8 problems

Associativity conjecture. The $\mathbb{Z}[t^{\pm1/2}]$-module $K(\mathscr{C})_t$ with this operation is a $\mathbb{Z}[t^{\pm1/2}]$-algebra, and hence gives a not necessarily commut…

For $(i,p)\in{\widehat{\triangle}}_0$ and $k\in\mathbb{N}$, define the KR-monomial … For a KR-monomial $m$, let $F_q(m)$ be its associated KR-polynomial, and let $L_q(m)$ be the co…

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 $m\in\mathcal{M}_+$, and let $L_q(m)$ be the corresponding canonical-basis element of the quantum Grothendieck ring. Call $L_q(m)$ real if, for every…

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…

For each $(i,r)\in\hat{I}$, let $\tilde{\chi}_{i,r}\in K_t(\mathcal{O}^+_{\mathbb{Z}})$ be the quantum cluster variable obtained from the initial quantum seed by the prescribed mut…

Let $\mathscr{C}_{\mathbb{Z}}$ be the finite-dimensional representation category and let $\mathcal{O}^+_{\mathbb{Z}}$ be the category whose quantum Grothendieck ring is…