Questions tagged [irreducible-representation]

An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.

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Properties of converging succession of functionals

Given a sequence of functionals $(\omega_n)_{n \in \mathcal{N}}$ on a Von Neumann algebra $\mathcal{W}$ converging to $\omega$, I have the following doubts: Is it true that $\omega_n$ is pure $\...
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Verifying that a vector is a highest weight vector

I'm trying to show that $(e_1 \otimes e_2 \otimes e_3 - e_3 \otimes e_1 \otimes e_2) \otimes e_n^*$ is a highest weight vector for the irreducible submodule of $V^{\otimes 3} \otimes V^*$ with highest ...
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Character table of S4

I am trying to understand the character table of $S_4$. I have obtained the trivial, signature and standard representations. The fourth one is the product of signature and standard. Now for the last ...
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Constructing the character table of Octahedral Group (order 48)

I am trying to construct the character table for the symmetry group of Cube, $O_h$, which as 48 elements. I figured out that there are 10 classes. Now, as a consequence of the great orthogonality ...
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Restriction of induced representation over a Young subgroup and Littlewood-Richardson coefficients

I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\...
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Matrices commuting with a completely reducible representation

By Schur's lemma, a matrix commuting with an irreducible representation (of a group over complex numbers, say) is a multiple of identity. What about a direct sum of irreducible representations (a.k.a. ...
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The fact that the space of matrix coefficients is a 2-side ideal in $C(G)$ implies Schur orthogonality.

Suppose that $G$ is a compact group and $C(G)$ is the space of continuous functions on $G$. For $f_1$, $f_2\in C(G)$, define the convolution by $$(f_1*f_2)(g)=\int_Gf_1(gh^{-1})f_2(h)\mathrm{d}h=\...
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Show that this irreducible representation is faithful

For a simple Lie algebra $\mathfrak{g}$ who contains a subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, I’m trying to show that a nontrivial irreducible representation $\pi:\mathfrak{g}\to\...
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N-dimensional linear representation of symmetric group S(N)

A beginner's question on the linear representations of the symmetric group $S(n)$: playing around with GAP, e.g.: gap> CharacterDegrees(SymmetricGroup(n)); for ...
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Moving between $Sp_{2n}(\mathbb{C})$ reps and $SL_{n}(\mathbb{C})$ reps

Say I have some irreducible $Sp_{2n}(\mathbb{C})$ representation, such as $\Gamma_{0,1,0,1}$. Consider the subgroup of $Sp_{2n}(\mathbb{C})$ isomorphic to $SL_n(\mathbb{C})$, consisting of matrices of ...
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Well-definedness of isomorphism in Schur's Lemma.

Proposition: (Schur's Lemma) Let $(\rho,V)$ and $(\tau,W)$ be irreducible representations of a finite group $G$. Then $$ \text{Hom}_G(V,W) \cong \begin{cases} \mathbb{C} & \text{ if $\rho \...
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A faster way to do induction from character table of subgroup to larger group by hand? $S_4$ to $S_5$ as example

The purpose of this question is to ask: for $H \leq G$ groups, when doing induction from the character table of $H$ to the character table of $G$, is there a faster method by hand than the one I ...
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Irreducible representation of the cyclic group over $\mathbb{Q}$

I'm trying to prove the following: Let $G$ be a cyclic group of order $n$. For each divisor $d$ of $n$, denote by $G_d$ the subgroup of $G$ of index $d$. Show that $G$ has an irreducible ...
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Finding decomposition of induced representation

Let $G = Af$ $f(\mathbb{F}_p)$ be the group of affine transformation, i.e., $G = \{f: \mathbb{F}_p \to \mathbb{F}_p, x \mapsto ax+b| a, b \in \mathbb{F}_p, a\ne 0\}$ Let $H$ be the cyclic subgroup of ...
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Method for classifying irreducible $\mathbb{C}$-representations of $D_{10}$ of dimension $2$

$D_{2n} = \{ r, s: r^n = s^2 = e, srs = r^{-1} \}$ is the dihedral group with order $2n$. I'm trying to classify the $2$-dimensional irreducible $\mathbb{C}$-representations of $D_{10}$ up to ...
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Decomposition of equivariant maps with symmetry-adapted basis

Let $(V, \rho)$ be a representation of a finite group $G$ whose irreducible representations over complex numbers are $(W_i, \pi_i)$ for $1 \le i \le m$. Suppose $\dim(\text{Hom}_G(W_i),V)=d_i$ and $\...
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Space of equivariant homomorphism from the space of G-linear maps to a vector space

