Linear Actions of Unipotent Groups

Abstract

Invariant theory originally concerned itself with groups of vector space transformations, so the linear algebraic $\mathbb{G}_{a}$-actions were the first $\mathbb{G}_{a}$-actions to be studied. The action of $SL_{2}(\mathbb{C})$ on the vector space V _n of binary forms of degree n has an especially rich history, dating back to the mid-Nineteenth Century. The ring of SL ₂-invariants, together with the ring of $\mathbb{G}_{a}$-invariants for the subgroup $\mathbb{G}_{a} \subset SL_{2}(\mathbb{C})$, were the focus of much research at the time. A fundamental result due to Gordan (1868) is that both $k[V _{n}]^{SL_{2}(\mathbb{C})}$ and $k[V _{n}]^{\mathbb{G}_{a}}$ are finitely generated [182]. Gordan calculated generators for these rings up to n = 6. These historical developments are discussed in Sect.  6.3.1 .

Appendices

Appendix 1: Finite Group Actions

The following fact is well-known, and is provided here for the readers’ convenience. The statement of the proposition and the proof given here are due to Daigle (unpublished).

Proposition 6.19

Suppose k is a field, and B is a finitely generated commutative k-domain. Let G be a group of algebraic k-automorphisms of B (i.e., G acts faithfully on B). Then the following are equivalent.

• (1)  G is finite

• (2)  B is integral over B ^G

• (3)  B is algebraic over B ^G

Proof

We first show that, for given b ∈ B, the following are equivalent.

• (4) The orbit $\mathcal{O}_{b}$ is finite

• (5) b is integral over B ^G

• (6) b is algebraic over B ^G

(4) ⇒ (5): If $\mathcal{O}_{b}$ is finite, define the monic polynomial f(x) ∈ k[x] by

$$\displaystyle{\,f(x) =\prod _{a\in \mathcal{O}_{b}}(x - a)\,\,.}$$

Then f ∈ B ^G[x] and f(b) = 0, and (5) follows.

(5) ⇒ (6): Obvious.

(6) ⇒ (4): If h(x) ∈ B ^G[x] and h(b) = 0 for nonzero h, choose $a \in \mathcal{O}_{b}$, and suppose a = g ⋅ b for g ∈ G. Then

$$\displaystyle{h(a) = h(g \cdot b) = g \cdot h(b) = g \cdot 0 = 0\,\,,}$$

i.e., every $a \in \mathcal{O}_{b}$ is a root of h. Since B is a domain, the number of roots of h is finite, and (4) follows.

Therefore (4),(5), and (6) are equivalent.

(1) ⇒ (2): Choose b ∈ B. Since G is finite, $\mathcal{O}_{b}$ is finite, and therefore b is integral over B ^G. So B is integral over B ^G.

(2) ⇒ (3): Obvious.

(3) ⇒ (1): Since k ⊂ B ^G, we have that B is finitely generated over B ^G. Write B = B ^G[x ₁, …, x _n ], and define a function

$$\displaystyle\begin{array}{rcl} G& \rightarrow & \mathcal{O}_{x_{1}} \times \cdots \times \mathcal{O}_{x_{n}} {}\\ g& \mapsto & (g \cdot x_{1},\ldots,g \cdot x_{n})\,\,, {}\\ \end{array}$$

Now each element x _i is algebraic over B ^G, meaning that each orbit $\mathcal{O}_{x_{i}}$ is finite. Therefore, the set $\mathcal{O}_{x_{1}} \times \cdots \times \mathcal{O}_{x_{n}}$ is also finite. In addition, the function above is injective, since the automorphism g of B is completely determined by its image on a set of generators. Therefore G is finite. □

Appendix 2: Generators for A ₅ and A ₆

This section reproduces the tables of generators for A ₅ and A ₆ given by Grace and Young in 1903 [185], and also gives the reader a method to decode these invariants, which are expressed symbolically in the form of classical transvectants. Table  6.3 gives generators for A ₅ and Table  6.4 gives generators for A ₆. A similar table for A ₈ is given in [336], and for A ₉ and A ₁₀ in [271].

On the polynomial ring k[x ₀, …, x _n ], D _n is the basic linear derivation, defined by D _n x _i = x _i−1 for i ≥ 1 and D _n x ₀ = 0. U _n is the up operator, a linear locally nilpotent derivation defined by:

$$\displaystyle{U_{n}x_{i} = (i + 1)(n - i)x_{i+1}\,\,(0 \leq i \leq n - 1)\quad \mathrm{and}\quad U_{n}x_{n} = 0}$$

Given f, g ∈ A _n = ker D _n and i ≥ 1, the symbol ( f, g)ⁱ denotes:

$$\displaystyle{(\,f,g)^{i} = [U_{ n}^{i}\,f,U_{ n}^{i}g]_{ i}^{D_{n} }}$$

See Sect.  2.11.1 for the definition of the transvectant on the right side of this equation. The theory developed in Sect.  7.4 shows that ( f, g)ⁱ ∈ A _n .

Recall that the degree of x _i equals (1, i). If f is homogeneous of degree (r, s), then the order of f (relative to n) is nr − 2s. Elements in A _n of order 0 are precisely the SL ₂(k)-invariants. If g is homogeneous of degree (u, v), then ( f, g)ⁱ is homogeneous of degree (r + u, s + v + i).

Example 6.20

To illustrate, we calculate f ₈ = ( f ₂, f ₄)² in the table for A ₅. First:

$$\displaystyle{\,f_{2} = (\,f_{0},f_{0})^{2} = [x_{ 2},x_{2}]_{2}^{D_{n} } = 2x_{0}x_{2} - x_{1}^{2}}$$

and

$$\displaystyle{\,f_{4} = (\,f_{0},f_{0})^{4} = [x_{ 4},x_{4}]_{4}^{D_{n} } = 2x_{0}x_{4} - 2x_{1}x_{3} + x_{2}^{2}}$$

(Note that these are equalities up to an integer multiple.) Next, we find that

$$\displaystyle{U_{5}^{2}\,f_{ 2} = 12x_{0}x_{4} + 3x_{1}x_{3} - 4x_{2}^{2}\quad \mathrm{and}\quad U_{ 5}^{2}\,f_{ 4} = 10x_{1}x_{5} - 16x_{2}x_{4} + 9x_{3}^{2}}$$

(again, up to integer multiples). Therefore:

$$\displaystyle\begin{array}{rcl} f_{8}& =& (\,f_{2},f_{4})^{2} = [U_{ 5}^{2}\,f_{ 2},U_{5}^{2}\,f_{ 4}]_{2}^{D_{5} } {}\\ & =& -150x_{0}^{2}x_{ 3}x_{5} + 48x_{0}^{2}x_{ 4}^{2} + 150x_{ 0}x_{1}x_{2}x_{5} + 54x_{0}x_{1}x_{3}x_{4} - 152x_{0}x_{2}^{2}x_{ 4} {}\\ & & +60x_{0}x_{2}x_{3}^{2} - 50x_{ 1}^{3}x_{ 5} + 50x_{1}^{2}x_{ 2}x_{4} - 57x_{1}^{2}x_{ 3}^{2} + 32x_{ 1}x_{2}^{2}x_{ 3} - 8x_{2}^{4} {}\\ \end{array}$$