6 problems

Tangle algebra conjecture. There is a tangle algebra $\widetilde{\mathcal P}$ with a basis labelled by set partitions of $\{1,2,\ldots,2r\}$ that specializes to the partition algeb…

Let … , let … and … such that … The conjecture proposes that these algebras are isomorphic for a suitable choice of parameter. The supplied text gives no proof or resolution.

Let $r>1$, let $d=1$, and let $m$ and $n$ be the parameters for the map … is an isomorphism if and only if $m n$. No resolution or supporting result is provided in the supplied tex…

Let $k$ be a positive integer, let $\mathcal{A}_{2k}^{\text{aff}}$ be the affine partition algebra, and let $z_l$ and $p_n$ denote the central elements introduced in the paper for…

Let $D_r$ and $D'_r$ be the partition-algebra objects containing the rank-one central idempotents $E(r)$ and $E'(r)$, respectively. Let $h_i$, $s_i$, and $p_i$ denote the generator…

Let ${\rm PTL}_n$ be the Partition Temperley–Lieb algebra, let $\mathsf{P}_n$ denote the set-partition monoid on $\{1,\ldots,n\}$, and let $F$ be a reduced word in the generators u…