A solvable counterexample to the Hambleton-Taylor-Williams Conjecture

TL;DR

This paper provides a counterexample to the Hambleton-Taylor-Williams conjecture by demonstrating that the solvable group SL(2, F_3) does not satisfy the proposed decomposition formula for algebraic K-theory.

Contribution

It shows that the HTW-decomposition conjecture fails for the solvable group SL(2, F_3), extending known counterexamples beyond symmetric groups.

Findings

SL(2, F_3) is a counterexample to the HTW conjecture

The rank of G_1(ZG) does not exceed the predicted expression

The conjecture does not hold for all solvable groups

Abstract

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group $G$ and noetherian ring $R.$ The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,\mathbb{F}_3)$ is also a counterexample to the conjectured HTW-decomposition. Furthermore, we prove that for any finite group $G$ the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition.