On the source algebra equivalence class of blocks with cyclic defect groups, I

Trivial source character tables of , part II

“…By 2.3, each such block contains precisely e trivial source kG -modules with vertex , where e is the inertial index of the block. Moreover, in order to determine these modules up to isomorphism, we need parameters (1), (2) and (3) of 2.3, namely the Brauer trees with their type function, which are given in Table 4, and the module , which is always trivial in our case by [8, Proposition 6.5(a)]. Thus, the characters listed in Table 5 are obtained by applying the classification of the trivial source modules given in [7, Theorem 5.3(b)(2) and Theorem A.1(d)], exactly as in [4, Lemma 3.3].…”

“…The arguments are analogous to those given in the proof of Lemma 4.3. In this case, the parameters (1), (2) and (3) of 2.3 necessary to apply the classification of the trivial source modules given in [7, Theorem 5.3] are the Brauer trees with their type function given in Table 10 and the module , which is also always trivial in this case by [8, Proposition 6.5(a)]. The characters are then obtained exactly as in the proof of [4, Lemma 4.3].…”