Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Abstract

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms b ˆ I , b I , b ¯ I , b I • : Z I × Z I → Z (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z -bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form q ¯ I ( x ) = b ¯ I ( x , x ) weakly positive (resp. with the reduced Euler form q I • ( x ) = b I • ( x , x ) weakly positive), and posets I, with the Tits form q ˆ I ( x ) = b ¯ I ( x , x ) weakly positive.