2
$\begingroup$

Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}_n(\mathbb{F}_p)$ acts on the polynomial ring $\mathbb{F}_p[x_1,\ldots,x_n]$ via $A\cdot x_i=\sum_j a_{ji}x_j$ for all $A=(a_{ij})\in H$, and $A\cdot f(x_1,\ldots,x_n):=f(A\cdot x_1,\ldots,A\cdot x_n)$.

How can we compute the $\mathbb{F}_p$-algebra of invariants $\mathbb{F}_p[x_1,\ldots,x_n]^H$ given generators of $H$?

For example, take $H=\operatorname{SL}_n(\mathbb{F}_p)$. Then if we let

$$L_{n,s}=\left|\begin{array}{cccc} x_1&x_2&\cdots&x_n\\ x_1^{p}&x_2^{p}&\cdots&x_n^p\\ \cdots&\cdots&\cdots&\cdots\\ \widehat{x_1^{p^s}}&\widehat{x_2^{p^s}}&\cdots&\widehat{x_n^{p^s}}\\ \cdots&\cdots&\cdots&\cdots\\ x_1^{p^n}&x_2^{p^n}&\cdots&x_n^{p^n} \end{array}\right|$$

where the $p^s$ row is omitted, the invariants are generated by the Dickson polynomials $\dfrac{L_{n,s}}{L_{n,n}}$ for $s=0,\ldots,n-1$.

However, if $H$ is arbitrary (described by a list of generators $h_1,\ldots,h_m$), is there some algorithm for computing the invariants? Or perhaps at least an upper bound on the degree of the generators of the invariants so that we can be sure to have found all of them via some computer search?

$\endgroup$
4
$\begingroup$

Let me add to Chris' answer that there is indeed a degree bound: $\mathbb{F}[x_1,\ldots,x_n]^H$ is generated in degree at most $n (|H| - 1)$. The bound was proved by Peter Symonds in: On the Castelnuovo-Mumford regularity of rings of polynomial invariants, Ann. of Math. (2) 174 (2011), 499–517.

And yes, a general algorithm can be found in Chapter 3 of the book with Derksen. It is implemented in the computer algebra system MAGMA.

$\endgroup$
2
$\begingroup$

I'm no expert, but I think the answer to your question is well known (unless I'm missing something):

For example, Corollary 3.3.3 in "Polynomial Invariants of Finite Groups" by L. Smith reads as follows:

Let $\rho\colon G\hookrightarrow GL({\mathbb F}^n)$ be a representation of a finite group $G$ of order $d$ over a field ${\mathbb F}$ of characteristic $p$ with $p=0$ or $p>d$. Then the invariant subring ${\mathbb F}[x_1,\ldots,x_n]^G$ is generated by at least $\binom{n+d}{d}$ elements, each of degree at most $d$.

It looks like the idea is to label your group elements $G=\left\{g_1,\ldots,g_d\right\}$ and simply take symmetric polynomials in the $n\cdot d$ "variables'' $\left\{\left.g_i\cdot x_j\right| 1\leq i\leq d, \ 1\leq j\leq n\right\}$...these are called "orbit Chern classes", and are guaranteed to generate your invariant ring, but some of these may be redundant...

I am not sure of a general algorithm to compute a minimal list of invariants--you may want to check out the book Computational Invariant Theory by Harm Derkson and Gregor Kemper. I haven't seen it myself, but it sounds like it might be useful for you.

$\endgroup$
  • 1
    $\begingroup$ That's an answer for $p>|G|$ but that's not all of it and, judging by the example given, he may want to consider the more general case. $\endgroup$ – Felipe Voloch Oct 14 '15 at 21:24
  • $\begingroup$ Thanks for this answer and the references. I will look into them. It would be nice if there was a similar result for small characteristic relative to the order of the group. $\endgroup$ – Jared Oct 16 '15 at 3:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.