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

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3
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1answer
46 views

How to compute $I(i)$?

Let $A=M_2(K)$ be the algebra of all $2\times 2$ matrices over $K$. Let $e_1=\left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)$ and $e_2=\left( \begin{matrix} 0 & 0 \\ 0 & 1 ...
2
votes
1answer
75 views

Length of modules.

Let $M, N$ be two $A$-modules. If there is a surjective $A$-map from $M$ to $N$, can we conculde that $\ell(M) \geq \ell(N)$. Here $\ell(M)$ is the number of modules in a composition series of $M$. ...
2
votes
1answer
63 views

Group representation scalar product

Let $\rho: G \rightarrow GL(V)$ be a finite dimensional complex representation of the group $G$. Show that there is an inner product on $V$ such that $G$ acts by unitary matrices. My approach so far ...
2
votes
1answer
37 views

Questions about epimorphisms.

Let $P, M, N$ be $A$-modules over a field $K$. If we know that $h:P\to M$ is surjective, $g:N\to P$ is a A-homomorphism such that $hg$ is surjective, can we have $\operatorname{im} g + \ker h = P$? I ...
1
vote
1answer
118 views
2
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2answers
225 views

Examples of projective modules which are not free modules. [duplicate]

Are there some examples of projective modules which are not free modules? Thank you very much.
3
votes
1answer
883 views

Definition completely reducible group representation

This should be a simple question, but i'm having a rather hard time though finding a explicit definition of a completely reducible group representation. Is it right to say that a presentation is ...
1
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0answers
77 views

Does a $C^*$ subalgebra of the centralizer of a unitary representation always contain the unit?

I am studying a theorem in Folland's "Course in Abstract Harmonic Analysis" where the following ingredients/assumptions are needed: $G$ a locally compact group, $\pi$ a unitary representation of ...
3
votes
1answer
109 views

Character on conjugacy classes

Let $V_j$, $j = 1,2$ be finite dimensional representations of a group $G$. Show: $\chi_{V_j}$ is a constant on each conjugacy class of $G$, where $\chi_{V_j}$ is the character of the representation. ...
4
votes
1answer
285 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
6
votes
1answer
323 views

how to get the injective envelope and projective cover of a given module

Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous ...
4
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0answers
154 views

Irreducible representation of $S_4$

Could one please point out an irreducible representation of degree 2 of the group $S_4$. Thank you.
1
vote
1answer
42 views

Question about radicals.

Let $A$ be a $K$-algebra, where $K$ is a algebraically closed field. Let $\text{rad} A$ be the radical of $A$, i.e. the intersection of all maximal right ideals of $A$. Let $g+\text{rad} A$ be an ...
3
votes
1answer
40 views

Question about maximal ideals of an algebra.

Let $A$ be a $K$-algebra. Let $\text{rad}A$ be the radical of $A$. That is the intersection of all maximal right ideals of $A$. Suppose that $A/\text{rad}A$ is isomorphic to $K$. How can we show that ...
1
vote
2answers
78 views

What are maximal ideals of $K[t]$?

Let $K[t]$ be the algebra of all polynomials in $t$. What are maximal ideals of $K[t]$? I know that $\langle t \rangle = \{tf \mid f \in K[t]\}$ is a maximal ideal. Are there other maximal ideals? ...
9
votes
3answers
585 views

Does every irreducible representation of a compact group occur in tensor products of a faithful representation (and its dual)?

Let $G$ be a compact (Hausdorff) group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. (Hence $G$ is a Lie group.) Is it true that every irreducible representation ...
2
votes
0answers
98 views

If $\mathfrak{g}$ admits a decomposition then it is semisimple

I want to show: If $\mathfrak{g}$ is a Lie algebra that has an abelian subalgebra $\mathfrak{h}$ such that $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha ...
4
votes
2answers
127 views

Characters of irreducible representations

Suppose $G$ is a finite group of odd order and $\chi$ is the character corresponding to some 2-dimensional representation of $G$. Must $\chi(x)\neq 0$ for every $x\in G$?
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0answers
51 views

How to show a $\mathbb{C}$-representation is not the complexification of a $\mathbb{R}$-representation?

For a homework problem in a course of representation theory, I have to show that some complex representation of a group $G$ is not the complexification of the real one. I don't really understand how ...
3
votes
0answers
77 views

What is the Induced Representation in Geometric Terms

As is well known, for $G$ a Lie group, and $H$ a subgroup of $G$ such that $G/H$ is homogeneous space (or maybe this is always a homogeneous space?), we have a correspondence between representations ...
1
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2answers
94 views

Question about idempotents.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $e$ be an idempotent of $A$ and ...
5
votes
1answer
121 views

A question of the book Elements of the Representation Theory of Associative Algebras: Volume 1

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . I have a question on page 9, line -7 (see Page 9 here). It is said that $$f_X(x_2) = x_2e_{21} = ...
3
votes
1answer
92 views

Fulton's Rep Theory, Notation, Transpose and Adjoints

I am reading Representation Theory by Fulton and Harris on my own to get an introduction to the topic. I have a question about notation which really boils down to a question about the adjoint and ...
2
votes
1answer
78 views

Radicals and direct sums.

