209 problems

Canonical-basis characterization conjecture.

Let $h$ be the quantum characteristic, let $\boldsymbol{S}^{\alpha}$ and $\mathbf{D}^{\beta}$ be the spin modules indexed by an $h$-strict partition $\alpha$ and a restricted $h$-s…

Berenstein–Zelevinsky conjecture. The product $b_1b_2$ belongs to the canonical basis up to a power of $q$ if and only if $b_1$ and $b_2$ $q$-commute.

Let $\mathfrak g$ be a simple Lie algebra of simply-laced type $\mathbb{A},\mathbb{D},\mathbb{E}$, and let ${\mathcal B}_q^*$ be the basis dual to the canonical basis of…

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 ${\cal M}$ 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…

Henn's conjecture. There always exists a rotation into a basis $\boldsymbol{J}(\boldsymbol{x},\varepsilon)$ in which the master integrals satisfy an $\varepsilon$-factorized differ…

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 $\mathcal{U}_q(\mathfrak{g})$ be the quantum group associated with the semisimple Lie algebra $\mathfrak{g}$, equipped with the braid group action extending Lusztig's action. A…

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 $l,l',l”$ be integral laminations indexing the quantum canonical-basis elements, and let $c^q(l,l',l”)$ denote the Laurent-polynomial structure constants in their product. Fock…

Let $\Pi$ be the indexing set of partitions, let $|\emptyset\rangle$ be the empty-partition vector in the $q$-deformed Fock space, let $b_\lambda$ denote its canonical basis elemen…

Let $\Lambda_V$ be the representation variety under consideration, let $G_V$ act on it, and let $\Lambda_b$ be an irreducible component indexed by a canonical-basis element $b$. A…

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 $\lambda$ range over dominant weights, and let $({\mathcal O}_\lambda,\rho_\lambda)$ be the nilpotent orbit and irreducible representation of…

Let $\mathbf{s}_l \in \mathbb{Z}^l(s)$, and let $\mathbf{b}_{\lambda_l,\mathbf{s}_l}$ be the canonical-basis element indexed by $\lambda_l$ in the crystal component…

Let $H_m(v)$ be the Hecke algebra of type $A_{m-1}$, let $S(\lambda)$ be its Specht module, and suppose that $\mu$ is an $n$-regular partition. Let $D(\mu)$ be the simple head of…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, let $\bu^+$ be the positive part of its quantized enveloping algebra, and let $w_0$ be the longest element of its Wey…

Let $V_{\lambda}=\bigoplus_{\mu}V_{\lambda}(\mu)$ be a finite-dimensional irreducible representation of $\mathfrak{sl}_m$, categorified by blocks of Ariki–Koike cyclotomic Hecke al…

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 ${\cal D}$ be the positive double, let ${\cal D}^{\vee}$ be its Langlands dual, and let ${\cal D}({\mathbb Z}^t)$ be the set of integral tropical points. Let…

Let ${\cal D}$ be the positive quantum double associated with a cluster ensemble, and let ${\Bbb L}_+({\cal D})$ denote the space of regular positive functions on ${\cal D}$. Canon…

Let $E_{\mathcal M}$ be the PBW-basis constructed using the representation theory of affine quivers, and let $L_{\mathcal M}$ be Nakajima's modified PBW-basis. Applying Lusztig's m…

Sharp stabilization-bound conjecture. Theorem 3.3 remains valid when its bound $N$ is replaced by $N'$.

Uglov's conjecture. The transition matrix of the homogeneous component of degree $m$ of the canonical basis of $\boldsymbol{\operatorname{F}}_q[\boldsymbol s_l]$ is equal to the de…