Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
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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 ...
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0answers
31 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 ...
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1answer
38 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 ...
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1answer
57 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 ...
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1answer
45 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 ...
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1answer
81 views
Finding irreducible representations of the following group using GAP
Given the following group of order 24, $$ G = \langle a,b \mid a^2=b^3=(abab^2)^2=1\rangle$$ how can one find (all) the irreducible representations using GAP? Since I have not installed GAP yet, I ...
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1answer
24 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 ...
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1answer
24 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 ...
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0answers
44 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 ...
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1answer
41 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.
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1answer
49 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 ...
3
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1answer
41 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 $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?
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1answer
33 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
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1answer
56 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 ...
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1answer
33 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 ...
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votes
1answer
44 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 ...
3
votes
1answer
53 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) ...
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1answer
39 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 ...
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1answer
43 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 -3 (see Page 9 here). It is said that $$h_1f_X = f_Y h_2.$$
I am ...
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1answer
56 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} = ...
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votes
1answer
41 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 ...
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3answers
41 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 = ...
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38 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 ...
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Characters of the symmetric group corresponding to partitions into two parts
Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
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85 views
Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$
I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189.
Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
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votes
2answers
112 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 ...
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votes
2answers
85 views
Projective modules of the path algebra
Let $Q$ be a finite and acyclic quiver. Let $a,b$ be vertices then $P(a)_{b}$, the projective indecomposable $kQ$-module is equal to the vector space having as a basis the set of all paths from $a$ to ...
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votes
1answer
143 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:
...
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1answer
53 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?
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1answer
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Are the irreps of SO(4) necessarily real?
I'm not a group theory buff, but I was confused when I used some code to generate the irreducible representatives of SO(4) and found the resulting matrix elements to be complex. Is this possible, or ...
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votes
2answers
91 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 ...
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1answer
187 views
Exercise on representations
I am stuck on an exercise in Serre, Abelian $\ell$-adic representations (first exercise of chapter 1).
Let $V$ be a vector space of dimension $2$, and $H$ a subgroup of $GL(V)$ such that ...
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21 views
The simplicity of $\bigwedge^i \mathbb{C}^{n+1}$ as a representation of $\mathfrak{sl_{n+1}}$ and its weight vectors
I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors.
So ...
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38 views
Induced character for D12
Let $H$ is subgroup of $G=D_{12}$ was generated by $R^3$ and $S$ ($R$ is rotation and $S$ is a reflection). We know that This subgroup is isomorphic to the dihedral group $D_4$. Suppose $F$ is the ...
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1answer
39 views
Representations of semisimple Lie algebra
Let $L$ a Lie algebra and $V$ a representation of $L$. We define
$$ V^{L}:= \{ v \in V \, | \, xv=0, \, \forall x \in L \} $$
and $V_{L}:=V/LV$. Let $\pi: L \rightarrow V_{L}$ be the quotient ...
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votes
3answers
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Exercise 6.1 in Serre's Representations of Finite Groups
I am trying to show that if $p$ divides the order of $G$ then the group algebra $K[G]$ for $K$ a field of characteristic $p$ is not semisimple. Now Serre suggests us to consider the ideal
$$U = ...
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1answer
47 views
Projective cover of $M$ equals projective cover of $M/\operatorname{Rad}(M)$
Take a module $M \in \Lambda$-mod, where $\Lambda$ is a finite dimensional algebra over a field $K$. Knowing that $M/\operatorname{Rad}(M) = S_1 \oplus \dots \oplus S_r$, with $S_i$ simple, and that ...
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1answer
60 views
Relation between a representative of a conjugacy class and corresponding irreducible character value
Is there a relation between the representative order of a conjugacy class and the corresponding irreducible character value?
Thanks in advance.
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0answers
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The center of a simply connected semisimple Lie group
I am learning about Lie groups, and I have the following basic question:
Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
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votes
3answers
71 views
Finding G- submodules
Let G be cyclic group of orfer 3, the generator being $\alpha$. Let V=$k^3$. Let $\alpha e_1=e_2$, $\alpha e_2=e_3$, $\alpha e_3=e_1$. How to find all $G$-submodules of V when
a) $K=\mathbb R$
b) ...
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1answer
102 views
Conjugate Representations
Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
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70 views
Question on Group rings
How to show that if $K$ is a field with char($K$)=2 and $G$ =$C_n$ (cyclic group of order $n$), then $K[G]$ is indecomposable $G$ - module, and is reducible, where n is not equal to 1
Moreover I ...
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1answer
40 views
When can you build up all representations from the fundamental and antifundamental ones?
Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I ...
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1answer
74 views
How can the Cartan-Weyl basis of su(2) be a basis if it does not consist of antihermitian operators?
Consider a Lie algebra. The ladder operators (i.e. root vectors, or eigenvectors of the Cartan subalgebra with respect to the adjoint representation) form a handy basis of the algebra called a ...
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0answers
58 views
Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$
Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
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1answer
33 views
How can I find a composition series of a module of a Hereditary algebra?
I have a problem in Representation Theory, I have to find a composition series of an indecomposable module. Consider this quiver $Q$: $$1 \rightarrow 2 \rightarrow 3 \leftarrow 4 \rightarrow 5$$
I ...
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1answer
229 views
The physical meaning of the tensor product
I have come across tensor products many times in physics, namely for matrices, vector-space elements, Hilbert-space elements (quantum states), and representations of groups and algebras. However, the ...
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1answer
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Role of the factor $\frac{1}{|G|}$ in the definition of this positive-definite hermitian form
Theorem: If $G$ is a finite group and $\rho: G \to GL(V)$ is a representation of $G$ on a hermitian space $V$, then there exists a $G$-invariant, positive-definite hermitian form $\langle ...
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1answer
59 views
Conjugate Representations of Lie Algebra of Lorentz Group
I'm trying to understand the Lie algebra of the Lorentz group and am almost there, but am stuck at the final hurdle! It's easy to prove that
$$\frak ...
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2answers
101 views
Representations of Direct Sum of Lie Algebras
I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
