Eigenvalues and elementary divisors of Cartan matrices of cyclic blocks with \(l(B)\leqslant 5\) and tame blocks. (Q706018)

In the paper under review, the author investigates connections between elementary divisors and eigenvalues of the Cartan matrix \(C\) of a block \(B\) of a finite group \(G\). In particular, he states the following decomposition conjecture: Suppose that the characteristic polynomial \(f(x)\) of \(C\) factors into irreducible integer polynomials \(f_1(x),\dots,f_r(x)\). For \(i=1,\dots,r\) let \(R_i\) be the set of roots of \(f_i(x)\). Then the set \(E\) of elementary divisors of \(C\) decomposes into disjoint sets \(E_1,\dots,E_r\) such that \(|E_i|=|R_i|\) and \(\prod_{e\in E_i}e=|f_i(0)|\) for \(i=1,\dots,r\). Moreover, the unique largest eigenvalue of \(C\) and the unique largest elementary divisor of \(C\) both belong to \(R_1\). Then, by a careful case-by-case analysis, the author verifies his conjecture for cyclic blocks with at most 5 irreducible Brauer characters, and for tame blocks.