Let $(V, \rho)$ be a representation of a finite group $G$ whose irreducible representations over complex numbers are $(W_i, \pi_i)$ for $1 \le i \le m$. Let $$H_i=\text{Hom}_G(W_i,V)= \{\tau \in \text{...
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Number of copies of irreducible representations for transitive actions

Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then $|X| \le |G|$ since $|X| \big| |G|$ by the orbit-stabilizer theorem. Let $\pi_i;\ \ 1\le i \le m$ be irreducible ...
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Faithful representations of a non-abelian group of order $p^3$

Question: Let $G$ be a non-abelian group of order $p^3$ ($p$ is a prime). (a) Determine the number of irreducible complex representations of $G$, and find their dimensions. (b) Which of the ...
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Decomposition of $(\mathbb C^d)^{\otimes n}$ into permutation modules

The Schur-Weyl duality says that $n$th tensor power of a $d$-dimensional vector space over $\mathbb C$ is isomorphic to the direct sum of the tensor product of the the Weyl modules $V_\lambda$ and the ...
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$\varphi:G\to GL(V)$ irreducible. If $H$ is a finite abelian subgroup of $GL(V)$ such that $H\leq C_{GL(V)}(\varphi(G))$, then $H$ is cyclic.

Question: Let $V$ be a finite dimensional vector space over a field $k$. Let $G$ be a finite group. Let $\varphi:G\to GL(V)$ be an irreducible representation of $G$. Suppose that $H$ is a finite ...
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Does an irreducible Markov chain have exactly one equivalence class?

I think I have a fundamental misunderstanding of equivalence classes and irreducibility of Markov chains. Consider this simple MC where States A and B communicate ($ A \leftrightarrow B $) and States ...
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Wiki claim - Irreducible representations with highest weight $(0,m)$ can be realized on the space of homogeneous polynomials of deg $m$ over 3 vars

In the last line of this Wikipedia article on representations of sl_3 there is the line: "There is also a simple pattern to the multiplicities of the various weight spaces. Finally, the ...
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GNS representation - A simple proof for 2x2 Matrices

I am learning the GNS representation for C* algebras, and I would like to make a first step by applying the GNS construction on the simple algebra of the 2x2 matrices on the complex field. I would ...
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Decomposing $\mathfrak{sl}_2$ as a sum of irreducible representations $U(sl_2)$

I want to find decompose $\mathfrak{sl}_2$ as a direct sum of irreducible representations of $U(\mathfrak{sl}_2)$ which is the universal enveloping algebra of $\mathfrak{sl}_2$. I have a theorem that ...
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Representation of symmetry group $D_4$ acting on a square

Consider the square with vertices $(\pm 1, \pm 1)$. The symmetry group $D_4$ of this square acts by permuting the vertices Show that each permutation comes from a linear transformation. Compute the ...
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Tableau which corresponds to alternating square representation

Recall that $S_5$ acts on $\mathbb C^5$ by permuting its coordinates and that $\mathbb C^5$ decomposes as $V\oplus W$ where $W$ is the trivial representation and $V$ has dimension $4$. Show that the ...
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The irreducible representation of $S_n$ corresponding to the partition $n = (n − 1) + 1$

Let a finite group $G$ act doubly transitively on a finite set $X$, i.e., given $x,y,z,w\in X$ such that $x\neq y$ and $z\neq w$, there is a $g\in G$ such that $g.x=z$ and $g.y=w$. So, we can write $$\...
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Example of an irreducible homogeneous Markov chain, that possesses an invariant measure but is not recurrent

I’m currently studying the topic of Markov chains and how invariant measures are connected to recurrence. I now know that an irreducible homogeneous Markov chain that possesses invariant measures must ...
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Show that $\theta$ is irreducible representation

Let a finite group $G$ act doubly transitively on a finite set $X$, i.e., given $x,y,z,w\in X$ such that $x\neq y$ and $z\neq w$, there is a $g\in G$ such that $g.x=z$ and $g.y=w$. So, we can write $$\...
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Prove that the degree of every irreducible representation divides $[ G : Z(G) ]$

The question is essentially the title. We assume that $G$ is a finite group and the irrep is over $\mathbb C$. $Z(G)$ is the center of $G$. The caveat is that I'm looking for an "elementary" ...
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Constructing a semidirect product in GAP using characters

I have a finite group $G$ and an elementary abelian group $E(p^h)$ where $p\not \mid |G|$. I just want to construct via GAP the semidirect product $E(p^h):G$ given by a character $\chi$ of $G$, not ...
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What do eigenvalues have to do with modules?

