On the number of simple modules of certain tame blocks and algebras (Q1106299)

The aim of this article is to give a bound for the number of simple modules over an algebra of quaternion type. Let \(\Lambda\) be a finite dimensional algebra over an algebraically closed field. Then \(\Lambda\) is said to be of quaternion type if (1) \(\Lambda\) is symmetric, indecomposable, representation-infinite and not wild. (2) Every projective-free \(\Lambda\)-module is \(\Omega\)-periodic, with period a divisor of 4. (3) The Cartan matrix of \(\Lambda\) is non-singular. The main theorem in this article states that, if \(\Lambda\) is of quaternion type, then \(\Lambda\) has at most three simple modules. - Examples of algebras of quaternion type are 2-blocks of a group algebra whose defect groups are generalized quaternion [see the author, Math. Ann. 281, No.4, 545-560 and 561-582 (1988)]. Therefore an immediate Corollary is \textit{J. Olsson}'s result [J. Algebra 36, 212-241 (1975; Zbl 0314.20009)] stating that, if B is a 2-block whose defect groups are generalized quaternion, then B has at most three simple modules.