23 problems

Let $\mathfrak{g}$ be a simple Lie algebra, let $\lambda$ be a minuscule dominant weight, and let $w_0^J$ be the corresponding Weyl group element. Write $C_{\mathfrak{g}}$ for the…

Let $L_n$ be the quadratic Lie algebra associated with the pure cactus group, let $\mathbf{L}_n$ be the corresponding braid Lie algebra, and let $\xi_n:L_n\to\mathbf{L}_n$ be defin…

Let $\widehat{\Gamma_n}$ be the prounipotent completion of the pure cactus group $\Gamma_n$, and let…

Let $\Gamma_n$ be the pure cactus group, let $\Gamma_n^k$ denote its lower central series, and let $\mathcal{L}_n$ be the associated graded Lie algebra after quotienting by $2$-tor…

Let $M_n$ be the real locus of the moduli space of stable genus-zero curves with $n$ marked points. A connected space is rational $K(\pi,1)$ if its rational completion is a…

Let $U_n=U(L_n)$, where $L_n$ is the quadratic Lie algebra generated by the elements $\mu_{ijk}$. Consider the natural morphisms $U_{n+1}\to U_n$ and $L_{n+1}\to L_n$ sending…

Let $L_n$ be the quadratic Lie algebra generated by the elements $\mu_{ijk}$, and let $U_n=U(L_n)$ be its universal enveloping algebra. Freeness conjecture. The $\mathbb{Z}$-module…

Suppose that $\mathtt{C}$ and $\mathtt{D}$ are coboundary monoidal categories, for example $\mathtt{C}=\mathfrak{g}\text{-}\mathtt{Crys}$ and…

Let $C_n$ be the cactus group acting on the set of standard Young tableaux of a partition, and let $A_N$ denote the alternating group on $N$ tableaux, where $N$ is the number of st…

Let $c$ be a constant. A partition $\lambda$ of $n$ boxes is called generic if its Young diagram fits into a $c\sqrt n$ by $c\sqrt n$ grid. Write $\operatorname{SYT}(\lambda)$ for…

A d-complete poset is a poset in the sense of the standard definition of d-completeness. A poset is LE-cactus when the Bender–Knuth involutions acting on its linear extensions sati…

Let $\breve D_n\to\widehat D_n$ and $\breve P_n\to\widehat P_n$ be the quotient maps, and define the two-sided mapping cylinders … … Let $\mathcal F_n(\mathbb{R})$,…

Let $\breve D_n$ be the cube complex whose cubes are indexed by $[n]$-labelled cyclic forests, and let $\overline M_{n+2}^\sigma(\mathbb{R})$ be the indicated real locus of the mod…

Let $P$ be a d-complete poset. A poset is LE-cactus when its Bender–Knuth group action on its linear extensions satisfies the cactus relations. The d-complete-poset LE-cactus conje…

Type $D_m$ cactus-generator conjecture. The action of the cactus group on $B(n\omega_1)$ is generated by the actions of cactus generators corresponding to one-element and two-eleme…

Let $J_{n+1}$ be the cactus group associated with the split real moduli space, and let the caterpillar points index the eigenbases of Bethe subalgebras on a tensor product of skew…

Let $\overline{M_{0,n+2}^{comp}}$ be the compact real moduli space, and let the eigenlines of Bethe subalgebras on a tensor product of Kirillov–Reshetikhin representations form the…

Let $V_i$, $i=1,\ldots,k$, be skew representations of $Y(\mathfrak{gl}_n)$, and consider the tensor product $\bigotimes_{i=1}^kV_i(x_i)$ in the split real setting, where the parame…

Etingof–Henriques–Kamnitzer–Rains conjecture. The holonomy Lie algebra is isomorphic to the Magnus construction:

Let $W$ be a Weyl group, let $C_\Delta$ be the cactus group acting on $W$, and suppose that the family of algebras $D_\chi$, parametrized by $\chi\in\mathfrak h^{reg}$, compactifie…

Let the $BK$-groups denote the groups generated by the Bender–Knuth involutions on standard tableaux, and let Krammer's groups be the groups defined in the cited presentation. Bend…

Let $\CB_{\ul{\l}}$ be the set of highest elements in the tensor product $\CB_{\l_1}\otimes\cdots\otimes\CB_{\l_n}$ of $\fg$-crystals, and let $B_{\ul{\l}}(\ul{z})$ be the set of B…

Let $\fg$ be a semisimple Lie algebra, and let the cactus group act on the set of Bethe vectors of the corresponding Gaudin magnet chain. The general coboundary category formalism…