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

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Reference Request - Spaces of Smooth Vectors

I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
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361 views

All irreducible representations of Pauli group

I'm supposed to find out all irreducible representations of Pauli group, that is, the group generated by Pauli matrices $\sigma_k(k=1,2,3)$. It has 16 elements: $\pm 1, \pm i, \pm \sigma_k, \pm i ...
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111 views

Existence of a 1-dimensional invariant subspace

Show the existence of a 1-dimensional invariant subspace for any 5-dimensional complex representation of the group $A_4$, where $A_4$ is the alternating group of degree 4. Any hints?
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67 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
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237 views

Examples of decomposition representation

Here is a question in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg. (Question 3.8(2), page 25) that I need some hints from you : Give an example of ...
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66 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
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111 views

Question about a Corollary of Engel's Theorem

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for ...
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122 views

induced representation of tensors of irreducibles

Let $V_{\lambda}$ and $V_\mu$ be representations of the symmetric groups $\mathfrak{S}_d$ and $\mathfrak{S}_m$ respectively where $\lambda$ is a partition of $d$ and $\mu$ is a partition of $m$. It is ...
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143 views

How to compute the character of a matrix group operating on homogeneous polynomials?

I have a little problem in representation and/or invariant theory which I need help with. Let's assume $G \leq \mathbb{C}^{n\times n}$ is a finite complex matrix group which operates linearly via ...
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480 views

Tensor products and irreducible representations

Again something from Fulton and Harris I'm having trouble with: Exercise 2.33 (c). If $U$, $V$, and $W$ Are irreducible representations, show that $U$ appears in $V \otimes W$ if and only if $W$ ...
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82 views

Minimal embedding of exceptionnal Lie groups into special orthogonal groups

Let $G$ be a Lie group. The set of all $N$ such that $G$ is a subgroup of $SO(N)$ has a minimum $N_{\min}(G)$. (If I am not wrong, $N_{\min}(G)$ is supposed to be less or equal to $\dim(G)$) What is ...
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90 views

symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
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82 views

radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or ...
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362 views

Irreducible finite dimensional representations of $SL(2,\mathbb C)$

Is there a book that finds all the irreducibile finite dimensional representations of $SL(2,\mathbb C)$ without considering the Lie algebra $sl(2,\mathbb C)$? For example, how can I show "directly" ...
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205 views

clarification on the definition of a group C*-algebra

I've been trying to understand the definition of a group C*-algebra. Given a topological group $G$ and a C*-algebra $A$, let $u: G \to A$ define a unitary representation $U(G)$ of $G$ on $U(A)$, the ...
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47 views

notation question $Aut_{Modk}(A)$

What is $Aut_{Modk}(A)$ if A is a finite generated algebra, k is a field? Is it a group of automorphisms of A as a vector space over k or what?
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16 views

Using Peter Weyl theorem to decompose an orbit

Let $G$ be a finite group and let $\pi$ be a unitary representation of $G$ on a Hilbert space $H$. Since $G$ is finite, we have that for every $v \in H$, the orbit of $v$, $\pi (G).v$, is of dimension ...
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12 views

Induced representation from subgroup to subgroup

I wonder why the third of the properties of the induced representation here (http://mathworld.wolfram.com/InducedRepresentation.html) holds. Does it follow from the universal property? I could not ...
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44 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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20 views

References about an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$.

Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times ...
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40 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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27 views

If two non-equivalent representations are irreducible then sum is 0

We have a finite group and two representations $D_1:G\to GL_n(\mathbb{C}),D_2:G\to GL_m(\mathbb{C})$ for some positive integers $m,n$. We define $$T=\sum_{g\in G}D_1(g)BD_2(g^{-1})$$ where $B\in ...
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37 views

Cohomology of permutation representation

Consider the action of $S_n$ over $\{1,...,n\}$ consider the associated representation with integral coefficients $X_n$. What are $H^r(S_n,X_n)$? More in general is there a nice way to predict the ...
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23 views

