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For a compact Lie group \(G\), a \(G\)-isovariant map \(f:X\to Y\) between two \(G\)-spaces \(X\) and \(Y\)
is a \(G\)-equivariant map preserving the isotropy subgroups.
In this talk, we consider isovariant maps between \(G\)-representation spaces.
First, we review Wasserman's results, as well as our recent results about the isovariant Borsuk-Ulam theorem.
Secondly, we consider bi-isovariant equivalent representations.
We say that two representations \(V\) and \(W\) are bi-isovariant equivalent
if there exist isovariant maps from \(V\) to \(W\) and from \(W\) to \(V\).
We show that if \(V\) and \(W\) are bi-isovariant, then their dimension functions coincide. Furthermore, if \(G\) is connected, these representations are isomorphic.
In order to give a proof, we use tom Dieck's and Traczyk's results in representation theory.
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