Three-partite subamalgams of tiled orders of finite lattice type (Q1295767)

Let \(D\) be a complete discrete valuation domain with quotient field \(F\). For a suitable subhereditary tiled order \(\Lambda\) in \(M_n(F)\), the author considers certain suborders \(\Lambda^\bullet\) of \(\Lambda\) by means of amalgamation modulo \(\text{Rad }D\). As the construction of \(\Lambda^\bullet\subset\Lambda\) depends on a partition \(n=n_1+n_2+n_3\), the order \(\Lambda^\bullet\) is called a three-partite subamalgam of \(\Lambda\). With any such \(\Lambda^\bullet\) the author associates a reduced Tits quadratic form \(q_{\Lambda^\bullet}\colon\mathbb{Z}^{n_1+2n_3+2}\to\mathbb{Z}\), and a two-peak poset \(I_{\Lambda^\bullet}\) with zero relations. The main theorem of the article asserts that \(\Lambda^\bullet\) is representation-finite if and only if \(q_{\Lambda^\bullet}\) is weakly positive, i.e. \(q_{\Lambda^\bullet}(z)>0\) for \(z\in\mathbb{N}^{n_1+2n_3+2}\). Furthermore, representation-finiteness of \(\Lambda^\bullet\) is characterized by a list of excluded minor suborders, and also by exclusion of certain two-peak subposets of \(I_{\Lambda^\bullet}\) with zero relations. With any three-partite subamalgam \(\Lambda^\bullet\) of \(\Lambda\), the author also associates a family of parabolic algebraic gropus \(G(v)\), \(v\in\mathbb{N}^{n_1+2n_3+2}\), acting on irreducible algebraic varieties \(M(v)\) over the residue class field of \(D\). Then \(\Lambda^\bullet\) is representation-finite if and only if \(M(v)\) has finitely many \(G(v)\)-orbits, that is, of \(\dim G(v)\geq\dim M(v)\) for all \(v\). In this case, a construction of the Auslander-Reiten quiver is given for \(\Lambda^\bullet\), and it is shown that the indecomposable \(\Lambda^\bullet\)-lattices are in one-to-one correspondence with certain positive roots of \(q_{\Lambda^\bullet}\).