14 problems

Let $B_n$ be the braid group on $n$ strands and let $K_n=[B_n,B_n]$ be its commutator subgroup. A subgroup has index $r$ if its coset action gives a transitive representation…

Let $B_n$ be the braid group on $n$ strands, let $K_n=[B_n,B_n]$ be its commutator subgroup, and let $S_r$ be the symmetric group on $r$ letters. A representation is a homomorphism…

Let $p\geq 5$ be prime, and let an exceptional group be a finite group $G$ having a normal subgroup $N$ such that $\mu(G/N)>\mu(G)$, where $\mu(G)$ is the least integer $n$ for whi…

Karn's permutation-module conjecture. Two copies of $cmathcal{S}_{cmathrm{OS}}(n+1,n-1)$ together constitute a permutation module.

Young-subgroup positivity conjecture. For $i\leq j\leq k\leq\ell$ with $i+\ell=j+k$, the element

Let $G$ be a comparability graph, meaning the graph of a partial order. Write $\mathcal{R}_2$ for the class of graphs with representation number at most two, and…

Let $G$ be a finite group and $Q$ a quotient of $G$. A quotient $Q$ is distinguished if its minimal faithful permutation representation degree satisfies $\mu(Q)<\mu(G)$, where…

Let $P$ be a natural unit interval order on $[n]$. Tymoczko's representation of $\mathfrak S_n$ on the cohomology $H^*({\mathcal H}(P))$ is the representation under consideration.…

Classification conjecture. Every good non-cyclic transitive homomorphism $B_k\to S_n$ is canonical when $n=k$, standard when $k\ne n$ and $k\mid n$, or derived from a canonical or…

Matei–Suciu divisibility conjecture. There is no non-cyclic transitive homomorphism

Standard-or-derived conjecture. Every good non-cyclic transitive homomorphism $B_k\to S_n$ is canonical if $n=k$, standard if $k\ne n$ and $k\mid n$, or derived from a canonical or…

Good homomorphism conjecture. Every non-cyclic transitive homomorphism $B_k\to S_n$ with $n>2k$ and $k>8$ is good.

Stable-degree conjecture. There exists a stably equivalent permutation representation $\rho'\colon G\to S_n$ such that

For a finite group $G$, let $4\Delta(G)4$ denote the invariant used in the paper, and let the sums range over isomorphism classes of groups of the indicated order. Numerical-sum co…