Invariant Theory

Chapter 9 Invariant Theory

This chapter is co-authored by Francesca Gandini, Sumner Strom, Al Ashir Intisar

Section 9.1 Invariant Rings Theory

Subsection 9.1.1 Finite Matrix Groups

Example: Consider

\begin{equation*} M = \begin{pmatrix} 1 \amp 0 \\ 0 \amp -1 \\ \end{pmatrix} \end{equation*}

and the vector \(\bar x = \begin{pmatrix} x\\ y\\ \end{pmatrix}\) This gives \(M \bar x = \begin{bmatrix} x \\ -y \\ \end{bmatrix}\text{.}\) Thus for the polynomial \(f(\bar x) = f(\begin{bmatrix} x \\ y \\ \end{bmatrix}) = x+y\) we have \(f(M\bar x) = f(\begin{bmatrix} x \\ -y \\ \end{bmatrix})= x-y\text{.}\)

Definition 9.1.1.

\(G \leq GLm(\mathbb{K}), |G| < \infty\text{,}\) then \(G\) is a finite matrix group. (In other words if \(G\) is a group of actions under which \(\mathbb{K}\) remains invariant under then it is smaller or equal to the total amount of group actions \(GLm(\mathbb{K})\) that would keep the polynomial invariant. AND the \(|G|\) is finite then \(G\) is a finite matrix group?)

NOTE: An action of a finite group \(G \curvearrowright \mathbb{K}^n\) given a realization of \(G\) as a finite matrix group.

Example: \(\langle \begin{bmatrix} 1 \amp 0 \\ 0 \amp -1 \\ \end{bmatrix} \rangle = \{ \begin{bmatrix} 1 \amp 0 \\ 0 \amp -1 \\ \end{bmatrix},\begin{bmatrix} 1 \amp 0 \\ 0 \amp 1 \\ \end{bmatrix}\} \cong C_2\)

Subsection 9.1.2 Invariant Rings

Notation \(\bar x = (x_1, x_2,..., x_n)\text{,}\) with \(R = \mathbb{K}[x_1,x_2,...,x_n]\)

Definition 9.1.2.

\(G\) is a finite matrix group within \(GLm(\mathbb{K})\) when? \(f\in \mathbb{K}[x_1,x_2,...,x_n]\) is invariant under the action of \(G\) if and only if \(f(A\bar x) = f(\bar x)\text{,}\) \(\forall A \in G\text{.}\)

Ex. \(f(\bar x)=x\) and \(f(\bar x) = x +y^2\) in \(\mathbb{K}[x_1,x_2,...,x_n]\) is invariant under \(C_2 = \langle\begin{bmatrix} 1 \amp 0 \\ 0 \amp -1 \\ \end{bmatrix} \rangle\) However \(f(\bar x)=x+y\) is not. What are others?

Definition 9.1.3.

\(R^G : = \{f \in R \, | f(A\bar x) = f(\bar x), \forall A \in G\} \subseteq R\) is the invariant ring for the action of \(G\)

Show this is a subring.

How does on find generators for \(R^G\text{?}\)

Is \(R^G\) even finitely generated?

Work through Hilbert’s proof.

Subsection 9.1.3 Reynolds Operator

Idea: "Averaging" over the action of \(G\) we get an invariant

Definition 9.1.4.

\(R_G: R \xrightarrow{} R^G\)

\begin{equation*} R_G(f) = \frac{1}{|G|} \sum_{A\in G} f(A \bar x) \end{equation*}

Exercise: \(R_G\) has many nice properties?

Example: \(C_2 = \langle\begin{bmatrix} 1 \amp 0 \\ 0 \amp -1 \\ \end{bmatrix}\rangle\)

\begin{equation*} R_G(x+y) = \frac{1}{2} ((x+y) + (x-y)) = x\in R^G \end{equation*}

Subsection 9.1.4 Nöether Degree Bound(NDB)

Theorem 9.1.5.

(Noether):

\begin{equation*} R^G = \mathbb{K} [ R_G(\bar x^{\bar \beta}) | \; |\bar \beta| \leq |G|] \end{equation*}

\(\implies\) NDB : The ring of invariants is generated in degrees \(\leq |G|\)

Note: This is a computational tool! We can apply \(R_G\) to all the finitely many monomials in degrees \(\leq |G|\) to get generators for \(R^G\text{.}\) Exercise: Try this for \(C_4\) ... show!

Subsection 9.1.5 Hilbert Ideal

Note: In general for \(\{ f_1,..., f_s\} \subseteq \R\text{,}\) \(\{f_1,...f_s\}\) and \(\R\) can be quite different objects Exercise?

Theorem 9.1.6.

