Invariant Theory of Matrices

Abstract

Let U _m be a vector space with basis {u₁,…, u_m} and let S(U _m ) be the symmetric algebra of U _m over K. This is the polynomial algebra K[U _m ] = K [u ₁ ,…,u_m] with m variables u₁,…,u_m. The algebra S (U _m ) is naturally graded. Its homogeneous component of degree p is the p-th symmetric powerS ^P (U _m ) of U _m i.e., the vector space spanned by the monomials of degreep. The general linear group $GL\left( {{U_m}} \right) \cong G{L_m}\left( K \right)$ acts on the vector spaceU _m and this action is extended diagonally to the group of homogeneous automorphisms of K[U _m ]:

g(f(u1,…,um )) = f (g((u ₁ ),…,g(um)), g ∈GL _m (K), f ∈K[U _m ].