Modular representation theory

In mathematics, modular representation theory is the branch of representation theory that studies linear representations of finite group G over a field K such that the characteristic of K divides the order of G. In other words, the number of elements of G is zero when considered as an element of K. Such a representation is known as a modular representation. An example of modular representation theory would be the study of representations of the cyclic group of two elements over F2, the field with two elements.

Related Topics:
Mathematics - Representation theory - Linear representation - Finite group - Field - Characteristic - Order - Cyclic group

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Modular representations are very different from when K is the complex numbers, or when the characteristic of K does not divide the order of G. In those cases, Maschke's theorem yields that every representation is a direct sum of irreducible representations. The key step in the proof of Maschke's theorem is to average over the elements of the group, which fails when the order of G is zero when treated as an element of K.

Related Topics:
Complex number - Maschke's theorem - Direct sum - Irreducible representation

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