stephen Doty

备注：: This is a revised version of a paper by the same title that appeared in Journal of Algebraic Combinatorics 16 (2002), 31-42. In particular 1.15-1.18 in that paper have been revised in 1.15-1.18 here and a few other related minor changes have been made in the first line of Section 2 and in Section 3. We are grateful to Alberto Elduque for alerting us to the mistake in the previous version
发表信息：: Journal of Algebraic Combinatorics 16 (2002), 31-42

英文摘要：

We obtain the decomposition of the tensor space $\mathfrak{sl}_n^{\otimes k}$ as a module for $\mathfrak{sl}_n$, find an explicit formula for the multiplicities of its irreducible summands, and (when $n \ge 2k$) describe the centralizer algebra $C=End_{\mathfrak{sl}_n}(\mathfrak{sl}_n^{\otimes k})$ and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of $C$ is given by the number of derangements of a set of $2k$ elements.

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