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The space U(n)/N(n), where N(n) is the normalizer of the maximal torus in U(n), arises naturally in many areas of mathematics and can be identified with the unordered flag manifolds. In this talk, we will introduce the concept of the n-fold extended symmetric power of a space X, and describe its cohomology as a Hopf ring. We will also demonstrate homological stability for the spaces {U(n)/N(n)}, and describe the stable cohomology ring of U(n)/N(n). If there is sufficient time, we may also see some of the low-dimensional computations that have been done in the case where n = 3, 4. This is joint work with Lorenzo Guerra.
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