Discrete Mathematics & Theoretical Computer Science, Vol 12, No 5 (2010)

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The cluster and dual canonical bases of ℤ[x11, …, x33] are equal

Brendon Rhoades


The polynomial ring ℤ[x11, …, x33] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group Uq(sl3(ℂ)). On the other hand, ℤ[x11, …, x33] inherits a basis from the cluster monomial basis of a geometric model of the type D4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.

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