Tag Archives: character theory

The Group Algebra (III)

Posted on January 6, 2015 by limsup

As alluded to at the end of the previous article, we shall consider the case where K is algebraically closed, i.e. every polynomial with coefficients in K factors as a product of linear polynomials. E.g. K = C is a common choice. Having assumed … Continue reading →

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The Group Algebra (I)

Posted on January 3, 2015 by limsup

[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading →

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Quick Guide to Character Theory (III): Examples and Further Topics

Posted on June 3, 2013 by limsup

G10(a). Character Table of S4 Let’s construct the character table for . First, we have the trivial and alternating representations (see examples 1 and 2 in G1), both of which are clearly irreducible. Next, the action of G on {1, 2, 3, … Continue reading →

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Quick Guide to Character Theory (II): Main Theory

Posted on May 31, 2013 by limsup

Reminder: throughout this series, G is a finite group and K is a field. All K-vector spaces are assumed to be finite-dimensional over K. G4. Maschke’s Theorem If is a K[G]-submodule, it turns out V is isomorphic to the direct sum of W and some other submodule W’. … Continue reading →

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Quick Guide to Character Theory (I): Foundation

Posted on May 28, 2013 by limsup

Character theory is one of the most beautiful topics in undergraduate mathematics; the objective is to study the structure of a finite group G by letting it act on vector spaces. Earlier, we had already seen some interesting results (e.g. proof … Continue reading →

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