27 problems

Let $cal C$ be a monoidal abelian length category, and let $cal A$ be a cluster algebra. A monoidal categorification of $cal A$ is an isomorphism of $bb Z$-algebras…

Monoidal categorification conjecture. If $C$ is symmetric, the pair

Let $cal C_{varepsilon,xi}$ be the full subcategory of finite-dimensional modules of the restricted quantum loop algebra $U_varepsilon^{{\rm res}}(Lmathfrak{sl}_k)$ determined by t…

Let $cal C_{varepsilon,xi}$ be the full subcategory of finite-dimensional modules of the restricted quantum loop algebra $U_varepsilon^{{\rm res}}(Lmathfrak{sl}_k)$ whose compositi…

Assume that ${\mathfrak{g}}$ is simply laced. Let $\mathcal{A}$ be the infinite-rank cluster algebra embedded in $K_0(\mathcal{O}^{\mathrm{sh}})$ by … and let $I^{-1}\circ i$ be th…

Let $\mathcal{Q}^{\leq \xi}_l$ be the principal quiver of $\Gamma^{\leq \xi}_l$, let $\mathcal{C}^{\leq \xi}_l$ be its associated cluster category, and let…

Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra under consideration, and let $\mathscr{C}^{\leq \xi}_l$ be its monoidal categorification. A module is real when its…

Let $\mathsf{g}$ be the simply-laced finite type Lie algebra associated with the quantum affine algebra $U_q'(\mathfrak{g})$, let $\mathsf{I}$ be the index set of simple roots of…

Let $M$ and $N$ be dominant monomials such that $L(M)$ and $L(N)$ are real prime modules, and let $\mathfrak{e}(\mathbf{g}_M,\mathbf{g}_N)$ be their generic symmetrized $E$-invaria…

Let $\mathcal{A}$ be as above, let $M_1,M_2\in\mathcal{P}_\ell^+$ be dominant monomials such that $L(M_1)$ and $L(M_2)$ are real, and let $\mathbf{g}_{M_1},\mathbf{g}_{M_2}$ be the…

Let $\mathcal{A}$ be a cluster algebra with monoidal categorification $\mathscr{C}_\ell$ and additive categorification $(\mathcal{F},\widehat{T})$, let $\mathcal{C}$ be the stable…

Let $\mathscr{C}_\ell$ be the Hernandez–Leclerc category for $\ell\in\mathbb{Z}_{\geq 1}$, let $\mathcal{P}_\ell^+$ be its monoid of dominant monomials, and let…

Let $W$ be the Weyl group of a Dynkin diagram, and let $w,v,u\in W$ satisfy … Let $\mathscr{C}_{w,v}$ be the subcategory associated with $w$ and $v$, let $K_q(\mathscr{C}_{w,v})$ d…

Let $\qbA{}\simeq\mathcal{K}_t(\catC_{\mathfrak{g}}^0)$, let $\mathtt{b}\in\bg^+$, and let $\qbA{}(\mathtt{b})$ be the associated subalgebra. The monoidal categorification conjectu…

Grothendieck-ring isomorphism conjecture. The isomorphism satisfies

Irreducible-object preservation conjecture. In general, the isomorphism maps the class of an irreducible object of $\mathEuScript C_{\bar i,\bar i}$ to the class of an irreducible…

Vanishing conjecture. We conjecture that

Let $\mathcal{A}$ be a cluster algebra and $\mathcal{C}$ an Artinian monoidal categorification of $\mathcal{A}$. Assume that the simple objects of $\mathcal{C}$ are parametrized by…

Let $\mathfrak g$ be a symmetric Kac-Moody algebra, let $w$ be a Weyl group element, and define … Let $\mathbf B^\mathrm{up}$ denote the upper global basis. Kimura's quantization c…

Let $\mathbf i$ be an adaptable word, let $\mathcal A^\dagger(\mathbf i)$ be the associated quantum cluster algebra, and let $\kappa$ be the map to the relevant Grothendieck ring w…

General monoidal categorification conjecture. For arbitrary choices of $xi$ and for every type $A$, $D$, or $E$, $\mathcal{C}_xi$ is the monoidal categorification of a cluster alge…

Let $\mathscr B(\infty)$ be the crystal indexing the dual canonical basis. A family is called compatible or strongly compatible as in the source, and $b^{[m]}$ and $S_m(b)$ denote…

Let $\mathscr B(\infty)$ be the crystal indexing the dual canonical basis. For basis elements $G^{\operatorname{up}}(b_i)$, write $b_i\bot b_j$ for the compatibility relation used…

Nakajima's monoidal categorification conjecture. The cluster algebra $\mathscr A$ has a monoidal categorification, given by the monoidal subcategory $\mathscr C_1$.

Arbitrary-level monoidal categorification conjecture. The assignment