Let A be a K-algebra and M, N be right A-submodules of a right A module L, $M \cap N =0$. How to show that $(M\oplus N) \text{rad} A = M \text{rad} A \oplus N \text{rad} A$? Let $m \in M, n\in N, x\in ...
4
votes
1answer
53 views

A question about the quotient of a $K$-algebra by its radical.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $e$ be an idempotent of $A$ and ...
1
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0answers
48 views

Isomorphisms betweenVerma modules over a semisimple Lie algebra

Fix a finite dimensional, semisimple Lie algebra $L$ and denote the Verma $L$-modules by $V(\lambda ')$ where $\lambda '$ are corresponding weights. Assume that there is an isomorphism between two ...
8
votes
2answers
368 views

What is an irrreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
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vote
1answer
53 views

Question about modules.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $I$ be an ideal of $B$ and $S$ be a ...
3
votes
1answer
88 views

Decomposition of semisimple Lie algebra via its roots

Exercise 14.33 in Fulton and Harris's Representation Theory claims that If the roots of a semisimple Lie algebra lie in a collection of mutually orthogonal subsets, one sees that the Lie algebra ...
4
votes
1answer
201 views

Irreducible Representations of Finite Coxeter Groups

The Coxeter group is defined as $$S = \langle s_i : s_i^2 = (s_i s_j)^{m_{ij}} = 1 \rangle $$ Does it have an irreducible representation of dimension >2 for $S$ finite? Is there a reference on ...
2
votes
1answer
45 views

Questions about $eAe$-modules.

Let $A$ be a $K$-algebra, $e$ be an idempotent, and $M$ be a right $A$-module. Let $f_M: \operatorname{Hom}_A(eA, M) \to Me$ be the map defined by $\varphi \mapsto \varphi(e)e$ for $\varphi \in ...
3
votes
1answer
78 views

On irreducible representations of an algebra.

Let $A$ be a complex algebra (with "nice" properties) and let $p : A \to \operatorname{End}(V)$ be an irreducible representation of $A$ with $V $ a finite dimensional complex vector space. Is it ...
1
vote
1answer
26 views

Why $e_1A=M_1$?

Let A be a ring with identity $1$ and $M_1, M_2$ submodules of $A$. We have $1=e_1+e_2$, where $e_i\in M_i$, $i=1, 2$. We can show that $e_i$ are idempotent and $e_1e_2 = e_2e_1 = 0$. We have ...
5
votes
1answer
282 views

Irreducibility and weights of a representation

For some reason I can't get a good hold of those topics (I'm reading Brian C. Hall's Lie Groups, Lie algebras and Representations. So it's matrices only). I'll try to narrow it a bit more: ...
0
votes
1answer
353 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
3
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0answers
89 views

Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all ...
0
votes
1answer
47 views

Why $(M/M \operatorname{rad} A) \operatorname{rad}A=0$?

Let $A$ be a ring and $M$ a right $A$-module. Why we have $(M/M \operatorname{rad}A) \operatorname{rad}A=0$? Thank you very much.
3
votes
1answer
102 views

Question about radical of a module.

Let $M$ be a right $A$-module. How to show that $m\in \operatorname{rad}(M)$ iff for any simple right $A$-module $S$ and any $f\in \operatorname{Hom}_A(M, S)$, $f(m)=0$? I think that if $m$ is ...
3
votes
1answer
55 views

How to show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?

Let $M, N$ be right $A$-modules. How can we show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?
5
votes
2answers
133 views

A basic question about group representation

Let $G$ be a finite group. If $\chi : G\to \mathbb{C}$ is a one dimensional representation, and let $\rho: G\to GL_n(\mathbb{C})$ be an irreducible representation of dimensional greater than 1. It's ...
1
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1answer
45 views

Is inversion of an irrep equivalent to inversion of the corresponding group element?

If $g\in G$ and $R:G\rightarrow GL\left(V\right)$ is the matrix form of an irreducible representation of $G$ then is the following statement true? $R^{-1}\left(g\right)=R\left(g^{-1}\right)$ Where ...
3
votes
1answer
109 views

Are the irreps of SO(n) orthogonal?

This seems like a trivial question, but I'm wondering if irreducible representations automatically inherit the properties of the group that they represent. Specifically, if I take an irrep of SO(4) ...
2
votes
1answer
65 views

Given a represenation $\rho:G \to Gl(V)$ and a subrepresentation $W \subset V$, is $\rho_V(g) = \rho_W(g)?$

I am asking for clarifications of the basic definitions in the representation theory of a subrepresentation and a character of a subrepresentation. Given a represenation $\rho:G \to Gl(V)$ and a ...
4
votes
1answer
596 views

Are two groups isomorphic if they have the same character table and each $|\chi| \leq 1$?

Suppose two groups have the same character table of complex representations. Also, all the entries in this character table have absolute value at most $1$. Does this imply that the two groups are ...
4
votes
1answer
142 views

General theory behind ladder operators

To derive the representation of SO(3) one uses the ladder operator method. What is the theoretical basis for this method? Often the ladder operators are simply stated in the textbooks of quantum ...
2
votes
1answer
47 views

How to show that $\operatorname{Hom}_A(M,N)$ is finitely dimensional?

Let $M, N$ be right $A$-modules and $A$ a ring over a field $K$. If $\dim_KM$ and $\dim_KN$ are finite, how to show that $\dim_K \operatorname{Hom}_A(M, N)$ is finite? I think that $\dim_K ...
4
votes
2answers
279 views

commuting algebra of an irreducible representation

Let $V$ be a finite-dimensional vector space and $\rho$ an irreducible abelian representation of $G$ on $V$. Is the centralizer of $\rho(G)$ in $End(V)$ necessarily a (commutative) field? (In ...
3
votes
3answers
108 views

$D_6$ as permutation group

I am trying to solve some exercises for a course in representation theory. We are studying finite groups and I have an exercise about the dihedral groups $D_n = ...
2
votes
0answers
56 views

Molecular vibrations and a generalisation of Wigner's rule for (non-finite) compact groups

years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to ...
5
votes
1answer
173 views

A proof of the Weyl Character formula via fixed point formula and

I've been looking all day for a reference or notes that prove the Weyl character formula via a fixed point formula and the Borel-Weil-Bott theorem. Does anyone know of these off hand?