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Tag Archives: representation theory
Polynomials and Representations IV
Power Sum Polynomials The power sum polynomial is defined as follows: In this case, we do not define , although it seems natural to set As before, for a partition define: Note that we must have above since we have … Continue reading
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Tagged partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations III
Complete Symmetric Polynomials Corresponding to the elementary symmetric polynomial, we define the complete symmetric polynomials in to be: For example when , we have: Thus, written as a sum of monomial symmetric polynomials, we have Note that while the elementary symmetric polynomials only go … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations II
More About Partitions Recall that a partition is a sequence of weakly decreasing non-negative integers, where appending or dropping ending zeros gives us the same partition. A partition is usually represented graphically as a table of boxes or dots: We will … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations I
We have already seen symmetric polynomials and some of their applications in an earlier article. Let us delve into this a little more deeply. Consider the ring of integer polynomials. The symmetric group acts on it by permuting the variables; specifically, … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Modular Representation Theory (I)
Let K be a field and G a finite group. We know that when char(K) does not divide |G|, the group algebra K[G] is semisimple. Conversely we have: Proposition. If char(K) divides |G|, then K[G] is not semisimple. Proof Let , a two-sided … Continue reading
The Group Algebra (I)
[ 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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Tagged character theory, group actions, group algebras, modules, representation theory, semisimple rings, simple modules
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Quick Guide to Character Theory (III): Examples and Further Topics
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
Quick Guide to Character Theory (II): Main Theory
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
Quick Guide to Character Theory (I): Foundation
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
Posted in Notes
Tagged character theory, dual spaces, fields, group algebras, groups, modules, representation theory
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Mathematics and Such 