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Questions tagged [representation-theory]

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

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Projection operator in group representation

Assume that we construct a representation for a group $G$ which is reducible. Then to block diagonlize it (decompose it to irreducible one), we first calculate the frequency of each irreps using (here ...
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2answers
26 views

Definition of the tensor product of representations

I'm a bit confused about the following definition: Let $\rho_1:G \to Aut(V_1)$, $\rho: G \to Aut(V_2) $ be two representations of the same group $G$. Then a tensor product of representations is ...
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43 views

Question regarding motivation of spectral theorem for unitary operators

$\newcommand{\mc}{\mathcal}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ Theorem B.4 in Einsiedler and Ward's [EW] Ergodic Theory with a view towards Number Theory states the following: Theorem 1. ...
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1answer
8 views

Incorrect modulus character computation in Casselman's notes?

In Proposition 6.3.3 in Casselman's notes on representation theory of $p$-adic groups, I believe there is an error in the statement of the Proposition. Let $G$ be the points of a connected, reductive ...
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1answer
24 views

Continuity(?) of induced group representation in the isometries of $L^p$

Let $G$ be a topological group which acts on a measure space $(X,\mu)$ by measure preserving transformations. It's well known that this induces a representation $\pi$ of $G$ in $O(L^p (X,\mu))$, the ...
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Affine $\mathfrak{su}(2)_k$ characters and Jacobi triple product

In this post, the Kac character formula for affine $\mathfrak{su}(2)_k$ $$\chi_{\ell}^{(k)}(\tau,z) = \frac{\Theta_{\ell+1,k+2}(\tau,z)-\Theta_{-\ell-1,k+2}(\tau,z)}{\Theta_{1,2}(\tau,z)-\...
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62 views

On a representation of $S_n$.

Fix a natural number $n$, and a complex vector space $V$ of dimension $d$. Consider the representation of $S_n$ on $V^{\otimes n}$ given by $\rho(\sigma)(v_1\otimes\cdots\otimes\cdots v_n)=v_{\sigma(1)...
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1answer
18 views

Grading of the character group of a maximal Torus in a reductive group

I am working through the paper "On the Algebraic $K$-Theory of Some Homogeneous Varieties" by Alexey Ananyevskiy and got stuck at the beginning of the second section. The set up is the following: Let ...
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26 views

Composition of Quotient Map and Irreducible Representation

This is exercise 4.7 in Chapter 10 of Artin. Let $\pi:G\rightarrow G/N$ be a canonical map of a finite group into a quotient group, and let $\rho'$ be an irreducible representation of $G$. Prove that ...
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1answer
30 views

No common eigenvectors then representation irreducible

Embed an equilateral triangle into $\mathbb{R}^2$ with vertices $(1,0), (\frac{-1}{2}, \frac{\sqrt{3}}{2}), (\frac{-1}{2}, -\frac{\sqrt{3}}{2})$. Counterclockwise rotation and reflection over the $x$ ...
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27 views

Why does Schur's Lemma imply $\dim \hom(V_1,V_2)^G=1$ if $V_1\cong V_2$?

Schur's Lemma says: Let $\rho_1:G\to GL(V_1)$ and $\rho_2:G\to GL(V_2)$ be two irreductible representations over $\mathbb{C}$ of a group $G$. If $f:V_1\to V_2$ is an equivariant linear ...
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1answer
12 views

Relation between the order of an element of a group and their character

I am struggling with a proof of a two part question: Let $\chi$ be a character of a finite group $G$. a) If $g$ has order 2, then $\chi(g) \in \mathbb{Z}$ and $\chi(g) \equiv \chi(1)$ (mod 2) ...
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1answer
53 views

Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here: Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\...
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Confusion about this notation for representation families $ R_1 \times_{H} R_2$

In a paper that I am reading I am given two families $R_1,R_2$ of representations of some groups $G_1,G_2$ into $SU(2)$. It follows that we can think of an element $\rho\in R_i$ as an homomorphism $$...
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1answer
56 views

Is the highest weight constructed irreducible representiation of a Lie Algebra unique?

In Georgis book I stumbled across the sentence: "It is true that every irreducible representation is finite dimensional and equivalent to one of the constructions we found with the highest weight ...
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38 views

Extensions of finite groups and group algebras

Let $G$ be a finite group, $H \subset G$ is a subgroup and $K = G/H$ is the quotient group, so we have extension of groups $$ 1 \to H \to G \to K \to 1. $$ Let $R$ be a commutative ring, can one ...
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How to test if a $\mathbb{R} G$-module is irreducible?

Let $V$ be a $\mathbb{C} G$-module with character $\chi$. We know that $V$ is irreducible if and only if the inner product $\left<\chi,\chi\right>=1$. But what if $V$ is a $\mathbb{R} G$-module?...
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If $V$ is a $\mathbb{C} G$-module whose character is real then dim$V$ and dim $V^G$ have the same parity.

