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Invariants of the hyperbolic action of the multiplicative group on the plane

Statement

Let $G_{m} = C^{\times}$ act on $C^{2}$ by $t \cdot (x, y) = (t x, t^{- 1} y)$ (Classical complex affine algebraic actions and rational modules). Then the invariant subring of $C [x, y]$ (The coordinate ring of a classical affine algebraic set) is $C [x, y]^{G_{m}} = C [x y]$ , a polynomial ring in one variable; explicitly, a polynomial $f = \sum_{a, b} c_{a b} x^{a} y^{b}$ is invariant if and only if $c_{a b} = 0$ for every pair with $a \neq b$ (Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ ).

Facts & Assumptions

Given: the multiplicative group $G_{m} = C^{\times}$ acting on $C^{2}$ by $t \cdot (x, y) = (t x, t^{- 1} y)$ , and a polynomial $f = \sum_{a, b} c_{a b} x^{a} y^{b} \in C [x, y]$ written in its unique finite monomial expansion.

[F1]

The action and the action on functions. The map $t \cdot (x, y) = (t x, t^{- 1} y)$ is an algebraic action of $G_{m}$ on the affine plane, and the induced left action on functions is $(t \cdot f) (p) = f (t^{- 1} \cdot p)$ , so that $(t \cdot f) (x, y) = f (t^{- 1} x, t y)$ (Classical complex affine algebraic actions and rational modules).

[F2]

Unique monomial expansion. Every polynomial in $F [x_{1}, \dots, x_{n}]$ has a unique finite expansion $f = \sum_{t} c_{t} x^{t}$ over finitely many multi-indices; in particular the monomials are linearly independent over $F$ and a polynomial is zero exactly when all its coefficients vanish (Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ ).

[F3]

The coordinate ring of the plane. For an affine algebraic set $X \subseteq k^{n}$ the coordinate ring is $k [X] = k [x_{1}, \dots, x_{n}] / I (X)$ ; for $X = C^{2}$ this is $C [x, y]$ , with $x$ and $y$ the coordinate functions (The coordinate ring of a classical affine algebraic set).

Proof

technique · direct

1.1F1

For $t \in G_{m}$ and a monomial $x^{a} y^{b}$ the function action of [F1] gives $(t \cdot (x^{a} y^{b})) (x, y) = (t^{- 1} x)^{a} (t y)^{b} = t^{b - a} x^{a} y^{b}$ , so the multiplicative group acts on each monomial by the scalar $t^{b - a}$ : monomials on the diagonal $a = b$ are fixed, while off-diagonal monomials are scaled by a nonconstant character.

2.1F1F2step 1.1

An element $f = \sum_{a, b} c_{a b} x^{a} y^{b} \in C [x, y]$ is invariant under $G_{m}$ if and only if $\sum_{a, b} c_{a b} (t^{b - a} - 1) x^{a} y^{b} = 0$ for every $t \in G_{m}$ ; by step 1.1 this is the difference between $t \cdot f$ and $f$ , and by uniqueness of the monomial expansion it holds if and only if $c_{a b} t^{b - a} = c_{a b}$ for all pairs $(a, b)$ and all $t \in G_{m}$ .

3.1F2step 2.1

The coefficient conditions hold if and only if $c_{a b} = 0$ for every pair with $a \neq b$ . If $f$ is invariant, fix a pair with $a \neq b$ ; evaluating the condition of step 2.1 at the nonzero complex number $t = 2$ gives $c_{a b} (2^{b - a} - 1) = 0$ , and $2^{b - a} \neq 1$ because $2^{b - a} > 1$ for $b > a$ and $0 < 2^{b - a} < 1$ for $b < a$ ; hence $c_{a b} = 0$ . Conversely, if $c_{a b} = 0$ whenever $a \neq b$ , then $f = \sum_{a} c_{a} x^{a} y^{a}$ is a sum of diagonal monomials, and by step 1.1 each such monomial satisfies $t \cdot (x^{a} y^{a}) = t^{a - a} x^{a} y^{a} = x^{a} y^{a}$ , so $t \cdot f = f$ for every $t$ and $f$ is invariant.

4.1F2F3step 3.1∎

Consequently a polynomial $f \in C [x, y]$ is $G_{m}$ -invariant if and only if its expansion has no off-diagonal terms, that is, if and only if $f = \sum_{a} c_{a} x^{a} y^{a}$ lies in the subalgebra $C [x y]$ generated by the invariant function $x y$ . Hence $C [x, y]^{G_{m}} = C [x y]$ . The substitution $φ : C [z] \to C [x, y]$ , $z \mapsto x y$ , is injective: if $φ (g) = 0$ for $g = \sum_{n} d_{n} z^{n}$ , then $\sum_{n} d_{n} x^{n} y^{n} = 0$ , and uniqueness of the monomial expansion forces every $d_{n} = 0$ , so $g = 0$ . Therefore $C [x y] ≅ C [z]$ is a polynomial ring in one variable, and the plane's invariant ring is $C [x y]$ .

Remarks

This is the invariant computation behind Brion's Example 1.27(2): the quotient of $C^{2}$ by the hyperbolic $G_{m}$ -action is the affine line with coordinate $x y$ . The companion example uses exactly this lemma to identify the invariant ring with $C [x y]$ and to describe the quotient map $(x, y) \mapsto x y$ ; that the raw level set ${x y = 0}$ contains both the origin and the two coordinate axes is the phenomenon behind the stable-locus analysis there.
No choice is used: a single test value $t = 2$ already separates every off-diagonal monomial from the invariant ones, so no infinite family of excluded values has to be avoided.

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Sources

Michel Brion, Introduction to actions of algebraic groups, Les cours du CIRM 1 (2010), no. 1, 1-22 (standard reference, not scraped)