3 problems

Koolen's conjecture. For almost all finite connected simple graphs, the Terwilliger algebras coincide with the full matrix algebras.

Let $n\geq 6$, and let $\mathfrak{X}(\operatorname{Sym}(n))$ be the conjugacy class association scheme of the symmetric group. Let $T$ be its Terwilliger algebra with respect to th…

Nomura–Terwilliger conjecture. With respect to this notation, the following hold: (i) $E^*_0\mathcal{T}E^*_0$ is generated by $E^*_0\mathcal{D}E^*_0$; and (ii) the elements of…