94 problems

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…

Canonical-basis characterization conjecture.

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 ${\cal M}$ be the set of multisegments. For ${\bf m},{\bf n}\in{\cal M}$, let $G^*({\bf m})$ and $G^*({\bf n})$ be the corresponding dual canonical basis elements, let…

Let $m^{(i)}[p,s]$ be a Kirillov–Reshetikhin monomial, and let $F_q(m^{(i)}[p,s])$ and $L_q(m^{(i)}[p,s])$ denote the associated KR-polynomial and canonical-basis element. KR-polyn…

Fock–Goncharov conjecture. The upper cluster algebra $\mathcal{U}$ possesses a basis parametrized by the tropical points.

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 $\mathcal{A}$ be a cluster algebra, and let a cluster monomial mean a monomial in cluster variables all belonging to the same cluster. Strong positivity conjecture. Any cluster…

Let $\Lambda_d$ be the relevant variety of $d$-dimensional modules over the preprojective algebra $\Lambda$, and let $Z$ be an irreducible component of $\Lambda_d$. Let $b_Z$ and…

Let $V_\theta(\lambda)$ be the symmetric-crystal representation for a dominant integral weight $\lambda$ fixed by $\theta$, and let $L_\theta(\lambda)$ be its crystal lattice with…

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 ${\cal D}$ be the cluster double, with Langlands dual ${\cal D}^{\vee}$, and let ${\mathbb Z}_+\{S\}$ denote the semigroup of finitely supported nonnegative-integer-valued func…

Let $B^*$ be the dual canonical basis of the quantum matrix algebra $O_q(M(n))$. Call $b\in B^*$ decomposable if there exist $m\in\mathbb Z$ and $b_1,b_2\in B^*$ such that…

Principal-subspace basis conjecture. The basis

Let $R$ be a reduced root system with equal parameters $t_s=t_\ell=:t$. For a weight $\lambda$, let $C'_{\lambda}$ be the canonical basis element and let $P_{v_\mu,v_\lambda}(t)$ b…

Let $B_n$ be the compatible bases of the level-one representation obtained by applying the canonical basis of the completed quantum loop-algebra positive part to the vectors $v_n$,…

Let $L_{\Lambda_0}$ be the level-one representation, let $W_0$ be its principal subspace, let $B_0$ be the canonical basis induced on $W_0$, and let $B^+$ be the Leclerc–Thibon can…

Let $I_0$ be the ideal associated with the principal subspace $W_0$ in the completed quantum loop-algebra positive part, let $\mathcal{C}_0$ be the class of sheaves…

Let $r\geqslant 2$, let ${\cal H}_m(t)$ be the relevant Hecke-Clifford superalgebra, and let ${\cal C}_{m,r}$ be its category of integral modules whose eigenvalues lie in…

Let $Q$ be a Dynkin quiver of type $cA_n$, $cD_n$ or $cE_{6,7,8}$, let $U_v^-$ be the negative part of the quantized enveloping algebra of the corresponding Lie algebra, and let…

Let $b_1,b_2\in{\bf B}^*$, where $b_1$ is real, meaning $b_1^2\in q^{\mathbb Z}{\bf B}^*$. Suppose that $b_1b_2\notin q^{\mathbb Z}{\bf B}^*$. Two-term expansion conjecture. The ex…

Let ${\mathcal{P}}$ be the category of semisimple perverse sheaves on ${\mathfrak{T}(\mathbf{d})}'$, let $\mathcal{D}:{\mathcal{P}}\to{\mathcal{P}}$ be Verdier duality, and let…

Let ${\mathcal{P}}$ be the category of semisimple perverse sheaves on ${\mathfrak{T}(\mathbf{d})}'$ constructible with respect to the stratification by the…

Let $w_0$ be the longest element of the Weyl group, let $\mathbf{i}$ and $\mathbf{j}$ be reduced expressions of $w_0$, let $C_{\mathbf{i}}$ be the Lusztig cone, and let … be the tr…

Suppose $\mathbf{g}$ is of type $A_n$, and let $\mathbf{i}$ be a reduced expression for the longest Weyl-group element $w_0$. Let $\mathbf{k}=(1,3,5,\ldots,2,4,6,\ldots)$ and…