Gross-Hacking-Keel-Kontsevich (<https://arxiv.org/abs/1411.1394>) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain combinatorial condition. This condition holds for the homogeneous coordinate ring of the Grassmannian (see e.g. <https://arxiv.org/abs/1712.00447> or <https://arxiv.org/abs/1803.06901>). On the other hand, the $m$th homogeneous component of the coordinate ring is isomorphic to the $GL$ representation $V(m \omega)^\*$ for the appropriate minuscule weight $\omega$, hence has a basis (the dual canonical basis of Lusztig/Kashiwara) coming from the theory of quantized enveloping algebras.


Could these bases be the same? There are some vague statements alluding to this possibility in GHKK, but nothing definitive.


I know the two bases share some interesting properties: e.g., the twisted cyclic shift symmetry of the Grassmannian permutes both bases according to promotion of semistandard tableaux (for the dual canonical basis this was proved by Rhoades, see <https://arxiv.org/abs/1809.04965>; for the theta basis this was proved by Shen and Weng <https://arxiv.org/abs/1803.06901>). Most naive bases (e.g. the standard monomial basis) do not satisfy this property.



Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?
I think there is good reason to think the answer is "no". 


In rank 2, the theta basis agrees with the greedy basis ([arXiv:1508.01404](https://arxiv.org/abs/1508.01404)). Greedy basis elements are indecomposable positive elements (see [arXiv:1208.2391](https://arxiv.org/abs/1208.2391)) (i.e., they cannot be written as a sum of two positive elements). If the indecomposable positive elements form a basis, this is what is now called the atomic basis. 


Consider the cluster algebra associated to the Kronecker quiver. Sherman and Zelevinsky showed [arxiv:math/0307082](https://arxiv.org/abs/math/0307082) that the Kronecker-type cluster algebra has an atomic basis. (Note that, confusingly, the term they used for what is now called the atomic basis is the "canonical basis".) If $x$ and $y$ are any two cluster variables from a common cluster, the atomic basis element associated to the imaginary root $\delta$ is $z=x/y + y/x + 1/xy$. The atomic basis element associated to $2\delta$ is $z^2-2$.


There is a cluster algebra with a dual canonical basis which is of type $\widetilde A\_1$. This is a quantum cluster algebra, so when we talk about the dual canonical basis of the corresponding classical (commutative) cluster algebra, we mean that $q$ has been set to 1. Inconveniently, this cluster algebra also has coefficients. If we simply set them to 1 as well, then the dual canonical basis element corresponding to $2\delta$ is $z^2-1$.


This follows from formula (31) in [arXiv:1002.2762](https://arxiv.org/abs/1002.2762). (It is a formula for the basis element associated to $n\delta$. Note that $p\_0$ and $p\_1$ are coefficients, which we are ignoring, while $u\_1$ and $u\_2$ are what I was calling $x$ and $y$.)


Even without the Sherman-Zelevinsky result, it is clear that $z^2-1$ cannot be in the greedy basis, since it is not an indecomposable positive element. It is easy to check that $z^2-1=(z^2-2)+1$ expresses it as a sum of two positive elements.


The thing I really wanted to mention, though, is the recent paper by Fan Qin, [arXiv:1902.09507](https://arxiv.org/abs/1902.09507), which gives a conceptual explanation for the existence of many good bases which disagree. He shows that there is in a sense a moduli space of good bases. The theta basis is a particular point in this moduli space, and the dual canonical basis is another.


