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

learn more… | top users | synonyms (1)

0
votes
0answers
36 views

For a nilpotent Lie subalgebra, $\mathfrak{h}$, is $ad(\mathfrak{h})$ simultaneously diagonalizable if each $ad(H)$ is diagonalizable?

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}\subseteq \mathfrak{g}$ be a nilpotent subalgebra such that for every $H \in \mathfrak{h}$, the adjoint map $ad(H): \mathfrak{g} \rightarrow ...
0
votes
1answer
34 views

Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
5
votes
3answers
511 views

Looking for texts in representation theory

I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
2
votes
0answers
42 views

A question on universal Coxeter group

In the set up of Lusztig's Hecke algebra with unequal parameters, let $W$ be a universal Coxeter group with finite many simple reflections, that is, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
0
votes
1answer
44 views

Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
1
vote
1answer
334 views

proof of basic fact that torus actions are diagonalizable

Suppose a torus $T=(\mathbb{C}^\ast)^n$ acts on a finite dimensional vector space $W$, and define for $m \in M$ ($M$ is the character lattice of $T$) the eigenspace $W_m$ by $$W_m = \{w \in W \mid ...
1
vote
0answers
50 views

Why is U(n) a real form of GL(n)

When $n=1$, we see that $U(1)$ is defined by the equation $z\bar z=1$, hence $a^2+b^2=1$ for $z=a+bi$. Taking complex $a,b$ we see that the solutions are nonzero complex points, hence $U(1)$ is ...
3
votes
1answer
126 views

Unitary Equivalence of Two Irreducible $ * $-Representations of a GCR $ C^{*} $-Algebra that Have the Same Kernel.

In general, if two irreducible $ * $-representations of a $ C^{*} $-algebra $ A $ have the same kernel, then we can say that they are approximately unitarily equivalent. When $ A $ is GCR, how can we ...
1
vote
0answers
29 views

Liesubgroups given by representations

Maybe it's a little odd question, but I encountered a classification of homogeneous spaces, and I'm stuck with the following description (c.f. Onishchick, "Topology of transitive transformation ...
4
votes
1answer
132 views

Characters on a torus.

Let $T = (\mathbb{C}^*)^n$. It is said that the characters on $T$ must be of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some $a_1,\ldots, a_n \in \mathbb{Z}$. Is it possible that $a_i ...
0
votes
1answer
26 views

Uniqueness of a map.

Let $f: \mathbb{Z}^n \to \mathbb{C}^*$ be a homomorphism. Where $(\mathbb{Z}^n,+)$ is considered as an additive group and $(\mathbb{C}^*$ is considered as an multiplicative group. Fix $b_1,\ldots, b_n ...
2
votes
1answer
56 views

Conjugacy class name of the product in ATLAS

Well, I'm trying to read "ATLAS of Finite Groups". To be more precise, I'm interested in character tables of some Weyl groups. Is it possible to determine the conjugacy class name of the product of ...
2
votes
2answers
55 views

What is the free algebra $A=k\langle X_1,…,X_n \rangle$? And why is it an algebra?

I'm reading a book, where they claim that the free algebra $A=k\langle X_1,...,X_n\rangle$ is an algebra. I've never seen this notation and I've never heard of the free algebra, so I wonder how this ...
1
vote
1answer
97 views

When does a short exact sequence of representations exist?

The context for this question is that I am trying to determine the Grothendieck group of finite-dimensional complex representations of $T = (\mathbb{C}^*)^n$, where $\mathbb{C}^*$ denotes the ...
2
votes
1answer
72 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
6
votes
1answer
79 views

What is known about the representation theory of the symmetric group over $\mathbb{F}_2$

There is a lot of material available about the representation theory of the symmetric group over $\mathbb{C}$ and fields of characteristic $0$. In particular, there is the decomposition of the group ...
5
votes
1answer
96 views

Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
3
votes
1answer
113 views

What is the central idempotent of a representation?

