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5
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Estimates on conjugacy classes of a finite group.
In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32.
Theorem:
Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
4
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2answers
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How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?
One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
5
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2answers
214 views
What is an irrreducible character of a finite group?
Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
2
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1answer
69 views
Sum of squares of dimensions of irreducible characters.
For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here:
(5.9) Theorem Let $G$ be a group of order $N$, let ...
10
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0answers
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Subgroups as isotropy subgroups and regular orbits on tuples
Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω?
Motivation:
In ...
6
votes
1answer
167 views
What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?
Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
3
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0answers
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Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.
I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question.
To start with we work with the $\mathbb{Q}$ version of ...
1
vote
1answer
83 views
Weyl character formula and finding the trace.
Let $v$ be a positive integer. I have a representation $\rho_v$ of $USp(4) = \{g\in M_2(\mathbb{H})\,|\,g^{T}\bar{g}\}$, where $\mathbb{H}$ is Hamiltons quaternions. The representation $\rho_v$ has ...
