The strong no loop conjecture for special biserial algebras

Let A be a finite dimensional algebra over a field given by a quiver with relations. Let S be a simple A-module with a non-split self-extension, that is, the quiver has a loop at the corresponding vertex. The strong no loop conjecture claims that S is of infinite projective dimension; see [1, 6]. This conjecture remains open except for monomial algebras; see, for example, [2, 6, 8, 11]. Under certain hypothesis on the loop, Green, Solberg and Zacharia have shown that ExtA(S, S) does not vanish for every i ≥ 1; see [5, (4.2)]. In this paper, we shall first present a short proof of this result, because not only is the original proof rather complex, but also our idea possibly works for other cases. Next we observe that this result reduces the conjecture to the case where some power of the loop is a component of a polynomial relation. This reduction works particularly well when A is special biserial, due to a combinatorial description of the syzygies of string modules; see (2.2). Our main result says that ExtA(S, S) does not vanish for every i ≥ 1 if the convex support of S is special biserial. We shall also prove that if S has an almost split self-extension, then the block of A supporting S is a local Nakayama algebra, in particular, ExtA(S, S) does not vanish for every i ≥ 1. In the course of its proof, we easily get a characterization of Nakayama algebras, strengthening the one stated in [1, (IV.2.10)]. Contrary to what will be seen in this paper, Happel’s example stated in [5, Section 4] shows the existence of a simple module S with a loop but Ext(S, S) = 0 for infinitely many i. Our motivation for studying special biserial algebras comes from the following two aspects. First of all, since their representations are completely understood, they form naturally a testing class for various well-known conjectures in the representation theory of algebras. Secondly, these algebras play an important role in the modular representation theory of finite groups; see [7], tracing back to the classification of the indecomposable HarishChandra modules of the Lorentz group by Gelfand and Ponomarev; see [4].