2006/05/26 by Frank D. Grosshans, Grosshans, Frank D. · 1 citation
Mathematics · #13A50 #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0605690
openalex publication_date 2006/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be an algebraically closed field of characteristic p > 0. Let H be a subgroup of GL(n,k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d vectors, d > n, are obtained from those of n vectors by polarization. This result is not true when char k = p > 0 even in the case where H is a torus. However, we show that the algebra of invariants is always integral over the algebra of polarized invariants and when H is reductive is actually the p - root closure of that algebra. We also give conditions for the algebras to be equal, relating equality to good filtrations and saturated subgroups. We conclude with examples where H is finite or a classical group or is a certain kind of unipotent subgroup of GL(n,k).