# The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal

Abstract : The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ 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. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials.
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Cited literature [11 references]

https://hal.inria.fr/hal-01186255
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• HAL Id : hal-01186255, version 1

### Citation

Brendon Rhoades. The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.995-1006. ⟨hal-01186255⟩

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