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# Splitting Universal Bundles over Flag Manifolds

R. E. Stong
Proceedings of the American Mathematical Society
Vol. 84, No. 4 (Apr., 1982), pp. 576-580
DOI: 10.2307/2044039
Stable URL: http://www.jstor.org/stable/2044039
Page Count: 5
Let $\mathbf{F}$ be one of the fields $\mathbf{R}, \mathbf{C}$, or $\mathbf{H}$ and correspondingly let $\mathbf{F}G$ be $O, U$, or $\operatorname{Sp}$, i.e. the orthogonal, unitary, or symplectic group. Over the flag manifold $\mathbf{F}G(n_1 + \cdots + n_k)/\mathbf{F}G(n_1) \times \cdots \times \mathbf{F}G(n_k)$ one has vector bundles $\gamma_i$ over $F$ of dimension $n_i, 1 \leqslant i \leqslant k$. This paper determines all cases in which $\gamma_i$ decomposes nontrivially as a Whitney sum.