Gluing of $n$-cluster tilting subcategories for representation-directed algebras

Given $n\le d<\mathrm{\infty }$, we investigate the existence of algebras of global dimension $d$ which admit an $n$-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras $A$ and $B$, a projective $A$-module $P$ and an injective $B$-module $I$ satisfying certain conditions, we show how we can construct a new representation-directed algebra $\mathrm{\Lambda }$ in such a way that the representation theory of $\mathrm{\Lambda }$ is completely described by the representation theories of $A$ and $B$. Next we introduce $n$-fractured subcategories which generalize $n$-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an $n$-cluster tilting subcategory for $\mathrm{\Lambda }$ by using $n$-fractured subcategories of $A$ and $B$. As an application of our construction, we show that if $n$ is odd and $d\ge n$ then there exists an algebra admitting an $n$-cluster tilting subcategory and having global dimension $d$. We show the same result if $n$ is even and $d$ is odd or $d\ge 2n$.