Separating invariants for the basic 𝔾ₐ-actions

Abstract

We explicitly construct a finite set of separating invariants for the basic Ga-actions. These are the finite dimensional indecomposable rational linear representations of the additive group Ga of a field of characteristic zero, and their invariants are the kernel of the Weitzenböck derivation Dn = x 0 ∂ ∂x 1 + . . . + x n−1 ∂ ∂xn .

Separating invariants for arbitrary linear actions of the additive group

“…In this paper, we describe a finite separating set for any finite dimensional representation of the additive group G a over a field k of characteristic zero, extending the results of Elmer and Kohls for the indecomposable representations (see [6]). Accordingly, from now on, k denotes a field of characteristic zero and G a its additive group.…”

“…for 1 ≤ j 1 = j 2 ≤ l ′ , where d = 0 and N is the least common multiple of n j1 and n j1 . • Π * n,⌊n/2⌋ (z j ) = x 3 0,j for l ′ + 1 ≤ j ≤ l, see [6,Lemma 5.4]. Showing that B is a separating set for k[x 0,j | 1 ≤ j ≤ l] C2 will end the proof.…”

“…Proof. The proof essentially carries over from the indecomposable case (see [6,Proposition 3.1]). The isomorphisms V * nj ∼ = S nj (V * 1 ) extend the G a -action on k[V ] to a SL 2 (k)-action when we identify V * 1 with the natural representation of SL 2 (k).…”

“…. , n k are ordered so that n j is even for 1 ≤ j ≤ l and odd for l + 1 ≤ j ≤ k, and further assume that n j ≡ 2 mod 4 for 1 ≤ j ≤ l ′ and n j ≡ 0 mod 4 for l ′ + 1 ≤ j ≤ l. As the problem of computing separating sets for indecomposable linear G a -actions was considered in [6], we assume throughout that k ≥ 2.…”

Invariants and separating morphisms for algebraic group actions

“…This part of our work was inspired by certain calculations done by Elmer and Kohls in [EK12]. An important tool is the geometric interpretation of the zero set of the plinth ideal .…”