# Tagged Questions

For questions about characters (homomorphisms from a group into the multiplicative group of a field).

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### For a semisimple algebra and two $M$- and $W$-homogeneous parts for $M \ncong W$, why we have $M(A)W(A) = 0$.

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$(cx)y = c(xy) = x(cy)$$ Then $A$ is called an $F$-algebra. If ...
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### Product of standard and sign representation of $S_5$

I am able to work out the sign representation of $S_5$ and standard representation of $S_5$. How do I compute the product of standard and sign representation of $S_5$? What kind of product do I need ...
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### Standard representation of $S_5$

I am trying to determine the standard representation of $S_5$. I understand that it will be a map from group elements to $\mathbb{C}^4$. The character table is as follows. I understand that the ...
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### Character table for $G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle$

Let $G$ be the group of order $20$ defined in terms of generators and relations: $$G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle.$$ Can anyone help me to derive the character table? ...
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### Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
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### Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
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### Exactly two irreducible characters of dimension 1

I've been working through Artin's Algebra on my own time, and I'm stuck on one of the questions, namely 10.5.3: Suppose that a group G has exactly two irreducible characters of dimension 1, and ...
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### why the column sums of character table are integers?

There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number. It is mentioned in Martin Isaacs Character ...
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### Relation of induced character with inflation of characters

I'm trying to prove the following exercise: If $G$ is a group and $N$ is a normal subgroup of $G$, then $$1_N^G = \sum_{\chi \in \text{Irr}(G/N)} \chi(1) \text{ Inf}^G_{G/N}\ \chi,$$ where $1_N^G$ is ...
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### Counter-examples in representations of associative algebra

Here are something well-known. $V \cong W \Rightarrow \chi_V=\chi_W$ holds for finite representations of arbitrary associative algebra. But $\chi_V=\chi_W \Rightarrow V \cong W$ is true only for ...
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### For simple $\mathbb C[G]$-modules is the representation unique

Let $R$ be a ring, a $R$-module is called simple if it has no proper, nontrivial submodules. Let $G$ be a finite group, and denote by $\mathbb C[G]$ the free vector space over $G$, with the product ...
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### Character representation of right regular representation as sum of irreducible characters

Let $G$ be a finite group acting on itself by the action $g \ast x := xg^{-1}$. Then this corresponds to an representation $\rho : G \to GL(L^2(G))$, where $L^2(G)$ denotes the space of function on $G$...
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### Proof that the induced class function $\theta^G$ is a character if $\theta$ is a representation on subgroup

In these lecture notes by Daniel Bump on Induced Characters I have a question on the proof of Theorem 2.5.1. If $H$ is a subgroup of the finite group $G$ and $(\pi, V)$ a representation of $H$, i.e. a ...
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### Order of group elements from a character table

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I'm trying to answer a question which goes the ...
### Inducing representation for groups of order $p^3$
For groups $G$ such that $|G|=p^3$ one can show that $(1)$ $Z=Z(G)\cong C_p$ $(2)$ $G'=Z$ $(3)$ $G/Z \cong C_p \times C_p$ Take any $x \in G/Z$. Then $N=\langle x,Z \rangle$ is an abelian normal ...