An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.
Questions tagged [irreducible-representation]
113 questions
7
votes
1 answer
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 Abelian it has only one dimensional irreducible…
Craig
- 974
6
votes
2 answers
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 $V$ must divide the order of $G$. See this…
Samuel Johnston
- 804
5
votes
1 answer
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 $g$ so that $\sigma(g) = T_g^{-1} \rho(g) T_g$. Does…
user1127
- 479
5
votes
0 answers
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
$$ V = \mathbb{C}\{111222, 112122, 112212, \ldots\} $$
(In this example, V is…
eti902
- 905
4
votes
1 answer
Representations of $\mathbb{Z}_n$ over $\mathbb{Q}$
I know that the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{C}$ sending $1$ to the roots of unity. There are $n$ of these and they're all $1$ dimensional since…
user1632940
- 43
- 3
4
votes
0 answers
Multiplicity of "dual" irreps of subgroups $H \subset S_n \times \mathbb{Z}_2$
Consider a subgroup $H$ of the group $G = S_n \times \mathbb{Z}_2$ given by the direct product of some permutation group $S_n$ and the group $\mathbb{Z}_2$. Denote a general element of $G$ by $\pm \sigma$, where $\sigma \in S_n$.
Given a unitary…
creillyucla
- 443
4
votes
1 answer
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, $\mathbb{C}^\times$ the complex numbers without…
Donson
- 303
3
votes
1 answer
Character-free derivation of the number of irreducible representations of finite group
I'm trying to prove as much as possible of classical results of finite groups representation theory without using the notion of character, especially orthogonal relations, since I want to work with representations over algebraic closures of finite…
Matthew Willow
- 762
3
votes
4 answers
Character table of semidirect product of $\mathbb{Z}/7\mathbb{Z}$ with $\mathbb{Z}/3\mathbb{Z}$
I am trying to compute the character table of $\mathbb{Z}/7\mathbb{Z} \rtimes \mathbb{Z}/3\mathbb{Z}$ given by $bab^{-1}=a^2$, where $a$ is the generator of $\mathbb{Z}/7\mathbb{Z}$ and $b$ the generator of $\mathbb{Z}/3\mathbb{Z}$.
I already have…
Gabriela Martins
- 459
3
votes
0 answers
Irreps of wreath product $SU(2) \wr S_2$
The irreps of $H = SU(2) \times SU(2)$ are of the form $V_{j_1} \otimes V_{j_2}$ where $V_j$ is the spin-$j$ irrep of $SU(2)$ of dimension $2j+1$ (e.g., $V_0$ is trivial, $V_{1/2}$ is fundamental, and $V_1$ is adjoint). What are the irreps of $G =…
Eric Kubischta
- 760
3
votes
1 answer
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 $\aleph_0,$ could you immediately deduce that any…
Miles Gould
- 1,284
3
votes
1 answer
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 product of the circle group $\mathbb{T}$ with the…
creillyucla
- 443
3
votes
1 answer
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 $S$, $S' = \mathcal{B(H)} \otimes \mathbb{C} I_{\mathcal{K}}$.
Is…
Anand O R
- 155
3
votes
0 answers
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 $V \cong \bigoplus_{i=1}^mW_i^{n_i}$. The projection of $V$…
khashayar
- 2,413
3
votes
0 answers
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 $V^{\lambda}_i$ denote the irreducible $S_i$-module…
Jose Brox
- 5,216