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Session 31. Representation Theory, Transformation Groups, and Applications
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Cartan matrices for restricted Lie algebras |
Jens Carsten Jantzen, Aarhus Universitet, Denmark
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A classical result by Richard Brauer says: The Cartan matrix \(C\) of a finite
group over a splitting field of prime characteristics is the product of its
decomposition matrix \(D\) with the transpose of the decomposition matrix:
\(C = {}^t D \cdot D\). This fact is a special case of the following general
result: Consider an algebra free of finite rank over a complete discrete
valuation ring. Then its Cartan matrix \(C_k\) over the residue field \(k\) is
related to the Cartan matrix \(C_K\) over the fraction field \(K\) via
\(C_k = {}^t D \cdot C_K \cdot D\) provided \(k\) and \(K\) are
splitting fields. Here \(D\) again is a decomposition matrix.
In my talk I'll show how this observation can be applied to the
representation theory of restricted Lie algebras.
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