46 problems

Let $\widetilde{\mathscr{D}}$ be a design and let ${\mathscr{D}}(\widetilde{\mathscr{D}},(e_1,e_2))$ denote the design obtained from it by the rectangular combination construction,…

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…

Wreath product conjecture. The algebra $J$ is a Jordan superalgebra. In this case, $J$ should be called the wreath product of $J_1$ with $H$.

Let $\Gamma$ be a finite subgroup of $SL_2(\mathbb{C})$, let $\Gamma_n=\Gamma^n\rtimes S_n$, and let $X_{\Gamma,n}$ be the union of connected components of the $\Gamma$-fixed locus…

Let ${\Gamma}$ and $G$ be finite groups. For each $n$, consider the complexified rings $\mathbb C\otimes_{\mathbb Z}\mathcal G_{\Gamma}(n)$ and…

Proposed colored-face model. The ideal $SI'_{n,r}$ is stable under ${\mathbb{Z}}_r\wr{\mathfrak S}_n$, and there is a module isomorphism

Sagan–Swanson's conjecture. The set of monomials ${\mathcal{A}}_{n,r}$ descends to a vector space basis of $SR_{n,r}$.

The LEF nilpotent-product conjecture. For each $k\geq 1$, the $k$-nilpotent product of a finite family of LEF groups is LEF.

Let $G$ and $F$ be groups, and define their wreath product by $G\wr F=(\prod_F G)\rtimes F$, with $F$ permuting the factors. The Full Farrell–Jones conjecture. A group $G$ satisfie…

Alspach et al.'s wreath-product conjecture. If $G$ and $H$ are hamiltonian decomposable directed graphs, then $G\wr H$ is also hamiltonian decomposable.

Wreath Macdonald norm conjecture. The wreath Macdonald polynomials satisfy

Let $G$ be a finite group, let $F_r$ be the free group of rank $r$, and let $w\in F_r$. For a stable irreducible character $\chi=(\chi_n)_{n\in\mathbb{N}}$ of $G\wr S_{\bullet}$, w…

Let $K$ be a Hilbertian field of characteristic $0$, and let $n\geq 2$. For a polynomial $f\in K[x]$, write $f^{-k}(0)$ for the set of $k$-th preimages of $0$, and let $[S_n]^k$ de…

Let $\Sigma_n\curvearrowright X_n$ be an asymptotically oligomorphic sequence of group actions, let $G$ be a compact group, and let $\phi\in\widehat G\setminus\\{\mathbf{1}\\}$. Fo…

Let $W_n$ be the ambient binary wreath-product group acting on $\{1,\dots,2^n\}$. For $n\geq 1$, let $H\leq W_n$ be a subgroup containing a permutation $\sigma\in W_n$ that acts tr…

Let $\Delta$ be a finite group. The wreath product $\Delta \wr \mathbb{Z}$ is locally stable. Finite-wreath-product local stability conjecture. Every finite wreath product of this…

Let $K/H$ be the set of equivalence classes of $K$ under the action of $H$, and let $N$ denote the length of a surjective strategy. Orbit-bound conjecture. If $G\wr H$ has a surjec…

Let $G\wr H$ be a wreath product with state group $K$, and call a surjective strategy minimal when it has the form $\{k_i\in K\}_{i=1}^{|K|-1}$. Minimal strategy conjecture. Whenev…

Let $K$ be the state group of a generalized spinning switches puzzle, and measure performance by the expected number of moves when the initial hidden state is chosen uniformly. Asy…

Let $G\wr H$ be a wreath product, and call a surjective strategy palindromic when its sequence of moves $S'=\{k'_i\in K\}_{i=1}^N$ satisfies $k'_i=k'_{N-i+1}$ for every $i$. Palind…

Let $G$ be a group generated by elements of order $p^n$, let $H$ be a $p$-group acting faithfully on a set of switches $\Omega$, and let $G\wr_\Omega H$ denote the corresponding wr…

Let $A_n$ be the alternating group on $n$ letters, let $C_3$ be the cyclic group of order $3$, and let $A_n\wr C_3$ denote their wreath product. Alternating-group wreath-product st…

Let $G$ be a finite group generated by involutions, and let $H$ be a $2$-group acting faithfully on the set of switches. The wreath product is denoted by $G\wr H$. Prime-power wrea…

Let a finite group be a group with finitely many elements, and call it a 2-group when its order is a power of $2$. Folklore conjecture. Almost all groups are $2$-groups. This conje…

Let $\mathcal{C}=(C_j)_{1\le j\le 2k}$ be a strongly generic tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$ satisfying the condition referred to…