${\tau }_d$-tilting theory for linear Nakayama algebras

Support $\tau$-tilting pairs, functorially finite torsion classes and $2$-term silting complexes are three much studied concepts in the representation theory of finite-dimensional algebras, which moreover turn out to be connected via work of Adachi, Iyama and Reiten. We investigate their higher-dimensional analogues via ${\tau }_d$-rigid pairs, $d$-torsion classes and $(d+1)$-term silting complexes as well as the connections between these three concepts. Our work is done in the setting of truncated linear Nakayama algebras $\mathrm{\Lambda }(n,l)=\mathbf{k}{\mathbb{A}}_n\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}{\mathbf{k}{\mathbb{A}}_n}^l$ admitting a $d$-cluster tilting module. More specifically, we classify ${\tau }_d$-rigid pairs $(M,P)$ of $\mathrm{\Lambda }(n,l)$ with $\mathrm{\mid }M\mathrm{\mid }+\mathrm{\mid }P\mathrm{\mid }=n$ via an explicit combinatorial description and show that they can be characterized by a certain maximality condition as well as by giving rise to a $(d+1)$-term silting complex in ${\mathrm{K}}^b(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}(\mathrm{\Lambda }(n,l)))$. We also describe all $d$-torsion classes of $\mathrm{\Lambda }(n,l)$. Finally, we compare our results to the classical case $d=1$ and investigate mutation with a special emphasis on the case where $d$ equals the global dimension of $\mathrm{\Lambda }$.