Group representations, \(\lambda\)-rings and the \(J\)-homomorphism.

*(English)*Zbl 0159.53301If \(E, F\) 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.

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