The positive-pair conjecture for SL_n-trace equivalence

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Let FrF_r be a free group, and call two pairs of words (u,v)(u,v) and (u′,v′)(u',v') SL⁡n\operatorname{SL}_n-trace equivalent when the corresponding words have equal traces under every representation into SL⁡n\operatorname{SL}_n. A pair is positive if, in a reduced representation, its words have only non-negative powers of the generators. Positive-pair conjecture. For every n≥2n\geq 2, there exist SL⁡n\operatorname{SL}_n-trace equivalent pairs (u,v)(u,v) if and only if there exist positive pairs (u′,v′)(u',v') that are SL⁡n\operatorname{SL}_n-trace equivalent. The source frames this as a more provocative conjecture after noting that non-conjugate palindromic pairs are expected, but not proved, never to be SL⁡3\operatorname{SL}_3-trace equivalent.

References

Primary source

Sean Lawton, Larsen Louder and D. B. McReynolds, “Decision problems, complexity, traces, and representations”, arXiv:1312.1261 (2016).

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