### The cluster and dual canonical bases of ℤ[x_{11}, …, x_{33}] are equal

*Brendon Rhoades*

#### 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_{11}, …, 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.Full Text: PDF PostScript