Isometries of combinatorial Tsirelson spaces

We extend existing results that characterize isometries on the Tsirelson-type spaces $T[\frac{1}{n},{\mathcal{S}}_1]$ ($n\in \mathbb{N},n\ge 2$) to the class $T[\theta ,{\mathcal{S}}_{\alpha }]$ ($\theta \in (0,\frac{1}{2}]$, $1⩽\alpha <{\omega }_1$), where ${\mathcal{S}}_{\alpha }$ denote the Schreier families of order $\alpha$. We prove that every isometry on $T[\theta ,{\mathcal{S}}_1]$ ($\theta \in (0,\frac{1}{2}]$) is determined by a permutation of the first $\lceil {\theta }^{-1}\rceil$ elements of the canonical unit basis followed by a possible sign-change of the corresponding coordinates together with a sign-change of the remaining coordinates. Moreover, we show that for the spaces $T[\theta ,{\mathcal{S}}_{\alpha }]$ ($\theta \in (0,\frac{1}{2}]$, $2⩽\alpha <{\omega }_1$) the isometries exhibit a more rigid character, namely, they are all implemented by a sign-change operation of the vector coordinates.