Skip to main content
×
Home
    • Aa
    • Aa

On the essential dimension of a finite group

  • J. BUHLER (a1) and Z. REICHSTEIN (a2)
Abstract

Let $f(x) = \Sigma a_ix^i$ be a monic polynomial of degree $n$ whose coefficients are algebraically independent variables over a base field $k$ of characteristic 0. We say that a polynomial $g(x)$ is generating (for the symmetric group) if it can be obtained from $f(x)$ by a nondegenerate Tschirnhaus transformation. We show that the minimal number ${\rm d}_k(n)$ of algebraically independent coefficients of such a polynomial is at least $[n/2]$. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13th problem.

Our approach to this question (and generalizations) is based on the idea of the ‘essential dimension’ of a finite group $G$: the smallest possible dimension of an algebraic $G$-variety over $k$ to which one can ‘compress’ a faithful linear representation of $G$. We show that ${\rm d}_k(n)$ is just the essential dimension of the symmetric group ${\rm S}_n$. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the essential dimension of a finite group
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      On the essential dimension of a finite group
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      On the essential dimension of a finite group
      Available formats
      ×
Copyright
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 54 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th September 2017. This data will be updated every 24 hours.