SO(2n,C)-character varieties are not varieties of characters

We prove that the coordinate rings of SO(2n,C)-character varieties are not generated by trace functions nor generalized trace functions for $n\geq 2$ and all groups Gamma of corank $\geq 2.$ Furthermore, we give examples of non-conjugate completely reducible representations undistinguishable by generalized trace functions. Hence, SO(2n,C)-character varieties are not varieties of characters. However, we also prove that any generic SO(2n,C)-representation of a free group can be distinguished from all non-equivalent representations by trace functions and by a single generalized trace function.

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