Let \(J_G = R(R^G)_t\text{,}\) ideal generated by all positive degree invariants. If \(J_G = (f_1,...,f_s)\) and \(f_i\in R^G, \,\, \forall i\) (apply \(R^G\) if it is not), then \(R^G = \mathbb{K}[f_1,...f_s]\)

Subsection 9.1.6 Presentations

Definition 9.1.7.

Definition: Let \(S = \mathbb{K}[f_1,...f_s] \subset R\text{.}\) A presentation of \(S\) is a map,

\begin{equation*} T=: \mathbb{K}[u_1,...u_s] \xrightarrow{\phi}S \end{equation*}

such that \(\frac{T}{ker(\phi)} \cong S\) With the syzygies of \(f_i\)’s giving the presentation ideal.

Proposition 9.1.8.

(Elimination Theory): In \(S \bigotimes \mathbb{K}[u_1,...,u_s] = \mathbb{K}[x_1,...,x_n,u_1,...u_s]\) consider the ideal,

\begin{equation*} I = (u_i - f_x(\bar x) | \, \langle f_i\rangle = S \end{equation*}

Then,

\begin{equation*} ker \phi = I \cap \mathbb{K}[u_1,...,u_s] \end{equation*}

Algorithm 9.1.9.

Compute a Groebner Basis \(G\) for \(I\) with elimination order for the \(x\)’s. Then, \(G \cap \mathbb{K}[y_1,...y_s]\) is the Groebner Basis for \(ker \phi\)

Subsection 9.1.7 Graph of Linear Actions

Definition 9.1.10.

Let \(G \leq GL_n(\mathbb{K}), \,\, G\curvearrowright \mathbb{K}^n =:V, \,\, |G|\infty\text{.}\) For \(A\in G\) consider,

\begin{equation*} V_A = \{(\bar v, A\bar v)|\,\,v\in V\} \subseteq V\bigotimes V \end{equation*}

Then \(A_G = \cup_{A\in G}V_A\) is the subspace arrangement associated to the action of G.

Note: \(V_A\) is a linear subspace, \(\mathbb{I}(V_A):=\) set of polynomials vanishing on \(\mathbb{V}_A\) is a linear ideal. Example:

\begin{equation*} V_{\begin{bmatrix} 1 \amp 0 \\ 0 \amp -1 \\ \end{bmatrix}} = \{(x_1,x_2,x_1,-x_2)\} = V(y_1,-x_1, y_2+x_2) \end{equation*}

Subsection 9.1.8 Subspace Arrangement Approach

Theorem 9.1.11.

(Dekseu): Let \(I_G = \mathbb{I}(A_G) = \cap_{A\in G}\mathbb{I}(V_A) \subseteq \mathbb{K}[x_1,...x_n,y_1,...y_n].\) Then

\begin{equation*} (I_G +(y_1,...,y_n)) \cap \R = J_G \end{equation*}

This uses elimination theory and the Hilbert ideal.

Note: The same approach works in the exterior algebra!

Theorem 9.1.12.

Let \(I_G^{'} = \cap_{A\in G} \mathbb{I}(V_A) \subseteq \Lambda(\bar x, \bar y)\text{.}\)\\ Then

\begin{equation*} (I_G^{'} +(y_1,...y_n)) \cap \Lambda(x_1,...,x_n) = J_G^{'} : = \Lambda(\bar x)(\Lambda(\bar x)^G)_+ \end{equation*}

Note: This approach is slow for polynomials, but might be fast for skew polynomials.

Subsection 9.1.9 Abelian GPS and Weight Matrices

Let \(G \cong \mathbb{Z}_d, \bigoplus....\bigoplus \mathbb{Z}_{dr}, \,\,\,\,\, d_i|d_{i+1}\) for \(1 \leq i \leq r-1\)\\

\begin{equation*} \langle g_1\rangle \bigoplus...\bigoplus\langle g_r \rangle, \,\,\,\,\, |g_i| =d_i \end{equation*}

A diagonal action of \(G\) on \(R\) is given by

\begin{equation*} g_i \cdot x_j = \mu_i^{\omega ij}x_j \end{equation*}

for \(\mu_i : d_i^{th}\) primitive root of unity and \(i \in [x]\text{,}\)\(j \in [n]\text{.}\) And encoded in the weight matrix \(W = (\omega_{ij})_{ij} = \begin{bmatrix} x_1 \amp \cdots \amp x_n \\ \vdots \amp \ddots \amp \\ x_n \amp \amp \end{bmatrix}\)

Theorem 9.1.13.

\(\bar x^{\bar \beta} \in R^G \iff W_{\bar \beta}\cong (0,...,0)\) for zeros being the weight of \(g_1\) acting on \(\bar x^{\bar \beta}\) and being modulo \(d_i\text{.}\)

Note: We can examine all monomials \(|\bar \beta| \leq |G|\) and sort them by their weight \(W\bar \beta\text{.}\) The ones with weight \(\bar 0\) will be invariant!

Question: Does this work for monomials in the exterior algebra?

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