Odd symplectic groups. (English) Zbl 0621.22009
Symplectic groups \(Sp_{2n+1}\) acting on odd-dimensional vector spaces are defined. These Lie groups are not simple or reductive. Weights of indecomposable trace free tensor representations of the groups are described with Young tableaux. A character formula and a dimension formula for these representations are found. These formulas are very similar in form to the usual formulas for simple Lie groups due to Weyl. The denominator of the character formula can be written as a sum or as a product, yielding a denominator identity for the unreduced root system \(BC_ n\). Various identities for \(Sp_{2n+1}\) characters are given, and it is indicated how these identities fill gaps in frameworks established by analogous identities for the groups \(Sp_{2n}\), \(O_{2n}\), and \(O_{2n+1}\). The dimension formula and the Young tableau description of weights are used to obtain product enumeration formulas for certain kinds of ordinary and shifted plane partitions.
MSC:
| 22E15 | General properties and structure of real Lie groups |
| 20G05 | Representation theory for linear algebraic groups |
| 05A17 | Combinatorial aspects of partitions of integers |
Keywords:
symplectic groups; trace free tensor representation; character formula; dimension formula; root system; Young tableau; weight; plane partitionsReferences:
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