Let $G$ be a finite group of odd order. Prove that if $V$ is a $\mathbb{C} G$-module whose character is real, then dim$V$ and dim$V^G$ have the same parity. where $V^G:=\{v \in V \mid vg=v \forall g \...
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Relation between $\dim V^G$ and the number of orbits of a given linear representation.

In the context of linear representations of finite groups, we know that $$\dim V^G=\frac{1}{|G|}\sum_{g\in G}\text{tr } \rho(g)= \frac{1}{|G|}\sum_{g\in G}|V^g|,\qquad\qquad\qquad (1)$$ where $V^G$ is ...
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Proof of Corollary of Clifford theorem

Let $H \trianglelefteq G$. Let $\theta$ be an irreducible character of $H$. Consider action of G on $\theta$ , $g.\theta=\theta^g$ where $\theta^g(x)=\theta(gxg^{-1})$. Let T be the stabilizer of ...
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1answer
37 views

The opposite of Weyl's theorem on Lie algebras

Let $k$ be an algebraically closed field of characteristic zero, and $\mathfrak{g}$ a semisimple Lie algebra over it. By Weyl's theorem, we know that any finite-dimensional representation $V$ of it is ...
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Quotient of an amenable group (non-discrete case) and Haar measure on the quotient

"Let $G$ be a locally compact amenable group with Haar measure $\mu$, $H$ a closed normal subgroup, then $G/H$ is amenable." I am trying to prove this fact, and there are two definitions I can use (...
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Is there an easy way to calculate $\sum_{s \in t^G} \rho(s)$ for a representation $\rho$ of $G$?

I am looking at ways to quickly decompose finite group representations into irreducibles. I am following Serre's book "Linear Representations of Finite Groups". He gives a formula for computing the ...
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1answer
29 views

What is the n-dimensional representation of a U(1) element?

What is the $n$-dimensional matrix representation of an element $g\in U(1)$? Is it simply the $n$-dimensional identity matrix times an exponential factor $\text{e}^{\text{i}\alpha}$? This would fit ...
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1answer
29 views

Equivalent representation definitions

A matrix representation of a group $G$ is a homomorphism $$\rho : G \rightarrow GL_{n}(\mathbb{C})$$ A G-space is a vector space $V$ equipped with an action of $G$ $$ G \times V \rightarrow V, (g,v) \...
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Chevalley-Shephard-Todd theorem

I'm studying Chevalley-Shephard-Todd theorem, in the version that states : let $G \subset GL(V)$ a finite group, where $V$ is a finite dimensional complex space. Let $S=S(V^*)$ indicates the symmetric ...
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1answer
51 views

Is there a notion of induced representation that works not only for subgroups?

Given a group $G$ an a subgroup $H<G$, a representation of $H$ on $V$ is a pair $(\rho, V)$ where $\rho \colon H \to \mathrm{GL}(V)$ where $V$ is a vector space over a field $K$. We can the ...
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Induction of representation commutes with dual

Suppose we have finite groups $H \leq G$ and a $\mathbb{K}[H]$ module $V$. I would like to say that $Ind_H^G(V^*) \cong (Ind_H^G(V))^*$. If I have good hypothesis on $\mathbb{K}$ (a.e char does not ...
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1answer
36 views

Let $V$ be a complex representation of a compact Lie group $G$. Prove that $\overline{V}$ and $V^*$ are isomorphic representations of $G$.

Let $G$ a compact Lie group and $V$ a representation of $G$. I have to proof that $\overline{V}$ $\cong$ $V^*$, where $V^*:=\text{Hom}(V,\mathbb{C})$ and $\overline{V}$=$V$ as a finite vector subspace ...
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criteria for reducibility or irreducibility of a tensor product of two irreducible representations

This is a follow-up question to Ines Institoris' question and the reply by Lord Shark the Unknown --- see irreducibility of a tensor product of two irreducible representations Suppose $U$ and $V$ are ...
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1answer
22 views

About the C*-algebra of the Schrodinger representation of the Weyl C*-algebra

We start with the Weyl C*-algebra $\mathcal{W}$ for a finite dimensional symplectic space and we consider the irreducible Schrodinger representation $\pi:\mathcal{W}\rightarrow \mathcal{B}(\mathcal{H})...
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Weak containment of trivial representation

Let $\sigma$ be a continuous unitary representation of the topological group $G$ on a Hilbert space $V$. Suppose $\sigma$ weakly contains the trivial representation, that is: for any compact subset $K$...
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1answer
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Locally closed subset of an l-space is also an l-space.

