Explicit Brauer induction for symplectic and orthogonal representations. (Q2575124)

The classical Brauer Induction Theorem expresses a complex representation of a finite group as an integral linear combination of representations induced from one-dimensional representations of subgroups. This paper uses algebraic methods to give a uniform generalization of Brauer induction to unitary, orthogonal and symplectic representations. For \(n\in\mathbb{N}\), let \(X(n)\) denote either \(U(n)\) (unitary case), \(O(2n)\) (orthogonal case) or \(\text{Sp}(n)\) (symplectic case). Let \(G\) be a finite group and \(\rho\colon G\to X(n)\) a representation. Then \[ \rho=\sum_{(H,\Psi)<(H_1,\Psi_1)<\cdots<(H_r,\Psi_r)}(-1)^r\tfrac{|H|}{|G|} m(\text{Res}_{H_r}^G\rho,\Psi_r)\text{Ind}_H^G\Psi\tag{*} \] where the sum runs over all chains of pairs \((H_i,\Psi_i)\) of subgroups \(H_i\leq G\) and representations \(\Psi_i\colon H_i\to X(1)\) (this is missprinted as \(X(i)\) on p. 100 of the paper), subject to the compatibility conditions \(\Psi_i|H_{i-1}=\Psi_{i-1}\). To define the coefficients, let \(c\colon R^OG\to R^UG\) and \(c\colon R^{\text{Sp}}_+(G)\to R^UG\) denote the respective complexification maps from the Grothendieck rings \(R^OG\) and \(R^{\text{Sp}}_+(G)\) of orthogonal and symplectic representations to those of unitary representations. Then \[ m(\Theta,\Psi)=\left\{\begin{alignedat}{2}2&\;\langle\theta,\psi\rangle_H \hphantom{\langle\theta,\psi\rangle_H/\langle\psi,\psi\rangle_H}\quad\text{with }\psi=\Psi\quad &&\text{unitary case} \\ &\;\langle\theta,\psi\rangle_H/\langle \psi,\psi\rangle_H\hphantom{\langle \theta,\psi\rangle_H}\quad\text{with }\psi=c(\Psi)\quad &&\text{symplectic case} \\ &\left.\begin{alignedat}{2} &\langle\theta,\psi\rangle_H\qquad&&\text{with }c(\Psi)=\psi\text{ irr.}\qquad \\ &\langle\theta,\lambda\rangle_H\qquad&&\text{with }c(\Psi)=\lambda+\overline\lambda\qquad \\ &[\langle\theta,\phi\rangle_H/2]\qquad&&\text{with }c(\Psi)=2\phi\qquad \\ &1 &&\text{with }c(\Psi)=\phi+\phi'\text{ odd}\qquad \\ &0 &&\text{otherwise}\qquad\end{alignedat}\right\}\quad &&\text{orthogonal case}\end{alignedat}\right. \] The formula (*) is functorial, but, in general, the coefficients need not be integers. However, integrality is shown for representations of Galois groups of certain local number field extensions. In section 1 the authors explain the topological motivation for this research, namely a proof of the existence of an exponential stable decomposition of the classifying space \(B\text{Sp}\) of the symplectic group Sp. However, they point out that there is still some uncertainty about the existence of such a decomposition. Another motivation (section 5) comes from sympletic local root numbers. The lengthy proof of the induction formula is developed combinatorially from scratch, and several explicit examples are given to aid the reader.