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Given $n\leq d<\infty$, we investigate the existence of algebras of globaldimension $d$ which admit an $n$-cluster tilting subcategory. We construct manysuch examples using representation-directed algebras. First, given tworepresentation-directed algebras $A$ and $B$, a projective $A$-module $P$ andan injective $B$-module $I$ satisfying certain conditions, we show how we canconstruct a new representation-directed algebra $\Lambda$ in such a way thatthe representation theory of $\Lambda$ is completely described by therepresentation theories of $A$ and $B$. Next we introduce $n$-fracturedsubcategories which generalize $n$-cluster tilting subcategories forrepresentation-directed algebras. We then show how one can construct an$n$-cluster tilting subcategory for $\Lambda$ by using $n$-fracturedsubcategories of $A$ and $B$. As an application of our construction, we showthat if $n$ is odd and $d\geq n$ then there exists an algebra admitting an$n$-cluster tilting subcategory and having global dimension $d$. We show thesame result if $n$ is even and $d$ is odd or $d\geq 2n$.

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