For an integer $n\geq 8$ divisible by $4$, let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$. Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$. Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$. We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$. We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$, $\operatorname {\mathrm {PSU_{2}}}(R_n)$, $\operatorname {\mathrm {PU_{2}}}(R_n)$, $\operatorname {\mathrm {PU_2^\zeta }}(R_n)$, and $\operatorname {\mathrm {SO_{3}}}(R_n^+)$.

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

Let $Y \rightrightarrows X$ be a finite flat groupoid scheme with $X$ a quasi-projective variety and let $S$ be its coarse moduli scheme. We associate to the groupoid scheme a coherent sheaf of algebras $\mathcal{O}_{X / Y}$ on $S$ which we call the non-commutative coordinate ring of the groupoid scheme. We show that when $X$ is a smooth curve and the groupoid action is generically free, the non-commutative coordinate rings which can occur are, up to Morita equivalence, the hereditary orders on smooth curves. This gives a bijective correspondence between smooth one-dimensional Deligne–Mumford stacks of finite type and Morita equivalence classes of hereditary orders on smooth curves.