85 problems

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

Let $\mathscr{A}_q$ be a quantum cluster algebra generated by cluster variables $\{Z_i\}_{i\in\mathsf{K}}$, contained in the non-commutative Laurent polynomial ring…

Let $\mathcal{A}(X,\mathbf{h},\Lambda,\widetilde{B})$ be a generalized quantum cluster algebra with initial extended cluster $\{X_1,\ldots,X_m\}$. A Laurent polynomial in this clus…

Let $Q$ be a valued acyclic quiver, and consider the associated quantum cluster algebra over a finite field. Rupel's quantum Caldero–Chapoton map assigns quantum Laurent polynomial…

Let $\mathcal A^q$ be a quantum cluster algebra associated with an acyclic quiver and a $z$-coefficient pattern. Let $C$ denote Berenstein–Zelevinsky's triangular basis, and let th…

Let $(B(t))_{t\in\mathbb{Z}}$ be the exchange-matrix sequence and let $(Y(t))_{t\in\mathbb{Z}}$ and $(y(t))_{t\in\mathbb{Z}}$ denote, respectively, the corresponding quantum and no…

Quantum upper cluster algebra conjecture. The homomorphism

Reduced skein–cluster conjecture. Under these assumptions,

Localized skein–cluster conjecture. Under these assumptions,

Let $\widetilde{\psi}\colon\mathcal{D}'_{\mathfrak{sl}_3}\to A_3(\mathbb{C}_{\epsilon}(t))[\mathbf{x}_1^{-1},\mathbf{x}_2^{-1},\mathbf{x}_3^{-1}]$ be the algebra homomorphism defin…

Let $h$, $\epsilon$, $q=e^{h\epsilon}$, and the quantum torus algebras $\mathfrak{D}_{\mathfrak{sl}_3}$, $\mathcal{D}_{\mathfrak{sl}_3}$, and $\mathcal{D}'_{\mathfrak{sl}_3}$ be as…

Skein–cluster equality conjecture. Under the same assumption as Theorem,

Let the affine Grassmannian slice carry its category of perverse coherent sheaves and equivariant K-theory, and let the corresponding quantum cluster algebra be equipped with Qin's…

Let $\mathfrak{S}$ be a triangulable pb surface, let $\mathcal V$ be the vertex set of the associated quantum seed, and let $\mathcal V_{\mathrm{mut}}$ be its mutable vertices. Wri…

Quantum Pfaffian–cluster interpretation conjecture. Quantum Pfaffians may admit interpretations in terms of cluster variables or canonical bases of $U_q(\mathfrak{g})$-modules.

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…

Let $I$ be the injective ring homomorphism from the cluster algebra $\mathcal{A}_{w_0}$ to $K_0(\mathcal{O}^{\textnormal{sh}}_{\mathbb{Z}})$, let $[\underline{P}]$ denote the coeff…

Quantum cluster structure conjecture. The algebra $\widehat{\mathcal{A}}(b)$ is a quantum cluster algebra with initial seed $\mathbf{t}_{\mathbf{i}}$, and its cluster monomials are…

Quantum T-system conjecture. For any $i$-box $[a,c]$, there exist constants $A$ and $B$ such that

Frozen-variable primeness conjecture. Every frozen cluster variable $x_j$ is a prime element of $\overline{\mathcal{A}}$.

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

Let $m_j\in \mathbb{Z}$, $\mathtt{b}_j\in\mathtt{B}$ for $1\leqslant j\leqslant r$, and $m'_k\in\mathbb{Z}$, $\mathtt{b}'_k\in\mathtt{B}$ for $1\leqslant k\leqslant r'$. For a posi…

Berenstein–Zelevinsky's positivity conjecture. For any quantum cluster variable $X$ of $\mathcal A_v$,

Positivity conjecture. The coefficients in the quantum Laurent expansion of every quantum cluster variable belong to $\mathbb N[q^{\pm 1/2}]$.

Let $\mathtt{b}\in\bg^+$, let $\pd=(\textbf{\textit{i}},\xi)$ be a PBW datum of $\mathtt{b}$, and let $\qbA{}(\mathtt{b})$ be the associated subalgebra of the bosonic extension. Th…