Questions tagged [irreducible-representation]
An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.
86 questions
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Character table for non abelian group of order $8$ over $\mathbb C$
I am trying to determine the character table for non abelian group of order $8$ over $\mathbb C$.
Let $G$ be the non abelian group of order $8$. I have proved that there are four $1$-dimensional ...
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irreducible factors in group determinant
I want to prove the irreducible part of Frobenius’ theorem, that is to say,
If $\rho$ is a irreducible representation, then $P(x_1,x_2,\cdots, x_n)=\text{det}(\sum_{s_i \in G} x_i\rho(s_i))$ is ...
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What do we know about higher degree representation of the Dihedral group $D_n$?
Let $l\geq 3$. What knowledge do we have about $l-$degree representation for the Dihedral group $D_n$ with $2n$ elements.
For one, two-degree representation, I found these:
Representation Theory of ...
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Tensor product of irreps projected onto fields
Some background:
I am trying to build a potential which consists of the $\varphi_i$ fields, which are $SU(2)$ doublets (if relevant). The form of the scalar potential should be
$$
V(\varphi) = A_{ij}\,...
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Integrable representations of affine Lie algebra and non-integrable ones
In the book Conformal Field Theory by Di Francesco, Mathieu and Sénéchal, they introduce the integrable highest weight representations of an affine Lie algebra $\widehat{\mathfrak{g}}_k$ associated to ...
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Uniqueness of the Degree of Faithful Irreducible Representations
Can a group have two faithful irreducible unitary representations with distinct degrees? For example, if you were to find a faithful irreducible unitary representation of a group $G$ of dimension $\...
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Is there a name for this phenomena: tensor products of representations behave like the underlying group?
Take the Klein four group
$V_4 = \mathbb{Z}_2 \times \mathbb{Z}_2$:
$$
\{e,a,b,ab\}, \quad \mathrm{with} \quad a^2=b^2=(ab)^2 = e
$$
Over an algebraically closed field of characteristic 0, being ...
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Choose a subset of n values from a set whereby the sum of the subset cannot be achieved with less than n values from the set
Not a mathematician (I'm a programmer) so please bear with me.
I've written a program to solve a Subset Sum and am trying to generate some test data to stress-test it. After attempting to do this ...
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Examples of proper irreducible subalgebra in $M_2 (\mathbb{R})$ and $M_4 (\mathbb{R})$
We known from Burnside theorem the following fact: For an algebraically closed field $\mathbb{F}$, the only irreducible subalgebra in $M_n (\mathbb{F}), (n \ge 2)$ is the algebra itself $M_n (\mathbb{...
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Irreps of subgroup of unit quaternions
In Proc. Amer. Math. Soc. 5 (1954), 753-768 (https://doi.org/10.1090/S0002-9939-1954-0087028-0), Taylor points out that there are three compact one-dimensional Lie groups with two components:
The ...
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Non-surjective representation of infinitely-generated simple module (see: Jacobson Density/Burnside's theorems)
The book I'm working through covers (on its way to the Artin-Wedderburn theorem) the Jacobson Density theorem:
Let $L$ be an irreducible left $R$-module, with $D := \text{End}_{R-}(L)^{\text{op}}$ (...
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Commutant of the tensor product of range of an irreducible representation with scalar operators.
Let $\mathcal{H,K}$ be Hilbert spaces. Consider $S = \mathbb{C} I_{\mathcal{H}} \otimes \mathcal{B(K)} \subseteq \mathcal{B(H \otimes K)}$. Then, it is known that the commutant of $S$, $S' = \mathcal{...
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Irreducible representations on commutative real $C^*$-algebra
Let $\mathcal{A}$ be a (complex) commutative $C^*$-algebra and $\mathcal{H}$ be a (complex) Hilbert space. If $\pi : \mathcal{A} \to \mathcal{B(H)}$ is an irreducible representation, then we know $dim(...
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Group representation over finite fields
In Isaacs' Character theory of finite groups, the theorem $9.14$ states that an absolutely irreducible over a field of prime characteristic can be realized over any subfield containing its character ...
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Changing the field of an irreducible representation leaves it irreducible.
I am currently learning representation theory of finite groups over arbitrary fields.
I've encountered an interesting result multiple times but never found a proof of it anywhere.
Let $E/F$ be a field ...
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Does an irr. rep. of finite $G$ have a basis of the form $\{hv : h \in H\}$ for $H$ subgroup of $G$?
Let $G$ be a finite group and $V$ be a (complex vector space) representation of $G$. Consider the following three facts:
Whenever a $V$ is an irreducible representation of $G$, the dimension of $V$ ...
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Generating a conjugate representation of an irreducible self-conjugate representation of $S_n$
Suppose we have a complex matrix representation $\Gamma_{ij}^\sigma \in \mathbb{C}^{d \times d}$ of dimension $d$ for the permutations $\sigma$ of the group $S_n$ of permutations of $n$ objects.
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Indecomposable complex finite dimensional representations of a compact Lie group are Irreducibles?
I am studying the book "Representations of Compact Lie Groups" by Theodor Brocker and Tammo tom Dieck. At page 68 they prove the following proposition:
Let $G$ be a compact group. If $V$ is ...
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Irreducible complex representations of some abelian Lie groups
I wanted to classify all irreducible complex representations of the following basic abelian Lie groups:
$\mathbb{S}^1$ the circle in the complex plane, $\mathbb{R}_{>0}$ the positive real numbers, $...
