Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets.~II

Zh. T. Chernousova ; M. A. Dokuchaev ; M. A. Khibina ; V. V. Kirichenko ; S. G. Miroshnichenko ; V. N. Zhuravlev

Algebra and discrete mathematics, no. 2 (2003), pp. 47-86.

Résumé

The main concept of this part of the paper is that of a reduced exponent matrix and its quiver, which is strongly connected and simply laced. We give the description of quivers of reduced Gorenstein exponent matrices whose number $s$ of vertices is at most 7. For $2\leq s\leq 5$ we have that all adjacency matrices of such quivers are multiples of doubly stochastic matrices. We prove that for any permutation $\sigma$ on $n$ letters without fixed elements there exists a reduced Gorenstein tiled order $\Lambda$ with $\sigma(\mathcal E)=\sigma$. We show that for any positive integer $k$ there exists a Gorenstein tiled order $\Lambda_{k}$ with $in\Lambda_{k}=k$. The adjacency matrix of any cyclic Gorenstein order $\Lambda$ is a linear combination of powers of a permutation matrix $P_{\sigma}$ with non-negative coefficients, where $\sigma= \sigma(\Lambda)$. If $A$ is a noetherian prime semiperfect semidistributive ring of a finite global dimension, then $Q(A)$ be a strongly connected simply laced quiver which has no loops.

Détail

Keywords: semiperfect ring, exponent matrix, tiled order, quiver, partially ordered set, index of semiperfect ring, Gorenstein tiled order
Mots-clés : global dimension, transition matrix.