# Tagged Questions

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### Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
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### Representation of $\mathbb C$

Let $M_2(\mathbb R)$ be the ring of $2\times 2$ matrices with real entries. Its group of multiplicative units is $GL_2(\mathbb R)$, consisting of the invertible matrices in $M_2(\mathbb R)$. (a) ...
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### weight of a group G

how can I found the weight of a group G ( I find the fundamental weights, but I don't know how found the linear combination of fundamental weights that give me the weights ). so we found the ...
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### Prove the following elementary properties of the group character

I'm reading the book appendix on group theory of the book Quantum Computation and Quantum Information by Nielsen & Chuang. I'm having trouble with exercises A2.11 and A2.12. This isn't homework ...
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### Irreducible representations of nonabelian group generated by $3$ elements

My question is rather commonplace, but nevertheless I'd like to discribe irreducible representations of the so called Heisenberg group (I suppose this one is just a special case of Heisenberg group). ...
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### Irreducible representations of group of order $pq$

There is the problem to describe dimensions of irreducible representations of a group of order $pq$, where $p$ and $q$ a distinct primes. I am doing it as follows: Suppose $p>q$. Then by the Sylow ...
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### Representation in Banach space and norms 'induced' by representation

By $G$ we denote some compact group, $X$ stands for some Banach space. Suppose $\pi\colon G\longrightarrow \mathrm{GL}(X)$ to be some representation in $X$. I'm trying to prove that there is an ...
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### Regular representation

I am stuck at the following question and dont know where to begin: Let $\rho$ be the permutation representation associated to the operation of $D_3$ of order 6 on itself by conjugation. Decompose ...
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### If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.

Let $H$ be a subgroup of a group $G$, and let $\rho :H\to GL(V)$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is at least ...
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### If an $H\leq G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of atleast dimension $d$. [duplicate]

Let $H$ be a subgroup of a group $G$, and let $\rho:H\rightarrow GL(V )$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is ...
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### Young diagram for exterior powers of standard representation of $S_{n}$

I'm trying to solve ex. 4.6 in Fulton and Harris' book "Representation Theory". It asks about the Young diagram associated to the standard representation of $S_{n}$ and of its exterior powers. The ...
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### Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...
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### Representation problem from Serre's book

I asked this question yesterday on the setting of an exercise problem (Ex 2.8) from Serre's book "Linear representations of Finite Groups" (I'm teaching myself representation theory...) Now that that ...
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### Martin Isaacs's exercise 3.7 (character theory of finite groups)

I would need some help with this exercise: Let $\chi\in{Irr(G)}$ be faithful, and suppose $\chi(1)=p^a$ for some prime p. Let $P\in{Syl_{p}(G)}$, and suppose that $C_{G}(P)\nsubseteq{P}$. Show that ...
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### Martin Isaacs's exercise 3.4 (character theory of finite groups)

I need some help with this: Let $G$ be a simple group and suppose $\chi\in{Irr(G)}$ with $\chi(1)=p$, a prime. Show that a Sylow $p-$subgroup of G has order p. Thanks a lot in advance.
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### Martin Isaacs's exercise 3.6 (character theory of finite groups)

I'm trying to solve this exercise, can anyone help me? Let $G$ be a p-group, and suppose $\chi\in{Irr(G)}$. Show that $\chi(1)^2$ divides $|G:Z(\chi)|$ Thanks a lot.
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### Martin Isaacs's exercise 3.5 (character theory of finite groups)

I need some help with this exercise: Suppose $A\subseteq{G}$ is abelian, and $|G:A|$ is a prime power. Show that $G'\lt{G}$ Thank you very much in advance.
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### Give a bijection between unitary, degree one representations of Z and elements of T.

Definition: My book defines $\mathbb{T}$ as the unit circle in $\mathbb{C}$, i.e. $\mathbb{T}=\{z \in \mathbb{C} : |z| = 1\}$ I'm trying to answer this question: "Give a bijection between unitary, ...
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### Completing a Character Table for a Group of Order 18

I have the following homework question: A group of order 18 has the following partial character table, where $y=-\frac{1}{2} + xi$: \begin{array}{c | c c c c c} \hline\hline & g_1 & g_2 ...
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### Induced representation, isomorphism between vectorspaces

I want to prove the following statement: The mapping $Ind_H^G{V}\rightarrow V^m: \psi\mapsto(\psi(g_1),\ldots,\psi(g_m))$ is a isomorphism. Here is $g_1,\ldots,g_m$ a representing system of $G/H$ ...
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### Show group is isomorphic to finite Heisenberg group

Show that the group $\langle x,y,z$ $|$ $z = xyx^{-1}y^{-1}$, $zx=xz$, $zy = yz$, $x^n = \mathbb{I},$ $y^n = \mathbb{I}$, $z^n = \mathbb{I} \rangle$, $(n \in \mathbb{Z_{>0}})$ is isomorphic to ...
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### Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
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### Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G$ is abelian if and only if every irreducible ...
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### Group representation scalar product

Let $\rho: G \rightarrow GL(V)$ be a finite dimensional complex representation of the group $G$. Show that there is an inner product on $V$ such that $G$ acts by unitary matrices. My approach so far ...
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### Character on conjugacy classes

Let $V_j$, $j = 1,2$ be finite dimensional representations of a group $G$. Show: $\chi_{V_j}$ is a constant on each conjugacy class of $G$, where $\chi_{V_j}$ is the character of the representation. ...
### How to show a $\mathbb{C}$-representation is not the complexification of a $\mathbb{R}$-representation?
For a homework problem in a course of representation theory, I have to show that some complex representation of a group $G$ is not the complexification of the real one. I don't really understand how ...