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

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Exercise on representations of the Dihedral group (Etingof 3.17)

I'm confronted once more with a problem on representation theory which I cannot fully solve (Problem 3.17 http://math.mit.edu/~etingof/replect.pdf): Let $G$ be the group of symmetries of a regular ...
2
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1answer
14 views

Exercise on formal deformations of representations (Etingof 2.24)

I'm trying to work out a few exercises in Etingof's book on representation theory of associative algebras (http://math.mit.edu/~etingof/replect.pdf) At the moment I'm looking at Problem 2.24. about ...
2
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25 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
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7 views

Rewriting a sum of Young Tableaux as Tensors

Is there a straightforward way (perhaps a software) that can write a direct sum of Young Tableaux in terms of tensors? For instance the direct product in $SU(3)$ (taken from this post) ...
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19 views

An integral with respect to the Haar measure over the unitary group

I am trying to find the answer of this integral: $$E:=\int dU \ (U^2 \otimes I) M (U^{ \dagger 2}\otimes I) $$ That is an integration with respect to the Haar measure and $U$'s are $d\times d$ ...
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17 views

$J(R/I) \cong J(R)/I$

Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition ...
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1answer
16 views

Conservation of bilinear forms and conjugation

Let $\omega,\omega'$ be non-degenerate skew-symmetric bilinear forms on $V$, a vector space over $\Bbb{C}$, preserved by $G,G' \subset GL(V)$ respectively. Must there be an element $\gamma \in GL(V)$ ...
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14 views

Functoriality of group homology

I understand that group homology $H_*(-)\colon \mathsf{Grps} \to \mathsf{Ab} $ is a functorial. In Weibel's homological algebra, there is an argument in 6.7 show this by using that $H_*(G;-)\colon ...
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13 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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1answer
23 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
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14 views

Filling in the last two rows of a character table?

I have a hopefully simple question about character tables. If I know all but two rows of a character table, and the character table is sufficiently large (say, at least 4 rows), do the orthonormality ...
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1answer
42 views

Classifying representations of $G = C_m \times C_n$

I have been set the following problem: Classify representations of $G = C_m \times C_n$; the direct product of two finite cyclic groups. My first thought is to rewrite $G$ in terms of its ...
2
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22 views

Nebentypus of contragredient representation

Let $k$ be a local non-archimedean field with ring of integers $\cal O$ and maximal ideal $\frak p$. Let $\pi$ be an irreducible admissible $\infty$-dimensional representation of $\text{GL}_2(k)$ ...
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48 views

Simplifying a direct sum $\mathbf{3}\oplus\mathbf{3}\oplus\mathbf{2}$ etc

In particle physics, one often uses the dimensionality of the irrep to label the irrep (apparently this is not a very good idea since the dimension does not unambiguously determine the rep.). What are ...
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14 views

(representation theoretic) meaning of sum over even rows of a Young tableau

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define ...
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23 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
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50 views

Degrees of irreducible complex characters of alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
2
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27 views

How do we know if a Young Tableau represents $3$ or $\bar{3}$?

Dimension three in the title was just an example...In general I want to know for any dimension $d$ and any SU(n). I am aware of the rule for tensors ; if more lower indices than upper then it is the ...
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25 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
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1answer
38 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$? [on hold]

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
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1answer
28 views

$V^{\oplus3}$, linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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1answer
112 views

Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$

I know that in $SU(3)$ $$\mathbf{8}\otimes \mathbf{8} = \mathbf{27}+\mathbf{10}+\mathbf{\bar{10}}+\mathbf{8}+\mathbf{8}+\mathbf{1}. $$ How can one use this to compute $$\mathbf{10}\otimes ...
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11 views

Dimension of intertwining operators on an irreducible representation

This is an exercise from Serre's Linear representations of finite groups: Let $H_{i}$ be the vector space of intertwining linear operators on the irreducible representation $W_{i}$ in $V_{i}$. ...
3
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1answer
28 views

The irreducibility of representations of parabolic induction over $\mathfrak{gl}_n(\mathbb{C})$

I have the following problem: Let $\mathfrak{g} = \mathfrak{gl}(n)$ be the general linear Lie algebra over $\mathbb{C}$. Denote $\Pi = \{\alpha_i:=\epsilon_i-\epsilon_{i+1}\}_{i=1}^{n-1}$ by the ...
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2answers
45 views

Group homology with coefficients vanish

Say $G$ is a group and $M$ is a $\mathbb ZG$-module with the property that $H_i(G;M)=0$ for all $i\ge 0$. Does this happen besides $M=0$?
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39 views

In what way is the representation not continuous

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf pp.7-8 Following the definition of a linear representation it states 'if V (an E-vector space) is equipped with a topological structure and if ...
2
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1answer
36 views

Representation theory of $\mathbb{Z}_k$ and complex roots of unity

Is there a natural relationship between the (characters?) of irreducible representations of $\mathbb{Z}_k$ and the $k$ complex-roots of unity? Can they be like thought of as characters of its ...
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48 views

Is a matrix element of a norm continuous representation always a trigonometric polynomial?

