- #1

- 10

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I understand the first two terms ##\mathfrak{o}(m-x,n-x) \oplus \mathfrak{gl}(x)## — i.e. a partition into a smaller ##\mathfrak{o}## part and a general linear part ##\mathfrak{gl}(x)## — but I am not so sure how the latter terms should be read/interpreted, especially not the ##(m+n-2x) \otimes (x \oplus \bar{x})## part. What "is", say, a ##9 \otimes (5 \oplus \bar{5})##?

If we represent ##\mathfrak{o}(m,n)## schematically as a matrix $$\left(\begin{array}{ccc|c|c} & & & & \\ & A & & E & F \\ & & & & \\\hline & G & & B & C \\\hline & H & & D & B \end{array}\right)$$ the ##A## block corresponds to the ##\mathfrak{o}(m-x,n-x)## subalgebra and the ##B##s to the ##\mathfrak{gl}(x)## subalgebra. I assume that the wedge terms in the direct sum correspond to the ##C## and ##D## blocks in this matrix, and that the ##(m+n-2x) \otimes (x \oplus \bar{x})## term is somehow associated with the blocks ##E##, ##F##, ##G## and ##H##, since ##(m+n-2x) \times x## is the block dimension of ##E## as well as ##F##.

Any interpretative help, references etc. would be greatly appreciated.