Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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57 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...
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15 views

Mackey's criterion and double cosets of $A_3$ and $S_3$

State Mackey's criterion $Ind_{H}^{G}$ is irreducible $\iff p$ is irreducible $p^s$ and $p$ are disjoint representations of $H \cap sHs^{-1}$ for any $s \in T $\ $ \{1\}$ Find the double ...
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24 views

Show $Res_H^G(sgn_G)=sgn_H$ where $G=S_4$ and $H=S_3$

Let $ G=S_3 $ and $ H=S_2 $. Show that $ Res^G_H(sgn_G)=sgn_H $ The symmetric group $G=S_3$ has three irreducible representations $ 1_G, sgn_G $ and $ V$ where $ 1_G $ denotes the trivial ...
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28 views

Comultiplication in a tensor algebra.

Let $V$ be a vector space. Then we have the tensor algebra $TV = \oplus_{i=0}^{\infty} T^i V$. In the webpage, it is said that the comultiplication $\Delta: TV \to TV \otimes TV$ is given by the ...
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24 views

Frobenius reciprocity and induced representations

In representation theory, we consider the restriction functor for any group $G$ and subgroup $H$. This is: $Res_H^G : Rep(G) \rightarrow Rep(H)$ This gives a representation of $H$ The Induced case ...
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Generators in adjoint representation are structure constants

Given that $g T_a g^{-1} = D^b_a T_b$ one can show that the generators in the adjoint representation of a group $G$ are the structure constants of the lie algebra satisfied by the $T_a$. Write $g$ ...
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25 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
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106 views

Matrix Proof of Schur Orthogonality

It seems to me like the coordinate statement of Schur's Orthogonality relations $$ \sum_{R \in G}^{|G|} \Gamma^{(\lambda)}(R)_{nm}^* \Gamma^{(\mu)}(R)_{n'm'} = \delta_{\lambda \mu} \delta_{n n'} ...
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25 views

Question about number of irreducible representation?

From the table I attached (which is from http://chemwiki.ucdavis.edu/Core/Theoretical_Chemistry/Symmetry/Group_Theory%3A_Theory), $\Gamma$ is the reducible representation. It is then deduced that ...
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Is character of a group representation the same as trace?

If so, why cannot the Klein group's character be zero? The group element of Klein group matrices can be traceless, right?
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27 views

Is character same as one dimensional irreducible representation?

Look at character table of the Klein Group: http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_Klein_four-group Is the plus or minus ones on each row the same as one dimensional ...
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11 views

How to show that antipode is anti-Poisson and counit is Poisson?

I am reading the book Algebras of Functions on Quantum Groups: Part I by Leonid I. Korogodski and Yan S. Soibelman. I have a question about the proof of that antipode is anti-Poisson and counit is ...
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59 views

A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

Let $G$ be a finite group and $H$ a subgroup. Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ ...
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29 views

Tilting object and mutation in Coh $(X)$

I'm studying the article "The cluster category of a canonical algebra" of Barot, Kussin, and Lenzing. I would like to understand an argument about mutation. I wrote the definition below: Let $T = ...
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64 views

composition series of a group algebra over finite field

Assume F is a finite field of characteristic 2 and G is the Klein's four-group. How many different composition series does the FG F-algebra have, as a module over itself? Is this number related ...
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33 views

Representation of a finite group and its Sylow $p$-subgroup

Let $G$ be a finite group with order $|G|=p^n \cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
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Question about proof on faithful character giving rise to abelian $P$-sylow subgroup

Faithful character of degree less than p gives abelian p-Sylow groups. I have a question about the above question answered by Derek Holts. As I don't have enough reputation I couldn't post my ...
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4answers
73 views

What is a representation?

I know the definition is given as follows: A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means Can someone help me gain some ...
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A question on Auslander-Bridger transpose

I am learning Auslander-Reiten Theory. When I read the book Frobenius Algebras I. Basic Representation Theory, I have some problems. On page 236-237, The Proposition Proposition 4.5. Let $M$ and ...
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15 views

Finite dimensional, irreducible representations of the Lie superalgebra gl(1|1)

I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial ...
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31 views

On a theorem of R. Brauer about character theory.

Let $\operatorname{Cl}(g_1),\ldots,\operatorname{Cl}(g_r)$ be the conjugacy class of a finite group $G$ and let $C_i \in C[G]$ be the sum of the element in $\operatorname{Cl}(g_i)$ where $C[G]$ is the ...
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The Lie algebra of the generalized unitary group $\{g \in GL_n(\mathbb{C}) : gS\bar{g}^t=S\}$ is $\{XS+S\overline{X}{}^t=0\}$

Let $ S \in M_n(\mathbb{C}) $ be a square matrix and let $ X$ be in the Lie algebra $\mathbb{\mu(S)} $ of the generalized unitary group, $$U(S):=\{g \in GL_n(\mathbb{C}); gS\bar{g}^t=S\} .$$ ...
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28 views

Adjoint matrix in $\mathbb{so_3}$

$\mathbb{so_3}$ has the following basis: $X_1=\begin{bmatrix} 0 & & \\ & &1 \\ & -1 & \end{bmatrix}$, s: $X_2=\begin{bmatrix} & & 1\\ & 0& \\ -1 & & ...
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32 views

Show that $ (\pi(g)\phi)(v)=\phi({^t}gv) $ defines a representation

Let $ G=SL_2(\mathbb{C}) $ and consider the action of $ G $ on the space of smooth functions on column vectors $ v \in \mathbb{C^2} $ given by: $ (\pi(g)\phi)(v)=\phi({^t}gv) $ Question 1: Show that ...
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34 views

Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g} $ we can define the adjoint representation as: $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $ I am ...
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22 views

Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R}) $ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
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70 views

Show that group has a nontrivial normal subgroup.