The article I am reading says Let $P_\lambda \in Z(G)$ be the central idempotent corresponding to the representation $\lambda$. Could someone explain what this sentence means?
2
votes
1answer
62 views

Differentiating a representation

I'm reading the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl_2}$. It describes a representation of the group algebra ...
1
vote
1answer
61 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
-1
votes
1answer
61 views

Is $\mathbb{C}[N]$ isomorphic to $U(\mathfrak{n})$?

Let $G$ be an algebraic group and $N$ its maximal unipotent subgroup consisting of all upper triangular unipotent matrices. Let $\mathfrak{n}$ be the Lie algebra of $N$. It is said that ...
1
vote
0answers
28 views

Help in finding real irreducible representation of $Q_8$

I have the following facts that I don't know how to prove: i) $\mathbb{R}[Q_8] = \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{H} $ where $\mathbb{H}$ is ...
3
votes
1answer
79 views

the representation of a free group

A group $G$ is generated by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\n&1\end{pmatrix}$, then we know $G\cong \mathbb{F}_2$ which is a free group generated by two ...
9
votes
0answers
171 views

The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
1
vote
2answers
138 views

Easy way to get real irreducible characters (reps) from complex irreducible characters?

For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
3
votes
1answer
360 views

Infinite abelian group counterexample [duplicate]

A finite group $G$ is abelian iff all its irreducible representation $\rho$ have dimension 1. I'm looking for a counter-example when $G$ is an infinite group. Are there any? EDIT We're dealing ...
2
votes
1answer
46 views

Some beginner facts on representaions of $\mathfrak{sl}_3(\mathbb{C})$

Beginning to learn about representations of $\mathfrak{sl}_3(\mathbb{C})$. One starts with a subspace $$\mathfrak{h}=\{\begin{pmatrix} a_1 & 0 & 0\\ 0 &a_2& 0\\ 0 & 0 & a_3\\ ...
1
vote
0answers
41 views

Decomposition of direct sum representation of a Lie Group

Suppose that $(\varphi,V)$ is an irreducible finite dimensional representation of a Lie group $G$ and let $\psi=\bigoplus_{i=1}^n{\varphi}$ the representation on $W=\bigoplus_{i=1}^n{V}$. I want to ...
1
vote
1answer
54 views

Inner product on the ring of representations of symmetric groups

I would like to ask what the euclidean inner product, defined on the ring of representations of $S_n$ is but first I am describing briefly the construction. Let $G$ be a group, then ...
2
votes
0answers
29 views

About the induced character of the principal character

In page 186 of Carter's Finite groups of Lie Type: Conjugacy Classes and Complex Characters, the induced representation is defined for a representation $\sigma$ of a group $H$ to a representation ...
2
votes
1answer
2k views

Proof of Schur's lemma

Can someone give me a simplified proof of Schur's lemma in group theory. Sorry if the question looks a standard textbook proof. But I find the proof complicated in books. It would be helpful if ...
2
votes
2answers
83 views

Decomposition of symmetric powers of $\mathrm{sl}_2$ representations

Let $\mathfrak{g}$ be the Lie algebra $\mathrm{sl}_2(\mathbb{C})$. There is a classification of irreducible representations of $\mathfrak{g}$: each of them is defined by the only natural number $n$, ...
1
vote
1answer
34 views

Question about weights of $\mathfrak{sl}_2 \mathbf{C}$

On p. 148 of Fulton and Harris' book "Representation Theory: A First Course", they write that "Moreover, by the same token, the $V_\alpha$ that appear must form an unbroken string of numbers of the ...
3
votes
1answer
128 views

A questions about the schur's lemma

Schur's lemma is this: If (ρ1,V1) and (ρ2,V2) are irreducible representations of a group G, then any nonzero homomorphism ϕ:V1↦V2 is an isomorphism. or Schur's Lemma. a. A unitary ...
10
votes
3answers
4k views

Symmetric and exterior power of representation

Is there exists some simple formula for characters $$\chi_{\Lambda^{k}V}~~~~\text{and}~~~\chi_{\text{Sym}^{k}V}$$ for some representation $V$ of finite group? Thanks.
3
votes
0answers
85 views

invariants of group action by algebra automorphism

I am trying to prove the following statement, but I'm having a lot of trouble with it: Let $k$ be an infinite field. Let $A$ be a commutative $k$-algebra. Let $G$, a group, act on $A$ by algebra ...
2
votes
1answer
144 views

Indecomposable quiver representations

Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
2
votes
1answer
98 views

Repetitive Algebra.

