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

TL;DR

This paper demonstrates that SO(2n,C)-character varieties are not generated solely by trace functions, providing examples of indistinguishable representations and establishing conditions where trace functions suffice to distinguish representations.

Contribution

It proves that SO(2n,C)-character varieties are not varieties of characters and identifies cases where trace functions can distinguish generic representations.

Findings

Coordinate rings are not generated by trace functions for n≥2.

Examples of non-conjugate representations indistinguishable by generalized trace functions.

Generic representations can be distinguished by trace functions and a single generalized trace function.

Abstract

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.