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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.

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    $\begingroup$ Great question! As Hugh and Chris explain below, the answer seems to be no. However, we also have the Mirkovic-Vilonen basis for these representations. In our recent paper, our computations suggest that perhaps the MV basis coincides with the GHKK basis. $\endgroup$ – Joel Kamnitzer Jul 28 '19 at 21:33
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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). Greedy basis elements are indecomposable positive elements (see arXiv: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 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. (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, 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.

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  • $\begingroup$ This is convincing to me, especially the reference to the paper of Qin explaining the "moduli space" of good bases. $\endgroup$ – Sam Hopkins Jul 10 '19 at 20:26
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Quoting from

Geiss, Christof; Leclerc, Bernard; Schröer, Jan, Preprojective algebras and cluster algebras.,

First, the cluster algebra structure, when it is known, is usually wellhidden, and its description requires a lot of difficult (but beautiful) combinatorics. As an example, one may consult the paper of Scott [34] and in particular the cluster structures of the Grassmannians Gr(3, 6), Gr(3, 7) and Gr(3, 8). Secondly, these cluster algebras are generally of infinite type so one cannot hope for a closed and finite description as in the above example. This is not too bad if one is mainly interested in total positivity, since one may not necessarily need to know all positive coordinate systems on X. But it becomes a challenging issue if one aims at a monomial description of the dual canonical basis of the coordinate ring, because that would likely involve infinitely many families of monomials. In fact such a monomial description may not even be possible, since, as shown in [25], there may exist elements of the dual canonical basis whose square does not belong to the basis. In any case, even if the cluster structure is known, more work is certainly needed to obtain from it a full description of the dual canonical basis

Reference [25] is

Leclerc, B., Imaginary vectors in the dual canonical basis of (U_q({\mathfrak n})), Transform. Groups 8, No. 1, 95-104 (2003). ZBL1044.17009.

That is, bases obtained by taking all cluster monomials cannot give the whole dual canonical basis: in particular it can't give those dual canonical basis elements whose square is not in the dual canonical basis.

This doesn't answer your question directly, though, since one would have to look more carefully at the definition of the theta basis to see exactly how it is constructed, and whether or not it has the same issue as above. Rather, I thought it would be helpful to you if you weren't aware of this potential stumbling block.

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