Congruences of bigraphs with applications to Grothendieck group recognition II. Coxeter type study
What is it about?
In this two parts article with the same main title we study a problem of Coxeter-Gram spectral analysis of bigraphs (a class of signed graphs). We ask for a criterion deciding if a given bigraph is weakly or strongly Gram-congruent with a graph. The problem is inspired by recent works of Simson et al. started in [SIAM J. Discr. Math. 27 (2013), 827-854], and by problems related to integral quadratic forms, bilinear lattices, representation theory of algebras, algebraic methods in graph theory and the isotropy groups of bigraphs. In this Part II we develop general combinatorial techniques, with the use of inflation algorithm discussed in Part I, morsifications and the isotropy group of a bigraph, and we provide a constructive solution of the problem for the class of all positive connected loop-free bigraphs. Moreover, we present an application of our results to Grothendieck group recognition problem: deciding if a given bilinear lattice is the Grothendieck group of some category. Our techniques are tested in a series of experiments for so-called Nakayama bigraphs, illustrating the applications in practice and certain related phenomena.
Why is it important?
Grothendieck group recognition problem can be treated as a first step, of a computational character, in recognizing and understanding the derived equivalence class of a finite-dimensional algebra, which is an important and difficult problem appearing in many branches of representation theory and related areas.