The Cluster and Dual Canonical Bases of Z
نویسنده
چکیده
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)) [8] [5]. 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 [3] [4]. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky [10].
منابع مشابه
The Cluster Basis of Z[x1,1,...,x3,3]
We show that the set of cluster monomials for the cluster algebra of type D4 contains a basis of the Z-module Z[x1,1, . . . , x3,3]. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases....
متن کاملThe cluster and dual canonical bases of Z [ x 11 , . . . , x 33 ] are equal
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 prov...
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