Linear representation of the cactus groups by automorphisms of free $\mathbb{Z}[t]$-–module

<jats:p>In the current work, a faithful linear representation of the cactus group $J_n$ that depends on the parameter $t$ was constructed in explicit form. This linear representation is based on result of R. Yu and presentation of the cactus group $J_n$ in a minimal system of generators. It was proven that this linear representation is reducible. In the case $n = 3$, the resulting reduced representation is reducible for all $t \in \mathbb{R}$.</jats:p>