Twisted forms of the determinant
Let f be a polynomial in n variables over the field k homogeneous of degree m. Think of f as a function on a k-vector space V of dimension n. Another homogeneous polynomial \(f_ 1\) on a k-space \(V_ 1\) is called a twisted form of f split by the extension L of k if there is an L-linear isomorphism \(V_ 1\otimes L\quad \to \quad V\otimes L\) transforming \(f_ 1\otimes id_ L\) to \(f\otimes id_ L\). Two twisted forms \(f_ 1,f_ 2\) are equivalent over k (up to scalar) if there is a k- isomorphism \(V_ 1\to V_ 2\) transforming \(f_ 1\) into \(f_ 2\) (respectively into a scalar multiple of \(f_ 2).\) In the paper under review the special case \(f=\det(x_{ij})\) is considered. It can be shown: Each twisted form of \(\det(x_{ij})\) can be split by a separable extension field and the equivalence classes of forms up to scalar correspond to the elements of \(H^ 1(Gal(k_ s/k),G(k_ s))\) where \(k_ s\) denotes the separable closure of k and G is the group of automorphisms leaving \(\det(x_{ij})\) semi-invariant, i.e. those automorphisms sending \(\det(x_{ij})\) to a scalar multiple of itself. For special fields k all equivalence classes of twisted forms of the determinant can be determined by the standard machinery of Galois cohomology. For instance over the reals the only real forms obtained from \(\det(x_{ij})\) by a complex linear change of variables are up to real equivalence the form \(\det(x_{ij})\), plus or minus the determinant on \(n\times n\) Hermitian matrices or plus or minus the reduced norm on quaternion matrices of suitable dimension.
equivalence classes of twisted forms of determinant
scientific article; zbMATH DE number 3841964
scientific article