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The cyclotomic KLR algebra is isomorphic to the Ariki-Koike algebra over a field and so admits a trace (this is used in Hu-Mathas' paper to define bases for the KLR algebra corresponding to Murphy and dual Murphy bases). The construction is not very explicit. In Kashiwara's paper "BIADJOINTNESS IN CYCLIC KHOVANOV-LAUDA-ROUQUIER ALGEBRAS" he seems to define an explicit trace on the KLR algebra, but in a manner which seems a bit opaque to a reader who doesn't understand how KLR algebras arise from 2-representation theory. Has anyone written this map down explicitly in terms of the $\psi_r$, $e_i$, and $y_r$ generators of the KLR algebra?

Also, Mathas and Hu's "Murphy" and "dual Murphy" bases are constructed uniformly up to a reflection in the combinatorics of Young tableaux, this seems to have something to do with re-orienting the underlying quiver. Can this reorientation be seen as being related to the trace map?

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