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Group representations, $$\lambda$$-rings and the $$J$$-homomorphism. (English) Zbl 0159.53301
If $$E, F$$ are representations of a finite group $$G$$, one may ask whether there exists a non-trivial $$G$$-diffeomorphism, $$G$$-homeomorphism, $$G$$-homotopy equivalence, or some other type of $$G$$-map from $$E$$ to $$F$$. One of the main theorems of this paper is:
If $$E$$ is an irreducible, unitary (or orthogonal) representation of a $$p$$-group $$G$$ $$(p\neq 2)$$, then there exists a continuous $$G$$-map of unit spheres $$S(E)\to S(F)$$ of degree prime to $$p$$ if, and only if, $$F$$ is conjugate to $$E$$.
(A unitary representation of $$G$$ may be realised in $$\mathbb Q(\omega)$$, where $$\omega$$ is a primitive $$N$$-th root of unity and $$N$$ is the order of $$G$$. The representations are said to be conjugate if they are conjugate by an element of $$\Gamma_N$$, the Galois group of $$\mathbb Q(\omega)$$ over $$\mathbb Q$$.)
Note that, in particular, this theorem exhibits non-trivial $$G$$-maps between non-isomorphic representations. More generally, two unitary representations $$E, F$$ of an arbitrary finite group $$G$$ are said to be $$J$$-equivalent if there are $$G$$-maps from $$S(E)$$ to $$S(F)$$ and from $$S(F)$$ to $$S(E)$$, both of degree prime to the order of $$G$$. Let $$R(G)$$ be the complex representation ring of $$G$$, and if $$E$$ is a unitary representation of $$G$$, let $$[E]$$ denote its class in $$R(G)$$. Define $$T(G)$$ to be the additive subgroup of elements $$[E] - [F]$$ where $$E, F$$ are $$J$$-equivalent and let $$J(G)=R(G)/T(G)$$. In standard notation let $$R(G)_{\Gamma_N}=R(G)/W(G)$$ be the quotient ring of coinvariants, where $$W(G)$$ is the additive subgroup generated by elements $$[E]-[\alpha E]$$, $$\alpha\in\Gamma_N$$. The main theorem states:
$$J(G) =R(G)_{\Gamma_N}$$ for a $$p$$-group of odd order.
The $$J$$-homomorphism $$\nu: R(G)_{\Gamma_N}\to J(G)$$ is defined by constructing explicit $$J$$-equivalences between conjugate representations. This is done for monomial transfer groups (which include $$p$$-groups). The map $$\nu$$ is then visibly an epimorphism and to show it is a monomorphism, algebraic invariants of $$J$$-equivalent representations are defined. The corresponding theorems for orthogonal representations follow from this result.
The techniques of proof are analogous to the work of J. F. Adams on the groups $$J(X)$$ [Topology 2, 181–195 (1963; Zbl 0121.39704); 3, 137–171 (1965; Zbl 0137.16801); 3, 193–222 (1965; Zbl 0137.16901); 5, 21–71(1966; Zbl 0145.19902)], but are considerably adapted to suit this particular ease. In particular, the purely algebraic part of the theory is isolated and a systematic study is made of special $$\lambda$$-rings and of their $$p$$-adic completions. This occurs in the first part of the paper and may be read independently. (A special $$\lambda$$-ring is defined, by Grothendieck, to be a commutative ring with identity, together with a family of operators $$\{\lambda^n\}_{n>0}$$ satisfying certain relations given by the formal properties of exterior powers. Examples include $$R(G)$$ for a finite group $$G$$ and $$K(X)$$ for a topological space $$X$$.) Theorems proved include an algebraic version of the splitting principle for vector bundles. Later developments in the paper also include an algebraic version of Adams’ theorem “$$J'(X)= J''(X)$$” which gives considerable insight into Adams’ original proof.
Reviewer: Michael F. Atiyah

##### MSC:
 20-XX Group theory and generalizations 55N15 Topological $$K$$-theory 19L20 $$J$$-homomorphism, Adams operations
##### Citations:
Zbl 0121.39704; Zbl 0137.16801; Zbl 0137.16901; Zbl 0145.19902
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