I am trying to prove that a locally closed subset of an l-space is also an l-space. An l-space is defined as a Hausdorff, locally compact, zero-dimensional topological space. I am having ...
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Dimension of vector space for irreducible representations

Let $G$ be a locally compact group, $\rho$ an irreducible unitary representation on some inner product space $V$. Is there a bound on the dimension of $V$ with respect to the cardinality of the group? ...
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1answer
50 views

The collection of (unitary representations on) Hilbert spaces is a set

Let $G$ be a locally compact group. I know that the collection of all unitary representations of $G$ is not a set, since there are unitary representations on inner product spaces with bases of any ...
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1answer
55 views

Schur's Lemma and $Z(G)$

Let $Z(G)$ be the centre of G. Let $V$ be an irreducible $G-$space with matrix representation $\rho_v$. Let $z \in Z(G)$, then I'm trying to show that $\rho_v(z)$ is multiplication by a root of ...
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1answer
46 views

Order of an element and its character

Let $G$ be a finite group and let $\chi$ be its character. Suppose $g \in G$ as order $3$ and $\chi(g) \in \mathbb{R}$. Show that, in fact, $\chi(g) \in \mathbb{Z}$ and $\chi(g) \equiv \chi(1)mod3$. ...
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1answer
23 views

$\mu$, $\nu$ are compositions with the same parts then for any $\lambda$, $K_{\lambda\mu}=K_{\lambda\nu}$ ($K$ Kostka number)

I want to show the following. If $\mu, \nu$ are compositions with the same parts (only rearranged) then for any $\lambda$ we have that $K_{\lambda\mu}=K_{\lambda\nu}$. I know that the Kostka ...
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Relationships between permutation and left representation of the same algebraic structure

A permutation group $P \leq S_n$ can be represented as a subgroup $H \leq GL(n,2)$ if a permutation $\sigma\in P$ acts in the indices of the vectors of $e_n$. Then every permutation of $P$ has a ...
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Representation of Lie group U(1)

I have some problems with understanding the following representation of the Lie group $U(1)$. For the irreducible representations have the following: $$T_l (e^{i\theta}) = e^{il\theta},$$ where $l \...
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Does character table unique?

When I calculate the character table of $Z_3$, I start from the orthogonal relations: $$ \matrix{Z_3 & (e) & (g) & (g^2) \\ \hline \chi_0 & 1 & 1 & 1 \\ \chi_1 & 1 &...
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1answer
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Proving that a representation is irreducible

Let $V_n = \mathbb{C}^n$ the representations of $SU(n)$ given by matrix multiplication $SU(n) \times V_n \rightarrow V_n, (A, v) \mapsto A \cdot v .\,$ Show that $V_n$ is irreducible. I tried to ...
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Confusion on Irreducible representations of a group

There are two questions which are interrelated hence I want to mention them over here. Given the irrep $\Gamma^{(3)}$ of group C3V, which is of 2 dim. It can be diagonalized further, into a simple ...
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How to find $S_3$-invariant vectors under a representation

How would you find all $S_3$-invariant vectors for the standard representation $\rho$ of $S_3$, where $\rho:S_3\rightarrow GL_2(\mathbb{C})$ is defined by its generators (with $S_3=\langle x,y:x^3=y^2=...
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From root and weight lattices of SU(N) to $\theta$-functions as sections of a line bundle and $CP$-space

I have troubles to digest the following messages/discussions in the following work in p.10-12; Which construct a map from the moduli space of flat connections $M_{\rm flat}=\mathbb{E} / {\mathfrak S}...
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$\chi$ is a character and $\chi(g) \in 2\mathbb{Z}$ then $\frac{1}{2}\chi$ is also a character.

Suppose $G$ is a finite group. Let $\chi$ be the character of some $\mathbb{C}G$-module with the property that $\chi(g)$ is an even integer for every $g \in G$. Is it true that $\chi/2$ defined by $\...
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57 views

Quotient of linear algebraic groups: dimension of a faithful representation

Let $k$ be an algebraically closed field (I am mostly interested in $k=\mathbb{C}$ if that matters). Let $G$ be an affine algebraic subgroup of $GL_n(k)$ (ie $G$ is in fact linear). Let $H$ be a ...
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1answer
96 views

Nilpotent Action of an Ad-Nilpotent Element of $\mathfrak{sl}_2(\mathbb{C})$

If you are familiar with Lie algebras and representation theory, then you can skip my introduction and get right to the two questions (conjectures) at the end. You might only need to read the next ...
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1answer
32 views

Does $Ind_{H}^{G}\mathbb{Q}\cong\mathbb{Q}[G/H]$?

I know that $Ind_{H}^{G}\mathbb{Z}\cong\mathbb{Z}[G/H]$, but I am unsure whether $Ind_{H}^{G}\mathbb{Q}\cong\mathbb{Q}[G/H]$ would hold? I don't understand why it would but at the same time can't ...
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2answers
50 views

Decomposition of tensor product of two representations in $S_3$.

Consider the group $S_3$. There are three irreducible representations, the trivial, $\varphi^{triv}$, the sign representation $\varphi^\epsilon$ (both 1-dimensional), and the two-dimensional one $\...