16 problems

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 $M$ 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…

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

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 $\mathbf i$ 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…

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