Exterior powers of the adjoint representation.

*(English)*Zbl 0878.20028Let \(g\) be a complex semisimple Lie algebra. A long standing problem in representation theory is to decompose the exterior algebra \(\wedge g\) as a representation of \(g\) under the adjoint action. The action of \(g\) on the symmetric algebra \(S(g)\) was understood by Kostant in a groundbreaking 1963 paper, and more recently Broer understood \(S(g)\) as a graded module. Reeder’s paper gives some interesting but incomplete results in the case of apparently much harder problem of \(\wedge g\).

We recall that the occurrences of the trivial representation in \(\wedge g\) correspond to rational cohomology classes on a compact group with Lie algebra \(g\). The cases of \(\wedge^ig\) for \(i=2\) and \(i=3\) are given explicitly by general formulas. In addition, a module \(V(\lambda)\) with highest weight \(\lambda\) is called small if \(\lambda\) is in the root lattice but no nonzero weight of \(V(\lambda)\) is twice a root. If \(V(\lambda)\) is small, the author proves that the multiplicity of \(V(\lambda)\) in the \(\wedge g\) equals \(2^l\) times the multiplicity of the zero weight space in \(V(\lambda)\), where \(l\) is the rank of \(g\). This is proved easily by establishing a recursive formula for computing multiplicities in \(\wedge g\) in general, and by observing that the recursion is trivial for small modules. The author points out that this follows from an old result proved independently by Kostant and Parthasarathy, Ranga Rao, and Varadarajan. A conjecture is given for graded multiplicities for small modules. In addition, the author gives a formula for certain multiplicities when the highest weight is related to a Levi factor. By using the above results plus a theorem of Stembridge for certain small modules in the case of \(gl(n)\), the author gives tables for the decomposition of \(\wedge g\) in all cases of rank at most 3 besides \(B_3\).

We recall that the occurrences of the trivial representation in \(\wedge g\) correspond to rational cohomology classes on a compact group with Lie algebra \(g\). The cases of \(\wedge^ig\) for \(i=2\) and \(i=3\) are given explicitly by general formulas. In addition, a module \(V(\lambda)\) with highest weight \(\lambda\) is called small if \(\lambda\) is in the root lattice but no nonzero weight of \(V(\lambda)\) is twice a root. If \(V(\lambda)\) is small, the author proves that the multiplicity of \(V(\lambda)\) in the \(\wedge g\) equals \(2^l\) times the multiplicity of the zero weight space in \(V(\lambda)\), where \(l\) is the rank of \(g\). This is proved easily by establishing a recursive formula for computing multiplicities in \(\wedge g\) in general, and by observing that the recursion is trivial for small modules. The author points out that this follows from an old result proved independently by Kostant and Parthasarathy, Ranga Rao, and Varadarajan. A conjecture is given for graded multiplicities for small modules. In addition, the author gives a formula for certain multiplicities when the highest weight is related to a Levi factor. By using the above results plus a theorem of Stembridge for certain small modules in the case of \(gl(n)\), the author gives tables for the decomposition of \(\wedge g\) in all cases of rank at most 3 besides \(B_3\).

Reviewer: S.Evens (Tucson)

##### MSC:

20G05 | Representation theory for linear algebraic groups |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

20C30 | Representations of finite symmetric groups |

22E10 | General properties and structure of complex Lie groups |

22E60 | Lie algebras of Lie groups |