Let $G$ be a group and assume that it has an irreducible (complex) character of degree $2$. How can I prove that then $G$ has a non-trivial normal subgroup? I tried to prove that for the ...
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Quick question on Pauli matrices and u(2)

The wiki page for Pauli matrices states "Together with the 2 × 2 identity matrix I (sometimes written as σ0), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the ...
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63 views

Show that any representation of $\mathfrak{sl}(2,\mathbb C)$ is a subrepresentation of $V^{\otimes m} \oplus V^{\otimes {(m+1)}}$ for some $m$

Suppose $M$ be a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb C)$, then there is a positive integer $m$ such that $M$ is isomorphic to a subrepresentation of $V^{\otimes m} \oplus ...
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1answer
43 views

How to check if two group representations are equivalent

If we have two representations of the same group $G$, say $\phi$ and $\psi$, they are called equivalent if a $U$ exists such that $U^{-1} \phi(g) U=\psi(g)$ holds for all $g$ in $G$. However, given ...
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Show $v\in FS_n$ is an $F$-multiple.

This is coming from Exercise 8 in Section 18.1 of Dummit and Foote. We are talking about representation theory and in particular focusing on Example 3 and 10 in this section. Let $n\in ...
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Question surrounding young symmetrizers

Let $\lambda$ be a partition of $n$, and let $T$ be the standard tableau associated to $\lambda$ (write the Young diagram of $\lambda$ down and fill in the boxes with $1$ through $n$ left to right, ...
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11 views

Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \oplus_{J}\textbf{r}_J $$ where $\textbf{r}_J$ are ...
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36 views

Dual representations of fundamental representations of a Lie algebra.

Let $g$ be a Lie algebra. Let $V(\omega_i)$, $i=1,\ldots,n$, be the fundamental representations. Are the dual representations $V(\omega_i)^*$ highest weight representations? The dual representation ...
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19 views

Relations of $S^2 V$ and heighest weight representations of Lie algebras.

Let $V$ be the natural representation of $sl_n$. Then $V = V(\omega_1)$, where $\omega_1$ is the first fundamental weight. We have $\Lambda^2 V = V(\omega_2)$. Is $S^2 V = V(\lambda)$ for some weight ...
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53 views

Do we have $\mathbb{C}[V^*] \cong S(V)$ or $\mathbb{C}[V] \cong S(V)$? [closed]

Let $V$ be a vector space over $\mathbb{C}$ and $V^*$ its dual vector space. Let $\mathbb{C}[V^*]$ (resp. $\mathbb{C}[V]$) be the coordinate ring of $V^*$ (resp. $V$) and $S(V)$ the symmetric algebra. ...
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26 views

Definition of coroots

I'm having a bit of trouble with the definition of coroots. From textbooks, we know that given a root system $\Phi$ with $\alpha \in \Phi$, there exists a coroot system $\Phi^{\vee}$ with ...
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1answer
21 views

$G \ne [G,G]$ from irreducible representations

For a group $G$ of order 24, how can I prove using restrictions on possible irreducible reps, that $G \ne [G,G]$? A priori of knowing how many conjugacy classes there are, I can get to $24 = 1^2 + 1^2 ...
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Prove it exists integers so to write character as linear combination.

Let $G$ be a group and suppose that $\chi$ is a character on $G$. Furthermore, suppose that $\chi(g)$ is constant for all $g \not=1$, prove it exists integers $a,b$ such that $$\chi=a1_G + b\chi_0$$ ...
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Prove $V$ is simple $\iff$ all non-zero vectors are cyclic

I am working on some Representation Theory practice questions and I think I have given a valid proof of : Prove $V \ne 0$ is a simple A-Module$\iff$ all non-zero vectors are cyclic $"\leftarrow"$ ...
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44 views

Find conjugacy classes of $G= \left\langle a, b \mid a^4, b^2=a^4, aba=b \right\rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, ...
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56 views

Character of $Sym^2(V)$ and decomposition into irreducible representations

Let $G=S_3$ be the symmetric group on three elements, whose character table is given as follows: Let $V$ be the unique irreducible representation of dimension $2$ Question 1: Compute the character ...
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1answer
28 views

Character regular representation

Consider the regular representation of a finite group $G$ and let $X_{reg}$ be its character. Let $(\pi, V)$ be any finite dimensional representation of $G$ with character $X$. Show that ...
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47 views

Character Table, Row and Column orthogonality, Conjugacy Classes

Let $G$ be a finite group with conjugacy classes $C_1, C_2, ..., C_k$ and let $g_i \in C_i$ be an element for each $i=1, ..., k$ Part 1: State the theorems on row and column orthogonality in the ...
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30 views

How can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\mathrm{End}V)[[x,x^{-1}]]$?

In the context of vertex operator algebras, if $V$ is a vector space, how can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\operatorname{End}V)[[x,x^{-1}]]$? The notation $V((x))$ is the ...
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1answer
74 views

Pushforward of a representation?

Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, ...
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16 views

Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$?

Let $g$ be a Lie algebra. Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$? Thank you very much.
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54 views

$G$-invariant complement to an infinite dimensional vector space

Let $G$ be a finite group and let $$\rho : G \to GL(V)$$ be a complex representation of $G$. Suppose we have an internal direct sum decomposition $$V=W \oplus U$$ where $W$ is infinite dimensional ...
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39 views

Closure of algebraic groups

Let $\phi: G\rightarrow V$ an embedding, with $G$ a complex algebraic group and $V$ a vector space (actually a $G$-representation). Is it true that the closure (in the Zariski topology) of $\phi(G)$ ...