20 problems

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 $m\in\mathcal{M}_+$, 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 $n,\tilde{n}\in\mathbb{Z}_{>1}$ satisfy $\tilde{n}/n\in\mathbb{Z}$. For the inflation homomorphism … given on fundamental modules by…

Let $\mathscr{C}_{\mathbb{Z}}$ be the relevant representation category, let $K(\mathscr{C}_{\mathbb{Z}})_t$ be its $t$-deformed Grothendieck module, and let…

For $n\geq2$, let $m_n\colon K(\mathscr{C})_t^n\to K(\mathscr{C})_t$ be the multilinear operation defined on filtration classes by … with $m_1=\mathsf{id}$ and $m_2(x,y)=x*y$. Stro…

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…

Let $\mathcal{K}_t(\mathscr{C}_\mathfrak{g})$ be the quantum Grothendieck ring, with canonical basis $\mathbf{L}_t$ consisting of elements $[L]_t$ indexed by simple modules…

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…

Let $d4ac_t$ be the quantum torus containing the quantum Grothendieck ring, and let $L_t(m)$ be its canonical-basis element associated with $md41ed50c$. Positivity conjecture for…

Let $d49e$ be the quantum Grothendieck ring of $d4a2_{d4a2}$, with canonical-basis elements $L_t(m)$ indexed by $md41ed50c$, and let $L(m)$ be the corresponding simple module with…

Let $\widehat{I}$ be the index set used for the quantum $T$-systems, let $m_{k,p}^{(i)}$ be the dominant monomial associated with $(i,p)\in\widehat{I}$ and $k\in\mathbb{Z}_{\ge 0}$…

Let $\mathcal{K}_t$ be the quantum Grothendieck ring, let $\mathcal{M}$ be the set of dominant monomials, and let $L_t(m)$ denote the canonical element associated with the simple m…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\mathcal{K}_t(\mathscr{C}_{\mathfrak{g}})$ be its quantum Grothendieck ring, and let $F_t(m)$ denote the elements used to formulate…

Assume that $\mathfrak{g}$ is non-simply-laced and that $\mathsf{g}$ is a simply-laced Lie algebra obtained by unfolding its Dynkin diagram, with $(\mathfrak{g},\mathsf{g})$ of typ…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\mathcal{K}_{t}(\mathscr{C}_{\mathfrak{g}})$ be its quantum Grothendieck ring, let $\mathcal{Y}_t$ be the ambient quantum torus, an…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\mathcal{K}_{t}(\mathscr{C}_{\mathfrak{g}})$ be its quantum Grothendieck ring, and let $\{L_t(m)\}_{m\in\mathcal{M}}$ be the canoni…

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…