Let $G$ be the cyclic group of order 4 , given by the presentation $$G = \left\langle a: a^4 = 1\right\rangle.$$ Let $F$ be $\mathbb{R}$. Let $V$ be a 2-dimensional vector space over $F$ with basis $\...
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Order of groups and dimensions of representations [closed]

Show that if $H$ is an abelian subgroup of order $p$ of finite group $G$ of order $n$, then every irreducible representation is of dimension $\leq n/p$. I'm really confused of how an order of a group ...
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Irreducible representations in group theory

I am currently working through a chapter of my quantum mechanics textbook about how group theory relates to quantum mechanics (I considered asking this question on the physics exchange, but figured ...
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A reductive lie algebra $L$ satisfies that $[L,L]$ is semisimple

Let $L$ be a reductive Lie algebra (i.e., $\mathrm{Z}(L) = \operatorname{Rad}(L)$). Then $\mathrm{ad} \colon L \to \mathfrak{gl}(V)$ is completely reducible, $L = [L, L] \oplus \operatorname{Z}(L)$ ...
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Unitary irreps of a semi-direct product

I am wondering how to construct unitary irreps of the group $$(SO(3) \times SO(3)) \rtimes \mathbb{Z}_2$$ where the $\mathbb{Z}_2$ acts by swapping the two copies. This is the isometry group of $\...
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Example of a primary representation that is not a multiple of an irreducible representation

Let $G$ be a group. A unitary representation of $G$ on a Hilbert space $H$ is called primary if for any invariant subspace $V\subseteq H$ one has that either: $V\in \{0, H\}$. There is a non-zero ...
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Ambiguity in unitary and completely reducible representations

I am reading the book "Group Theoretical Methods and Their Applications" by A. Fassler and E. Stiefel. There is a part that I do not understand. Suppose $\mathcal{v}$ is a finite ...
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Decomposing $Sym^2\mathbb{C}[X]$ without using characters

There is a nice and not too long way to decompose $Sym^2\mathbb{C}[X]$ (permutation representation of $S_5$ acting on $\{x_1,\cdots,x_5\}$) in irreducible without using characters? I know a way using ...
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Intuition behing Mackey's irreducibility criterion for irreducibility of the induced representation.

We consider $\mathbb C$-linear representations of finite groups. If $H$ is a subgroup of $G$, and $(W,\theta)$ is an $H$-representation, irreducibility Mackey's criterion states that $Ind_H^G W$ (the $...
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Irreducible representations of $\operatorname{SO}(3)$ from $\operatorname{SU}(2)$

I‘m having trouble understanding what exactly the irreducible representations of $\operatorname{SO}(3)$ are. I know that the irreducible representations of $\operatorname{SU}(2)$ are given by: $\rho(A)...
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How to complete this proof of the Schur-Weyl duality theorem using commutants?

I have a question on a proof from the following book :"Symmetry, representations and invariants by Roe Goodman and Nolan R. Wallach. Specifically on section 4.2.4 on Schur Weyl duality and the ...
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A question related to the spectrum of Gelfand-Tsetlin Algebra

Let's take the chain of permutation subgroups $S_1=\{1\}\subseteq S_2\subseteq .....\subseteq S_n$,where $S_{i-1}$ sits inside $S_i$ canonically as the stabilizer of $\{i\}$.Then for this inductive ...
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Group algebra, cyclic group, ring isomorphism

Consider the cyclic group $G=\langle g\mid g^{3}=1\rangle$,his group algebra $\mathbb{C}G$ and $\mathbb{R}G$. As a complex algebra, how to construct an isomorphism between $\mathbb{C}G$ and $\mathbb{C}...
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Non atomic ring example

Can someone provide me with an example of a ring that is not atomic, that is, there is an element that has no finite factorization in irreducibles?
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Decomposition of the representation generated by words with m copies of n letters

Let $V$ be the (complex) vector space generated by words of length $nm$ where each letter from $1$ to $m$ appears exactly $n$ times. For example, if $m=2$ and $n=3$, then $$ V = \mathbb{C}\{111222, ...
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Unique irreducible complex representation of Clifford algebra implies isomorphism with matrix algebra

Consider the Clifford algebra $\mathrm{Cl}(n)$ over Euclidean space $\mathbb{R}^n$ (with the standard inner product). Now, in the case that $n$ is even, it is known (cf. [1]), then $\mathrm{Cl}(n)$ ...
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Tensor product of two representations of $D_4$

Let $(\tau,\mathbb C^2)$ be the irreducible representation of $D_4$ by matrix multiplication, namely for every $v\in\mathbb{C}^2$: $$\begin{bmatrix}\tau(s)\end{bmatrix}v=\begin{bmatrix}-1 & 0\\ 0 &...
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