Cells and cacti

Cédric Bonnafé (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

Let $(W,S)$ be a Coxeter system, let $\varphi$ be a weight function on $S$ and let ${\mathrm{Cact}}_W$ denote the associated {\it cactus group}. Following an idea of I. Losev, we construct an action of ${\mathrm{Cact}}_W \times {\mathrm{Cact}}_W$ on $W$ which has nice properties with respect to the partition of $W$ into left, right or two-sided cells (under some hypothesis, which hold for instance if $\varphi$ is constant or if $W$ is finite of rank $< 5$). It must be noticed that the action depends heavily on $\varphi$.

Domaines: Théorie des représentations [math.RT]

Cédric Bonnafé. Cells and cacti. International Mathematics Research Notices, 2016, 19 (1), pp.5775-5800. ⟨10.1093/imrn/rnv324⟩. ⟨hal-01168198v3⟩