Let $\pi:G\to {\mathcal B}(H)$ be a unitary representation of a compact group $G$ in a Hilbert space $H$. Consider a matrix element of $\pi$, i.e. a function of the form $$ ...
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+50

Why a tensor product of 2x2 unitaries cannot implement a 3x3 unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
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1answer
30 views

Counterexample for Maschke's lemma

I'm trying to relax conditions and come up with a counterexample to Maschke's lemma in such a case. For example, with $k = \mathbb Z/p\mathbb Z$, I'm considering the two dimensional representation of ...
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22 views

Showing that a subrepresentation is isomorphic to the trivial representation

I'm considering $V$ to be the regular representation of a group $G$ and $W$ to be the 1-dimensional subspace of $V$ generated by the element $x=\sum_{s\in G} e_s$. I'm trying to show that $W$ is a ...
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3answers
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if $\rho: H \to \text{GL}_n({\bf C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful

How do I show that if $\rho: H \to \text{GL}_n(\mathbb{C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful?
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2answers
50 views

Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?

Let $G$ be a nonabelian group with center $Z(G)$. Let $\rho: Z(G) \to \text{GL}_n({\bf C})$ be an irreducible representation. Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not?
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39 views

Frobenius reciprocity, proof if $V$ irrep of $G$ then multiplicity of $V$ in regular rep of $G$ is $\dim(V)$

I know the standard proof of the fact that if $V$ is an irreducible representation of $G$ then the multiplicity of $V$ is the regular representation of $G$ is $\dim(V)$. Does there exist a proof using ...
2
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1answer
33 views

exists homomorphism of $G$-representations $\pi: V \to V$ with image $X$

Let $G$ be a group (not necessarily finite) and $F$ a field. Let $V$ be a $G$-representation. Suppose $V$ is isomorphic to a direct sum of irreducible representations. Let $X \subset V$ be any ...
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1answer
52 views

An example of $_AM$ is simple but $M^\star_A$ is not?

Let $A$ be a finite dimensional $k$ algebra, $k$ is a field. It is evident that the duality functor ($(-)^\star=hom_k(-,k)$) preserves the simplicity in the case of finitely generated $A$- modules ...
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12 views

When are all Gorenstein projective also pure-injective?

For an artin algebra of finite global dimension, each Gorenstein projective module is projective then is pure-injective. Are there any other examples having this property? That is, all Gorenstein ...
3
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1answer
41 views

Fixed point subspaces $V^B$ and $V^G$ for a Borel subgroup $B\subset G$ coincide

Assume that $G$ is a linear algebraic group, and let $B \subset G$ a Borel subgroup of it. Let $(V,\rho)$ a rational $G$-module. Define $$V^G := \{ v \in V \mid g \cdot v = v \quad \forall g \in G ...
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2answers
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$n-1$ dimensional permutation module for $S_n$

Say $n \ge 5$. Let $P$ be the $(n-1)$ dimensional permutation module for $S_n$, i.e. the permutation representation on $\{(x_1, \dots, x_n) \in {\bf C}^n: \sum x_i = 0\}$. Prove that: $\wedge^2P$ ...
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61 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.
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45 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
2
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1answer
30 views

Why it is central in $\mathbb {Z}[G]$?

In proposition 4.17, why is $P$ an central element?
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2answers
42 views

extracting the middle term of $ (z \cos \theta + w\sin \theta )^m(- z\sin \theta + w\cos \theta )^m $

Is there a systematic way to extract the middle term of the following expression? $$ (z \cos \theta + w\sin \theta )^m(- z\sin \theta + w\cos \theta )^m $$ This is homogeneous polynomial of degree ...
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1answer
25 views

adjoint representations

I am trying to work out the adjoint representations of $$H=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), X = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} ...
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representations of Lie algebras

I am studying irreducible representations of Lie algebras when our filed is of positive characteristic, I need an explicit explanation with example (or an article) which describes the differences what ...
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28 views

An example contradicting the validity of Maschke's theorem for infinite groups.

I am learning representation theory of finite groups and I am in doubt about a homework problem: Let $G = \mathbb{Z}$ and $V = \{(a_1 , a_2 , . . . )|a_i ∈ R\}$ be a vector space of infinite ...
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26 views

On the contragredient representation

Let $\pi$ be a representation of group $G$.Then its contragredient representation $\pi^{\vee}$ is defined by $\pi^{\vee}(g)=^{t}\pi(g^{-1})$. (here $^t$ means the transpose) But I heard that it is ...
2
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1answer
36 views

$E_6$ lie algebra and its representation

I've just started learning about Lie theory (only just finished up to basic classification of semisimple lie algebras) and I've got the following questions: How do I show that the complex lie algebra ...
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1answer
54 views

Representations of symmetric groups of order $2n$ and $n$

Background: Denote by $S_n$ the symmetric group of order $n$. There are many ways to embed $S_n$ as a subgroup into $S_{2n}$. Given a symmetric group, we can use Young diagrams to classify all ...