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The cluster and dual canonical bases of Z[x(11), ..., x(33)] are equal

Abstract : The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 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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Brendon Rhoades. The cluster and dual canonical bases of Z[x(11), ..., x(33)] are equal. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, Vol. 12 no. 5 (5), pp.97-124. ⟨10.46298/dmtcs.515⟩. ⟨hal-00990455⟩

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