53 problems

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…

Quantum cluster intersection conjecture. If $w_{1}\leq w_{2}$ in the Bruhat order, this triple intersection contains a quantum cluster algebra, and its quantum cluster monomials ar…

Let $\Delta$ be an ideal triangulation of $\frak{S}$ and let $\mathbb{I}^q_\Delta$ be the proposed quantum duality map. Quantum Laurent coefficient positivity conjecture. For every…

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 $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, let $W$ be its Weyl group, and let $w\in W$. Denote by ${\mathsf{A_q}}[N_{-}(w)]$ the corresponding quantum unipotent subgr…

Let $b,c$ be positive integers, and let $X[a_1,a_2]$ denote the quantum greedy element of the rank $2$ quantum cluster algebra $\mathcal{A}_v(b,c)$ associated with…

Localized skein–cluster conjecture. Under these assumptions,

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…

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…

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

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

Let $s\in\mathbb{Z}$, let $\mathfrak{S}_s$ be the indicated quantum seed, and let ${}^{s}\mathfrak{u}_{i,p}$ denote its seed variables. Let…

Let $(m^{(i)}_{k,r},m^{(j)}_{l,t})$ be a pair of KR-monomials, and let $F_q(\underline{m}^{(i)}_{k,r})$ and $F_q(\underline{m}^{(j)}_{l,t})$ be the associated KR-polynomials. Let…

Let $(i,p)$ and $(i,s)$ be vertices of the indicated folded Dynkin diagram with $p<s$. Let $m^{(i)}[p,s]$ be the corresponding dominant monomial, let $\underline{m^{(i)}[p,s]}$ be…

Braid-group preservation conjecture. The maps $\sigma_i$ and $\sigma_i^{-1}$ also preserve $r$-term quantum Plücker relations for $r\geq 4$ in…

LLRZ's support conjecture. The coefficient $e(p,q)$ of $C[a_1,a_2]$ is nonzero if and only if $D(p,q)\ge0$.

Tropical parametrization conjecture. The basis $\mathbb{I}_{\mathcal{X}}^q(\mathcal{L}_{\mathfrak{sl}_3}^p(\Sigma,\mathbb{Z}))$ is parametrized by tropical points. Namely, for every…

Compatibility conjecture. For any ${\widetilde{w}}$ satisfying this condition, the pair $(\Lambda^{\widetilde{w}},\widetilde{\mathsf{B}}^{\widetilde{w}})$ is compatible, namely

Positivity conjecture. The coefficients $c_n$ for cluster monomials are contained in $\mathbb{N}$ in the classical case, or in $\mathbb{N}[v^{\pm}]$ in the quantum case.

Let $\mathbb{I}^q_\Delta$ be the quantum duality map and let $c^q(\ell,\ell';\ell”)$ denote the coefficients in the product expansion of its basis elements from the stated quantum-…

Let $\Delta$ be an ideal triangulation of a triangulable generalized marked surface $\frak{S}$, let $\mathbb{I}^q_\Delta$ be the proposed quantum duality map, and let…

Let ${{\mathbf L}}$ be a common triangular basis of a quantum cluster algebra. A basis element $b\in{{\mathbf L}}$ is real if $b^{2}\in{{\mathbf L}}$. Multiplicative reachability c…

Let $w$ be a Weyl group element, let ${\mathsf{A_q}}[N_{-}(w)]$ be the corresponding quantum unipotent subgroup, and let $\mathsf{B}^{{\mathsf{up}}}(w)$ be its dual canonical basis…

Let ${{\mathbf L}}$ be a common triangular basis of a quantum cluster algebra. A basis element $R\in{{\mathbf L}}$ is real if $R^{2}\in{{\mathbf L}}$. The analog of Leclerc's conje…