Permutation module $M^\lambda$ as induced module

If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module: $M^\lambda := ...
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20 views

Dimension of the restricted representation

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
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52 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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7 views

How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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19 views

A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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26 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
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Relation between induced and coinduced spaces

Let $G$ be a compact Lie group and $H$ a closed subgroup of it. Let $X$ be a $G-$space. The induced $G-$space is defined to be $$G\times_H X$$ with the equivalence $(gh, x)=(g, hx)$, for any $g\in G, ...
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45 views

$ \mathbb{C} $-characters of $ A_5 $

How can we find all of five irreducible $ \mathbb{C} $-characters of $ A_5 $? Precisely how can we construct the character table with the aid of orthogonality relations?
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18 views

Why do we need the Dynkin Basis to compute Branching Rules?

Given a representation $R$ of some Lie algbra $g$, we can compute the corresponding representation $R'$ (in general reducible) for some subgroup Lie algebra $ g \supset g'$ by utilizing the weights in ...
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27 views

Why $\int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn$?

I am reading the lecture notes. On page 5, formula (1.24) is $$ \int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn, $$ where $dn$ is the Haar measure on $N$, $U \subset N$ is ...
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40 views

What values can a character $\psi$ take on an element of order $2$?

If $\psi$ is the character of a degree $2$ complex representation $\varphi\colon G\to GL_2(\mathbb{C})$, and $x\in G$ has order $2$, then $\psi(x)=0,\pm 2$. I noticed this by seeing $\varphi(x)$ ...
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25 views

Infinitely many direct sum decompositions of $M$ into direct sum of irreducible $\mathbb{C}G$-modules?

I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4. Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, ...
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43 views

Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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24 views

Irreducible unitary representation of a solvable lie group

Determine the equivalence classes of irreducible unitary representations of a solvable lie group. $$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$$ for $a,b\in \mathbb{R}, t\in ...
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27 views

How does $\mathrm{Res}_{\{I\}}\chi = \mathbb{I} \oplus \dots \oplus \mathbb{I}$?

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
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20 views

Does every Young diagram have a unique minimal major index?

Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ ...
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27 views

Complex irreducible representation of solvable lie algebra

How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1? What I know is that one ...
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27 views

How can I find the Weights of a Subalgebra

I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group. For example, if we start with the 16-dimensional representation of ...
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Completing the character table

Let $G=\{a,b|a^6=1,a^3=b^2,b^{-1}ab=a^{-1}\}$ be a group of order 12. $G$ has 6 conjugacy classes $$\{1\},\{a^3\},\{a,a^{-1}\},\{a^2,a^{-2}\},\{b,a^2b,a^4b\},\{ab,a^3b,a^5b\}.$$ Name them ...
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30 views

Natural representation of GL(V)

Let $V$ be a vector space over some field. Is the natural representation $V$ of the group $GL(V)$ irreducible? Is it absolutely irreducible? Is the span of $GL(V)$ inside $End(V)$ all of $End(V)$? I ...
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24 views

Complete the character table

Let $G$ be a group of order 10 having 4 conjugacy classes and the above character table. Complete the table. It's easy to get the degrees of the remaining 2 irreducible characters to ...
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30 views

Decomposition of the Regular Q8 Module

For a worksheet we were asked to find the decomposition of the regular $Q_8$ module into a direct sum of simple modules. This isn't me asking for help on homework though, the problem is that I already ...
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64 views

Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
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46 views

hard lemma from a paper

I was reading a paper, there its mentioned as lemma that : Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $\phi_1 : G\to GL(V)$ is a irreducible representation of $G$ and $\phi_2 : H \to ...
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25 views

Convolution product in Borel-Moore homology

I have a question about Exampla 2.7.10 from the book "Representation theory and complex geometry" by N. Chriss and V. Ginzburg. It concerns the convolution product. In the example we have $M_1 = M_2 ...
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41 views

Understanding restriction in $S_3$

Let $G=S_3$. \begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & 1 & -1 \\ \hline \chi_2 & 2 & -1 & 0 \\ ...
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37 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...