Vector Invariants in Arbitrary Characteristic

Abstract

Let k be an algebraically closed field of characteristic p 0. Let H be a subgroup of GLn(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 the p-root closure of the algebra of polarized invariants. We also give conditions for the two algebras to be equal, relating equality to good filtrations and saturated subgroups. As applications, we discuss the cases where H is finite or a classical group.

Mapping toric varieties into low dimensional spaces

“…Now suppose S gives a separating algebra A. As this algebra is a graded subalgebra, it follows that the extension A ⊆ k[x] T is finite and k[x] T is the purely inseparable closure of A in k[x] (see [6,Remark 1.3] or [13,Theorem 4]). Hence, for any x α ∈ k[x] T , there exist m α ∈ N such that (x α ) p mα ∈ A.…”

Characteristic free description of semi-invariants of 2 × 2 matrices

“…In the special case m = 4, S 4 is even a generating system by Theorem 2.1 and Theorem 4.3. Since dim F (F 2×2 ) = 4, a general result of Draisma, Kemper and Wehlau [17] or Grosshans [20,Theorem 7] implies that the polarizations of a separating system in O((F 2×2 ) 4 ) constitute a separating system in O((F 2×2 ) m ) for an arbitrary m. Now for m ≥ 4 the polarizations of the subset S 4 are contained in the subalgebra generated by S m .…”

Groups with large Noether bound

“…This is based on Weyl's theorem on polarization (see [42]). If char(F) > 0, then Weyl's theorem on polarizations fails even in the non-modular case; instead of that, if char(F) does not divide |G| then by a result of Grosshans in [22] for any G-module W containing V reg as a submodule, the ring F[W ] G is the p-root closure of its subalgebra generated by the polarization of F[V reg ] G . Corollary 2.21 is an improvement of Pawale's result who proved in [33] in characteristic 0 that β(G, W ) = 9 for n + , n − = 2, and from this he concluded β(G) = 9 using a version of Weyl's Theorem on polarization.…”