9 problems

Berenstein–Zelevinsky's conjecture. Quantum double Bruhat cells admit a structure of quantum cluster algebras via quantum minors.

Goncharov–Shen's conjecture. In the case of double Bruhat cells, the Donaldson–Thomas transformation of the cluster Poisson variety is a slight modification of the Fomin–Zelevinsky…

Let $G$ be a split semisimple algebraic group with Weyl group $W$, and let $L^{u,v}=N\overline{u}N\cap B_-vB_-$ be the reduced double Bruhat cell for $u,v\in W$. For a reduced word…

Quantum upper cluster algebra conjecture. The homomorphism

Let $G$ be a simply-connected, simple, complex algebraic group with Lie algebra $\mathfrak{g}$ of simply-laced type. Let $\mathcal{A}_{t,w_0}$ be the quantum cluster algebra struct…

Berenstein–Zelevinsky conjecture. For every signed word $\underline{\mathbf{i}}$ for $(u,w)$,

The double Bruhat-cell cluster algebra is the cluster algebra defined by Berenstein, Fomin, and Zelevinsky; its cluster variables are the elements obtained through the cluster-alge…

Fix $m$, and let $G^{w_0,w_0}$ denote the big open double Bruhat cell in $SL_m$. For $v\in S_n$, let $\operatorname{Imm}_v X(R,C)$ denote the associated dual canonical basis immana…

Cluster conjecture for inverse-map functions. The set $\{f_1,\ldots,f_n\}$ is a cluster for $L^{e,w}$; equivalently, the fraction field generated by $\{f_1,\ldots,f_n\}$ is the fun…