Automorphisms of tiled orders (Q1611850)

The automorphism group of a tiled order \(\Lambda\) is calculated. If \(Q(\Lambda)\) denotes the link graph of \(\Lambda\), every automorphism of \(\Lambda\) induces an automorphism of \(Q(\Lambda)\). Since, in general, a tiled order \(\Lambda\) is not fully determined by its link graph, \(\Aut(\Lambda)\) is mapped to a subgroup \({\mathcal O}_\Lambda\) of \(\Aut(Q(\Lambda))\) which may be proper if \(\Lambda\) is not, e.g., a \((0,1)\)-order. The authors show that the kernel of \(\Aut(\Lambda)\twoheadrightarrow{\mathcal O}_\Lambda\) is the group \(\text{Inn}(\Lambda)\) of inner automorphisms, and thus \(\Aut(\Lambda)=\text{Inn}(\Lambda)\rtimes{\mathcal O}_\Lambda\). Their explicit description of \({\mathcal O}_\Lambda\) can be interpreted as follows. There are finitely many representations of \(\Lambda\) by an exponent matrix \(E\) with non-negative entries, and these exponent matrices correspond bijectively to the irreducible representations of \(\Lambda\). The group \({\mathcal O}_\Lambda\) of outer automorphisms consists of exactly those permutations of the rows and columns of \(E\) which lead again to an exponent matrix of \(\Lambda\). In other words, if \({\mathfrak S}_\Lambda\) denotes the lattice of irreducible \(\Lambda\)-representations in a fixed simple module over the ambient matrix algebra of \(\Lambda\), then the outer automorphisms of \(\Lambda\) correspond to symmetries of \({\mathfrak S}_\Lambda\) modulo translations.