37 problems

Suppose that $\tilde{\Phi}$ is not of type $\mathsf{B}\mathsf{C}$. Let $\tilde{\mathbf A}$ be the Cartan matrix whose realization is given by the vector space ${}^e\!{\mathfrak s}$…

Chain-independence conjecture. The colored directed graph defined by the action of root operators on the admissible subsets corresponding to any $\lambda$-chain does not depend on…

Let \text{bf g} be a symmetrizable Kac-Moody algebra, let $G$ be the (minimal) Kac-Moody group, and let $M_{drr}$ be the monoid associated to the category of integrable…

Let $\mathfrak g$ be a Kac–Moody algebra, let $\Lambda_1,\dots,\Lambda_n$ be dominant integral weights, let $z_1,\dots,z_n$ be points, and let $\boldsymbol b$ be symmetric paramete…

Kac's conjectures. For every root $d\in\mathbf{N}I$, there exists a polynomial $P_d(t)$ with nonnegative integer coefficients such that

Let $\mathbb{X}_{\mathbf{p},\underline{\lambda}}$ be a weighted projective line, let $\mathfrak{Lg}$ be its corresponding loop Kac–Moody algebra, and use the identification…

Let $M_{0,\lambda}(d,v)$ be the quiver variety in question, and set … Let $(L_d)_0$ denote the $0$-weight space of the Kac–Moody algebra associated to the quiver, for the represent…

Let ${\mathfrak{e}}_{10}$ be the hyperbolic Kac–Moody algebra, and let $\mathcal{U}_{2}^{{\mathbbm{Z}}_{n}}$ be the automorphism defining the orbifold action. Let…

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody Lie algebra, let $\delta$ be its null root, and let $V(\gamma)$ denote the irreducible highest weight representation of highest…

Let $\mathfrak{g}$ be a simple Lie algebra or an untwisted affine Kac–Moody Lie algebra, let $\rho$ denote the sum of the fundamental weights, and let $V(\gamma)$ be the irreducibl…

Holomorphic potential conjecture. Assuming the limiting-measure and operator conjecture, there exists a set of full $\nu_\kappa$-measure

Limiting-measure and operator conjecture. The family $\hat\nu_\kappa^\sigma$ converges weakly as $\sigma\to0$ to a measure $\hat\nu_\kappa$ on \hat\mathcal A. Moreover, the opera…

Let $\widehat{\mathfrak g}$ be the completed Kac–Moody algebra, with graded pieces $\mathfrak g_k$. Define … and let $\widehat{{U}}=\widehat{{U}}_1$ and $\mathcal H$ be as above. D…

Normalization conjecture. The group $\mathcal H$ normalizes $\widehat{{U}}$. This is one of the conjectures proposed for extensions of Scott's definition in the Kac–Moody setting;…

Kac–Moody extension conjecture. The analogues of the equations giving the Chevalley formulae for $\lambda$ and $-\lambda$ hold for $W$.

Let $\Gamma=(\Gamma_0,\Gamma_1)$ be a finite quiver with no loops, let $\alpha\in\mathbb{N}^n$ be a dimension vector, and let $A_\Gamma(\alpha,q)$ be its Kac polynomial. Let the Ka…

Let $\mathcal M(\lambda,\mu)$ be the Coulomb branch associated with a quiver gauge theory, let $\chi$ be the one-parameter subgroup defined by a character of the gauge group, and l…

Let $\mathcal M(\lambda,\mu)$ be the Coulomb branch associated with a quiver gauge theory, let $\chi\colon\mathbb C^\times\to\pi_1(\mathbf G)^\wedge$ be the one-parameter subgroup…

Bijection conjecture. The assignment $(X,\tau,\chi) \mapsto \theta(X,\tau,\chi)$ induces a bijection

Let $X \subseteq I$ be of finite type and let $\tau \in \operatorname{Aut}_X(A)$ satisfy $\tau^2=\operatorname{id}_I$ and $\tau|_X=\operatorname{id}_X$. For…

Let $\mathfrak g$ be a symmetrizable complex Kac–Moody algebra with Weyl group $\mathcal W$ and simple reflections $\mathcal S$. Fix a Weyl vector $\rho$, let…

Let $\dot{\mathfrak g}$ be an untwisted affine Kac–Moody Lie algebra, and let $\ddot{\mathrm{U}}_q(\dot{\mathfrak g})$ denote its double quantum affinization. Existence and uniquen…

Boundary fixed-point conjecture. The fixed-point set $\mathcal M(\lambda,\mu)^{\chi_i}$ is either empty or isomorphic to a Coulomb branch $\mathcal M_{A_1}(\lambda',\mu')$ of an…

Stronger minimal-generator conjecture. The ring $\hat R_{\mathrm{gen}}$ is generated by the four representations $W(\mu)$ corresponding to

Minimal-generator conjecture. The ring $\hat R_{\mathrm{gen}}$ is generated by the six representations $W(\mu)$ corresponding to