21 problems

Let $J_n$ be the cactus group on $n$ strands, and let $k$ be a nonnegative integer. A free abelian subgroup of rank $k$ is a subgroup isomorphic to $\mathbb{Z}^k$. Free abelian sub…

Let $J_n$ be the cactus group on $n$ strands. A finite group embeds into $J_n$ precisely when it embeds into an iterated permutational wreath product of copies of…

Let $\lambda=l_1\omega_1+l_2\omega_2\in P^+$, and let $N^{i;\lambda}$, for $i\in\{1,2\}$, be the matrices of the cactus-group generators $\sigma^i$ with respect to the basis…

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…

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…

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…

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 $\breve D_n$ 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 $W$ 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 $\fg=\mathfrak{gl}_N$, let the $V_{\l_i}$ be symmetric powers of the standard representation, and let $PJ_n$ act on the associated Gelfand–Tsetlin polytope through Bethe-vector…

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…