Mathematics

Questions tagged [irreducible-representation]

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

120 questions
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Intuition behind the representation theory of semi-direct products (Mackey's theorem)

I've been attempting to understand the irreducible representations of semi-direct products involving abelian groups; this is primarily motivated by a desire to understand Wigner's classification of elementary particles. It seems that these…
Baylee V
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The sum of an irreducible representation over a subset of a finite group

Let $G$ be a finite group and $S\subseteq G$ . Let $\rho$ be an irreducible representation of $G$ and $\chi$ be its (irreducible) character. Define $$\rho(S):=\sum_{s\in S}\rho(s),$$ $$\chi(S):=\sum_{s\in S}\chi(s).$$ By Schur's lemma, it can be…
SPDR
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Eigenvalues of a sum of representation matrices.

Let $G$ be a finite group and $H\le G$ . Let $\lbrace{\rho_1,\rho_2,\ldots,\rho_t}\rbrace$ be the set of all inequivalent, irreducible representations of $G$ . Consider the sum $$T:=\sum_{g\in G\setminus H}\rho_i(g)\otimes\rho_j(g).$$ I was trying…
SPDR
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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 Abelian it has only one dimensional irreducible…
Craig
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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 $V$ must divide the order of $G$ . See this…
Samuel Johnston
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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 $g$ so that $\sigma(g) = T_g^{-1} \rho(g) T_g$ . Does…
user1127
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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 $$ V = \mathbb{C}\{111222, 112122, 112212, \ldots\} $$ (In this example, V is…
eti902
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Whether the division algebra of morphisms of a SO3 representation on symmetric trace zero 3 by 3 matrices , is Real, Complex or Quaternions

Let $V$ be the space all symmetric matrices in $M_{3}(\mathbb{R})$ with trace zero. Consider the representation $\rho$ of $SO_{3}(\mathbb{R})$ on $V$ given by, $$ \rho : SO_{3}(\mathbb{R}) \to GL(V) \\ Q \mapsto (A \mapsto QAQ^{t}) $$ I got that…
Neville S Vempeny
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Existence of irrational eigenvalues of a sum of representation matrices

Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$ . It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $\rho(G\setminus H):=\sum_{g\in G\setminus…
Black Widow
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Sum of even powers of basis elements of su(2)

Let $iS_x$ , $iS_y$ , and $iS_z$ be a basis for an irreducible unitary representation of $\frak{su}(2)$ , the Lie algebra of the Lie group $SU(2)$ . In the physics context, $S_x$ , $S_y$ , and $S_z$ are spin operators corresponding to a spin- $j$…
Ralph Sabbagh
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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
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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
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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, $\mathbb{C}^\times$ the complex numbers without…
Donson
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Complete reducibility of infinite-dimensional representations

We have Maschke’s theorem stated in the following form (I am working over a field $k$ of characteristic $0$ ): Theorem . Let $G$ be a finite group (or a compact Lie group), and let $V$ be a (possibly infinite-dimensional) $G$ -representation. Then…
Viktor
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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
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