94 problems

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 ${\cal M}$ 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 $w_0$ 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…

Let $G_{m,n}$ be the degree-$m$ part of the grove algebra, and let $c$ generate the cyclic action … The basis elements are to be indexed by height $m$ staircase plane partitions of…

Let $(V,\mathbb{B}_V)$ be a based $U_{-q^2}$-module, for example a finite-dimensional irreducible type I representation with Lusztig canonical basis $\mathbb{B}_V$. Let…

Positivity conjectures. The Laurent polynomials $Q_{x,w}\in\mathbb{Z}[v,v^{-1}]$ have non-negative coefficients for all $x\in W_{\mathrm{aff}}$ and $w\in W_{\mathrm{aff}}^+$, and t…

Let the skein algebra of a marked surface have frozen variables, and consider its localization at those frozen variables. The skein algebra carries a (quantized) band basis and a c…

Consider the $\imath$quantum group of type ${\rm AIII}_{2r}$ and its dual canonical basis. Type-${\rm AIII}_{2r}$ positivity conjecture. The positivity results for the dual canonic…

Let $\widetilde{{\mathbf U}}^\imath$ be a right coideal subalgebra of $\widetilde{{\mathbf U}}$, with embedding $\iota$, and let $\mathcal{B}^\imath$ and $\mathcal{B}$ be the respe…

Let $\widetilde{{\mathbf U}}$ be the quantum group with double (equivalently, dual) canonical basis, and let its coproduct be the corresponding bialgebra coproduct. Berenstein–Gree…

Canonical-basis characterization conjecture.

Limit-vector and prime-element conjecture. All limit g-vectors for $\mathbb{C}[\operatorname{Gr}(3,9)]$ and $\mathbb{C}[\operatorname{Gr}(4,8)]$ can be obtained by the braid-group…