Sadek Alharbat (University of Leeds) – m-commutativity, m-avoidance and an old-new algebra of P. Martin

Abstract: 36 years ago, P. Martin defined  $M_m$  as an $m$-quotient of the Hecke algebra, where $1<m$ is an integer. $M_2$ being the Known Temperley-Lieb algebra. On the way towards understanding the structure of $M_m$, we find ourselves in the intersection of the two worlds: $A_n$ Coxeter groups and its representation $\operatorname{Sym}_{n+1}$.  I shall try to explain, by means of commutativity, which elements fit to be a linear basis for $M_m$ in a very primitive way and to show some algebraic and combinatorial properties of these newly born elements.