I am studying the category of finitely generated left modules over the repetitive algebra and I'm using the book of Happel: Triangulated categories in the representation theory of finite dimensional ...
3
votes
1answer
47 views

Representation of a subgroup

I'm trying to solve the following problem. Suppose there is a $V$, representation of $G$, and a subgroup $H\leq G$ with index $|G:H|=3$. Given that $V$ seen as a representation of $H$ is a direct sum ...
0
votes
0answers
26 views

G is a finite group , U and V are CG modules with characters $\chi_U$ and $\chi_V$, dim(U)=dim(V). Show $\chi_U-\chi_V$ is not a character of G.

Suppose that G is a finite group and that U and V are CG modules whose characters are $\chi_U$ and $\chi_V$ respectively. Suppose that dim(U)=dim(V). Show that $\chi_U-\chi_V$ is not a character of G. ...
1
vote
0answers
27 views

Set $V_0=\{v \in V : hv=v , \forall h \in G\}$.Show that $V_0$ is a submodule of V.

Set $V_0=\{v \in V : hv=v ,\forall h \in G\}$. (a) Show that $V_0$ is a submodule of V. You may assume that V is a vector space. [4 marks] I thought I had done this previously, but now the question ...
5
votes
2answers
75 views

Action of $H$ in representations of $\mathrm{sl}_2$

Let $X,Y,H$ be the standard base for the Lie algebra $\mathrm{sl}_2({\mathbb{C}})$, i.e. $H=\begin{pmatrix} 1 & 0\\ 0 &-1\end{pmatrix}$, $X=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$, ...
2
votes
1answer
24 views

Show that the map defined by $\sigma(g)$=$p(g^{-1})$ is a representation.

Suppose G is abelian. Show that the map $\sigma : G -> GL(n,F)$ defined by $\sigma(g)=p(g^{-1})$ for all g in G is a representation of G. I think I have done this I would just like to check my ...
0
votes
2answers
76 views

Find a 1 dimensional submodule of the regular FG module and justify your answer.

Find a 1 dimensional submodule of the regular FG module and justify your answer. At first I thought the identity, e, could be a 1 dimensional submodule of the regular FG module, but now I am not so ...
0
votes
1answer
39 views

Write down the character of W.

I have done part (ii)(a) of this but I am stuck with how to do part (b) write down the character of W, and I have yet to try (c) I don't really understand the tensor product. Many thanks!
1
vote
0answers
43 views

Suppose that G is a subgroup of the symmetric group $S_n$ where n≥2.

Suppose that G is a subgroup of the symmetric group $S_n$ where n≥2. (a) Give the definition of the permutation module W of G over the complex numbers. Let G be a subgroup of $S_n$ and let V be a ...
1
vote
0answers
75 views

Finding the adjacency matrix for any given quiver and some collection of words.

For a directed graph (quiver) $Q$ with $n$ vertices and without multiple arrows, we have the adjacency matrix $A$, in which $A(i,j)=1$, if there is an arrow from $i$ to $j$, and $0$ elsewhere. This ...
0
votes
0answers
74 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...
2
votes
0answers
93 views

Is every unitary representation a direct sum of irreducible subprepresentations?

I've read that any unitary representation of a compact group decomposes as a Hilbert space direct sum of irreducible representations. In the book I'm reading this is stated as a prong of the ...
2
votes
1answer
22 views

Suppose that G is abelian and that V is an irreducible CG module.

Suppose that G is abelian and that V is an irreducible CG module. Let k exist in G. Show that the map $\theta$$_k$:V -> V defined by $\theta$$_k$ $(v)=kv$ is a homomorphism for all v in V. So I just ...