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Why is the set of linear combinations of irreps closed under matrix products?
I'm studying Barry Simon's book on representation theory. There's this one result he proves stating that the set of linear combinations of $D_{ij}^{(\alpha)}$ is closed under taking products. The $D_{...
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Further decomposition of isotypic components in a representation
Let $(V,\rho)$ be an orthogonal (resp. unitary) representation of finite group $G$ whose irreducible representations over the same field as $V$ are $W_i$ with character $\chi_i$.
We have $V \cong \...
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Scalar extension of endomorphism ring of representation
I am currently trying to learn representation theory of finite group over an arbitrary field and i stumbled on a statement that seemed very intuitive to me but i could not find a proof of it anywhere.
...
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Given a representation of a group, how does one determine the multiplicity of the irreps?
Let G be a semi-simple Lie group and $(\pi, V)$ be a representation of $G$. $V$ is decomposed by $G$'s irreps as
$$
V = \oplus_{\mu,\lambda} V_{\mu,\lambda},
$$
where $\mu$ labels different irreps and ...
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A question about inducing of group representations
Let $G$ be a compact Lie group, $K$ its closed subgroup, $\rho$ a finite-dimensional real irreducible representation of $K$, $c\rho$ its complexification, $\mathrm{Ind}^G_K(\rho)$
the real ...
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Extension of the base field of a simple module over a finite dimensional algebra stays completely reducible/ semisimple
As I was thinking about a previous question of mine, i asked myself if there was a way to prove the following statement:
Let $L\setminus K$ be a field extension and $A$ be a finite dimensional $K$-...
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Matrices invariant under rotations are always proportional to the identity?
Is this proof true?
Suppose we have a $3\times 3$ matrix $M^{ab}$ satisfying
$$M^{ab}=R^a\,_cR^b\,_dM^{cd},$$
i.e.
$$M=RMR^T,$$
for all rotations $R\in \mathrm{O}(3)$. Now, if denote representations ...
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Extension of the base field of an irreducible representation of a finite group stays completely reducible
I was reading chapter $9$ of "Character theory of finite groups" by Isaacs in which he explores the theory of representation of finite groups over arbitrary fields.
In his theorem $(9.2)$, ...
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A brief explanation on Representation Theory
I'm trying to read this beautiful paper Regularity and cohomology of determinantal thickenings by Claudiu Raicu but I'm getting in trouble with Representation Theory, since I have no knowledge of it.
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Projection of real representations onto the isotypic components
Let $(V, \rho)$ be a representation of a finite group $G$ over field $\mathbb{F}$ and $W_i$ be irreducible representations (irreps) of $G$ over $\mathbb{F}$ with dimension $d_i$ and character $\chi_i$....
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What can be said of general representations of solvable Lie algebras?
I'm curious about the representation theory of solvable Lie algebras. Consider the following quote from Fulton & Harris:
[To] study the representation theory of an arbitrary Lie algebra, we have ...
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Non trivial subspaces of the permutation module $\mathbb{C}[X]$
Let $X={1,2,...,n}$. Consider the vector space $\mathbb{C}[X]=\{c_1\cdot1+\dots+c_n\cdot n : c_i \in \mathbb{C}\}$
Viewing this as an $S_n$ module, we call this the permutation module. I want to ...
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Locally compact group whose unitary irreducible reps are one dimensional
It's known that if G is a finite group such that all its irreducible unitary representations are one-dimensional, then G is abelian. This uses the fact that the left regular representation decomposes ...
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Reducible and indecomposable modules/representations
This is perhaps a soft question regarding terminology but I am somewhat confused by the similar posts I've seen. Note: I am not necessarily considering representaitons of finite groups (i.e. ...
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what's the relationship between the eigenvalues of $\phi_g$ and a certain set of $H$-representations $(W,\rho)$
Let $g\in G$ be an element in a finite group G and H the subgroup generated by g. For a given irreducible G-representation $(V,\phi)$ over $\mathbb{C}$, what's the relationship between the eigenvalues ...
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is it true that a certain induction of an H-irrep $\psi$ is irreducible iff there's a G-conjugacy class on which $\chi_\psi$ is nonconstant
Assume $H\subset G$ is a subgroup of index $2$. Let a G-conjugacy class mean the elements conjugate to a fixed h in G. Prove or disprove whether the induced representation $Ind_H^G (\psi)$ of an H-...
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When is the isotypical decomposition of a representation not unique?
I am reading some notes on representation theory, and I have found myself confused by the following proposition:
Suppose $G$ is a finite abelian group, then every complex representation $V$ of $G$ has ...
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does there exist $T\in GL(n,\mathbb{C})$ so that $\sigma(g) = T^{-1} \rho(g)T$ for all $g\in G$?
Suppose that $\rho$ and $\sigma$ are degree $n$ irreducible representations of a group $G$ over $\mathbb{C}$ and that for every $g\in G,$ there is a matrix $T_g\in GL(n,\mathbb{C})$ depending on $g$ ...
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About two Standard Representations of S3
https://groupprops.subwiki.org/wiki/Standard_representation_of_symmetric_group:S3
Gives two 2-D irreducible representations for S3. One in the basis (e1-e2) and (e2-e3) and another "viewed as a ...
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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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