Archiv der Mathematik

, Volume 103, Issue 3, pp 201–210 | Cite as

Big elements in irreducible linear groups

  • Nikolai Gordeev
  • Ulf Rehmann


Let V be a linear space over a field K of dimension n > 1, and let \({G \leq {\rm GL}(V)}\) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank \({(g - \alpha E_{n}) \geq \frac{n}{2}}\) for every \({\alpha \in K}\), where E n is the identity operator on V. This estimate is sharp for any \({n = 2^{m}}\). The existence of such an element implies that the conjugacy class of G in GL(V) intersects the big Bruhat cell \({B\dot{w}_{0}B}\) of GL(V) non-trivially (here B is a fixed Borel subgroup of G). The latter fact is equivalent to the existence of a complete flag \({\mathfrak{F}}\) such that the flags \({g(\mathfrak{F}), \mathfrak{F}}\) are in general position for some gG.

Mathematics Subject Classification

Primary 20C99 14L40 Secondary 20G05 20H20 


Linear groups Conjugacy classes Bruhat cells Complete flags 


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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsRussian State Pedagogical UniversitySaint PetersburgRussia
  2. 2.Department of MathematicsBielefeld UniversityBielefeldGermany

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