The bipartite Cambrian congruence conjecture for transformed cluster fans

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Let RuR_u and RvR_v be regions of the Coxeter arrangement, and let the transformed cluster fan be the fan whose maximal cones are under consideration. Write u≡vu\equiv v when uu and vv are equivalent under the bipartite Cambrian congruence. Bipartite Cambrian congruence conjecture. The regions RuR_u and RvR_v are contained in the same maximal cone of the transformed cluster fan if and only if

u≡v.u\equiv v.

This conjecture identifies the maximal cones of the transformed cluster fan with the classes of the bipartite Cambrian congruence. According to the supplied status evidence, it has been proved in types AnA_n and BnB_n, so the conjecture is recorded as solved.

References

Primary source

Sergey Fomin and Nathan Reading, “Root systems and generalized associahedra”, arXiv:math/0505518 (2008).

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