Definition. General linear gauge action [ftip-00AX]
Definition. General linear gauge action [ftip-00AX]
For \(A\in \mathrm {GL}(k)\), define the general linear gauge action by
\[ \gamma _A(U,V)=(UA,VA^{-\mathsf T}). \]The inverse transpose on the second factor is essential for a general invertible \(A\). When \(A=Q\in \mathrm O(k)\), one has \(Q^{-\mathsf T}=Q\), and the action becomes \((UQ,VQ)\).