Brauer induction and equivariant stable homotopy (Q5956894)

Let \(R(G)\) denote the complex representation ring of a compact Lie group \(G\). When \(G\) is finite, R. Brauer proved in 1946 that the Abelian group \(R(G)\) is generated by representations induced from one-dimensional representations of subgroups of \(G\). This improved an earlier induction theorem of E. Artin concerning generation of the rational vector space \(R(G)\otimes\mathbb{Q}\). In 1951 Brauer proved an explicit version of Artin's result -- that is, given \(x\in R(G)\otimes\mathbb{Q}\) Brauer gave a canonical formula for \(x\) as a rational combination of representations induced from cyclic subgroups of \(G\). He posed the problem of doing the same for his induction theorem. In 1985 I discovered an explicit natural formula which solved Brauer's problem and appeared under the title of ``Explicit Brauer induction'' [in Invent. Math. 94, No. 3, 455-478 (1988; Zbl 0704.20009)]. My original formula was found by means of a Segal-conjecture-like result which I proved about stable homotopy classes of maps from the classifying space of \(G\) to that of a torus. This result was generalised by my student Piotr Zelewski to replace the torus by any compact Lie group and appeared generalised to equivariant stable homotopy [in \textit{J. P. May, V. P. Snaith} and \textit{P. Zelewski}, Q. J. Math., Oxf. II. Ser. 40, No. 160, 457-473 (1989; Zbl 0694.55013)]. Once the naturality properties of my canonical formula were understood \textit{R. Boltje} derived a different, superior, natural explicit formula [in Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)] and soon after \textit{P. Symonds} proved the existence of Boltje's formula by a topological method similar to my original one [Comment. Math. Helv. 66, No. 2, 169-184 (1991; Zbl 0797.20008)]. From my point of view the author's elegant article castigates me for not reading the literature. He shows how to go directly from an equivariant of a 1973 stable homotopy theorem of G. B. Segal (which I certainly did know of) appearing in [\textit{A. Kono}, Publ. Res. Inst. Math. Sci. 17, 553-556 (1981; Zbl 0506.55012)] to Symond's results using the methods (of the Oxford school, I believe) of equivariant fibrewise stable homotopy theory. In fact the author explained this castigation to me in person about a decade ago but has been diplomatically tardy in writing it up! Oblivious of Brauer's long-standing problem (as well as of Kono's paper and of fibrewise topology in the Cotswolds) I can only excuse my characteristic lapse of scholarship by saying that I was preoccupied with a number of applications of explicit Brauer induction which I had lined up in number theory, ramification theory and algebra, which appeared in [\textit{V. P. Snaith}, Explicit Brauer induction. With applications to algebra and number theory, Cambridge Studies in Advanced Mathematics 40, Cambridge University Press (1994; Zbl 0991.20011)].