6 problems

Let $\mathbf{F}_n$ be the representation of the loop braid group $\mathrm{LB}_n$ on $V^{\otimes n}$ defined by the displayed braid and loop generators in the source, and suppose it…

Quotient structure conjecture. The structure of $L\!H_n/\chi^{(j+1)}$ is given by the $j\times j$ truncation of ${\mathcal M}^p_n$.

Semisimple quotient conjecture. For every $n$, the semisimple subalgebra of $L\!H_n(t=1)$ is precisely $\mathbb{C}S_n$.

Faithfulness conjecture. The extended super representation is faithful on $L\!H$ unless $t^2=1$.

Let $\rho$ be an irreducible $d$-dimensional matrix representation of the braid group $\mathcal B_3$, with $\rho(\sigma_1)=A$ and $\rho(\sigma_2)=B$. The matrices are in ordered tr…

Let $\Vloop=\Thomp{\mathit{LB}_*}$ and $\Floop=\Thomp{\mathit{PLB}_*}$ be the generalized Thompson groups containing the loop braid groups and pure loop braid groups, respectively.…