A framework for Coxeter spectral classification of finite posets and their mesh geometries of roots.
Summary: Following our paper [\textit{D. Simson}, Linear Algebra Appl. 433, No. 4, 699-717 (2010; Zbl 1196.15022)], we present a framework and computational tools for the Coxeter spectral classification of finite posets \(J\equiv(J,\preceq)\). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by \textit{Yu. A. Drozd} [Funct. Anal. Appl. 8, 219-225 (1974); translation from Funkts. Anal. Prilozh. 8, No. 3, 34-42 (1974; Zbl 0356.06003)]. We are mainly interested in a Coxeter spectral classification of posets \(J\) such that the symmetric Gram matrix \(G_J:=(1/2)[C_J+C_J^{\text{tr}}]\in\mathbb M_J(\mathbb Q)\) is positive semidefinite, where \(C_J\in\mathbb M_J(\mathbb Z)\) is the incidence matrix of \(J\). Following the idea of Drozd mentioned earlier, we associate to \(J\) its Coxeter matrix \(\text{Cox}_J:=-C_J\cdot C^{-\text{tr}}_J\), its Coxeter spectrum \(\mathbf{specc}_J\), a Coxeter polynomial \(\text{cox}_J(t)\in\mathbb Z[t]\), and a Coxeter number \(\mathbf c_J\). In case \(G_J\) is positive semi-definite, we also associate to \(J\) a reduced Coxeter number \(\check{\mathbf c}_J\), and the defect homomorphism \(\partial_J\colon\mathbb Z^J\to\mathbb Z\). In this case, the Coxeter spectrum \(\mathbf{specc}_J\) is a subset of the unit circle and consists of roots of unity. In case \(G_J\) is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets \(J\) with the Coxeter spectral properties of a simply laced Euclidean diagram \(DJ\in\{\widetilde{\mathbb D}_n,\widetilde{\mathbb E}_6,\widetilde{\mathbb E}_7,\widetilde{\mathbb E}_8\}\) associated with \(J\). Our aim of the Coxeter spectral analysis of such posets \(J\) is to answer the question when the Coxeter type \(\mathbf{Ctype}_J:=(\mathbf{specc}_J,\mathbf c_J,\check{\mathbf c}_J)\) of \(J\) determines its incidence matrix \(C_J\) (and, hence, the poset \(J\)) uniquely, up to a \(\mathbb Z\)-congruency. In connection with this question, we also discuss the problem studied by \textit{R. A. Horn} and \textit{V. Sergeichuk} [Linear Algebra Appl. 389, 347-353 (2004; Zbl 1068.15007)], if for any \(\mathbb Z\)-invertible matrix \(A\in\mathbb M_n(\mathbb Z)\), there is \(B\in\mathbb M_n(\mathbb Z)\) such that \(A^{\text{tr}}=B^{\text{tr}}\cdot A\cdot B\) and \(B^2=E\) is the identity matrix.
positive semidefinite symmetric Gram matrices
scientific article; zbMATH DE number 6165369
scientific article