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Category: group theory

Some examples of algebraic groups

In this post I want to give a few examples of the known “pathological” behavior of algebraic groups defined over general bases. In particular, this post contains examples of the following.

  • A smooth group scheme over a DVR with generic fiber \(\mathbf{G}_m\) and special fiber \(\mathbf{G}_a\),
  • An affine smooth group scheme
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Posted by Sean CotnerSeptember 6, 2021Posted in algebraic geometry, group theoryLeave a comment on Some examples of algebraic groups

The Steinberg Representation

In this post I want to describe a remarkable representation associated to finite groups of Lie type. For this, let \(G\) be a connected reductive group over a finite field \(k\) with \(q = p^r\) elements, and let \(U\) be the unipotent radical of some Borel \(k\)-subgroup \(B\) of \(G\). Steinberg constructed an irreducible representation \(\mathrm{St}\) of \(G(k)\) of …

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Posted by Sean CotnerMarch 27, 2021Posted in group theoryLeave a comment on The Steinberg Representation