A note on the Hambleton-Taylor-Williams conjecture (Q2001407)

Let \(G\) be a finite group \(G\) and \(R\) noetherian ring. Consider the \(G\)-theory \(G_i(RG) = K_i(\mathcal{M}(RG))\), \(i \ge 0\), where \(\mathcal{M}(RG)\) is the category of all finitely generated left \(RG\)-modules. \textit{I. Hambleton} et al. [J. Algebra 116, No. 2, 466--470 (1988; Zbl 0658.16019)] conjectured a general formula for the decomposition of \(G_n(RG)\), shown hold for some large classes of finite groups. \textit{D. L. Webb} [Invent. Math. 84, 73--89 (1986; Zbl 0589.55008); J. Pure Appl. Algebra 39, 177--195 (1986; Zbl 0589.16018)] and \textit{D. Webb} and \textit{D. Yao} [\(K\)-Theory 7, No. 6, 575--578 (1993; Zbl 0806.19001)] discovered that the conjecture does not hold for the symmetric group \(S_5\). In this paper, the author shows that the conjectured HTW-decomposition also fails for the solvable group \(\mathrm{SL}(2, \mathbb{F}_3)\). This is proved by comparing the rank \(R(G) = \operatorname{rank} G_1(\mathbb{Z}G)\) with the rank \(P(G)\) of \(G_1(\mathbb{Z}G)\) as predicted by the HTW-decomposition. Nevertheless, the author proves a general inequality estimating the number of modular irreducible representations of \(G\) in terms of the rational irreducible representations of \(G\), and as a corollary, it follows that \(P(G) \ge R(G)\) for any \(G\). This gives an explanation of the failure of the HTW-decomposition for \(G_1(\mathbb{Z}G)\) There are also some positive results: The HTW-decomposition correctly predicts the torsion for \(G_1(\mathbb{Z}G)\) and the rank of \(G_n(\mathbb{Z}G)\) for \(n \neq 1\).