Tagged Questions

1
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1answer
19 views

Invariant hermitian forms and irreducible representations

Let $V$ be a vector space over $\mathbb{C}$ of finite dimension $n$, $G$ is a finite group and $T:G\rightarrow GL(V)$ its irreducible representation that sends each $g$ into $T_g$. Let $E:V^{\bigoplus ...
4
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1answer
46 views

Group homomorphisms into a field

Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
6
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1answer
106 views

Decompose $P$ into the direct sum of irreducible representations.

Note: I need help with part (c). Consider the representation $P: S_3 \rightarrow GL_3$ where $P_{\sigma}$ is the permutation matrix associated to $\sigma$. a) Determine the character $\chi_P : S_3 ...
7
votes
1answer
44 views

Specific projective dimension of a module over bound quiver

Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver $$\require{AMScd} \begin{CD} 1 @>>> 2\\ @V{}VV @V{}VV \\ 3 @>>> 4 @>>> 5 ...
4
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1answer
58 views

How to compute Nakayama functor explicitly?

I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
3
votes
3answers
63 views

Book recommendation for associative algebras

Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ...
3
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1answer
19 views

multiplicity of irreducible components of S3 modules

Let V denote the 2 dimensional irreducible standard module for $S_3$. I want to find multiplicity of each of irreducible components of $V^{\otimes ^{10}}$ , by writing the character for $V^{\otimes ...
4
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0answers
26 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
1
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1answer
34 views

Image of the projection map onto an irreducible module

Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
2
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2answers
36 views

Group ring is not isomorphic to 2 by 2 matrices

Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the k-algebra of 2 by  2 matrices over k, is not isomorphic to the group ring of any finite group G over k.
7
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2answers
104 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
2
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0answers
29 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
3
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1answer
38 views

Braid Group of a Weyl Group

I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman. The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of ...
3
votes
3answers
114 views

Are Clifford groups very *non-commutative*?

Clifford groups seem to be very non-commutative by the relation \begin{equation} \gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}. \end{equation} But is it really so? Can we put this degree of ...
4
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0answers
56 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
3
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0answers
35 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
3
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2answers
53 views

Frobenius reciprocity

I would like to ask a question on Theorem 8.6 on page 246 in this book. There is the claim that the multiplicity of $F$ in $E^G$ is equal to the multiplicity of $E$ in $F_H$. Why is this just ...
2
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1answer
23 views

Question about the top of a bound representation of a bound quiver.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1. I have a on page 77. In (d) of Lemma 2.2 on Page 77, it is said that $$ L_a=\sum_{\alpha: a\to b} ...
2
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0answers
33 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
2
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1answer
53 views

Questions about representation theory of associative algebras.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1. I have two questions on page 85. On Line 18 of Page 85, it is said that $\ker p_i \subseteq ...
2
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1answer
42 views

Characters of elements under every representation equal implies conjugacy

If $G$ is a group, suppose that for every $G$-module $V$ we have $$\chi_V(g_1)=\chi_V(g_2).$$ How can I be sure $g_1$ and $g_2$ are conjugate in $G$? Its easy to the reverse implication; ...
2
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1answer
81 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
3
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1answer
29 views

Induced representation is isotypical?

Is there a theorem like this for the induced representation? Let $N$ be a normal subgroup of a finite group $G$ and $\rho$ be an irreducible linear representation over any field $k$. Then one of ...
3
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1answer
27 views

Different induced representations - same simples?

is the following case possible: $\pi_1, \pi_2$ two simple representations of the same subgroup over an arbitrary field. $\operatorname{Ind}(\pi_1)$ and $\operatorname{Ind}(\pi_2)$ have equal ...
3
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2answers
72 views

Question about the radical of the Jacobson radical.

I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
7
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0answers
64 views

Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
2
votes
2answers
72 views

Isomorphism of faithful representations

Let $G$ be a group and $f,g: G \rightarrow GL(V)$ be two faithful representations over some field $K$ with $f:x\mapsto f(x)$ and $g:x \mapsto f(x^{-1})$. I would like to find out if $f$ and $g$ are ...
13
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1answer
196 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
0
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0answers
43 views

Do we have $e(\operatorname{rad}^2 A)=\operatorname{rad}^2 (e A)$?

Let $e$ be a primitive idempotent of $A$, where $A$ is a finite dimensional algebra over an algebraically closed field $K$. Do we have $e(\operatorname{rad}^2 A)=\operatorname{rad}^2 (e A)$? Here ...
3
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0answers
60 views

Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
3
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0answers
55 views

Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
0
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0answers
27 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
4
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0answers
42 views

Induction from normal subgroup, problem with degrees

Suppose $K$ is an arbitrary field, $G$ a finite group and $N$ a normal subgroup. If one know all the irreducible representations of $N$ and then form the induced representations, one can use Mackey ...
1
vote
3answers
65 views

Show $U \otimes V$ is an irreducible G-module

Let $G$ is some group and $U$ is an irriducible $G$-module over the complex numbers. Now if $V$ is a $G$-module of dimension 1, I would like to prove $U \otimes V$ is an irriducible $G$-module. My ...
0
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1answer
32 views

Why $A/\operatorname{rad}A$ is generated by $e_a$?

Let $A$ be an algebra over an algebraically field $K$ and $(Q_A)_0$ be its ordinary quiver. Let $\{e_a \mid a \in (Q_A)_0\}$. Then $\{e_a \mid a \in (Q_A)_0\}$ is a complete set of primitive ...
0
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1answer
39 views

How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem?

Let $A$ be a $K$-algebra and $K$ an algebraically closed field. How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem? Thank you very much. ...
7
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0answers
81 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
2
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1answer
35 views

Why $\operatorname{Hom}_A(e_jA, e_iA) \cong \operatorname{Hom}_A(e_jA, e_i\text{rad}A)$?

On line 6 of the proof of Corollary 3.4 of page 62 of the book Elements of Representation Theory of Associative Algebras, Volume 1, it is said that $\operatorname{Hom}_A(e_jA, e_iA) \cong ...
1
vote
1answer
22 views

Question about the connectivity of the ordinary quiver of a connected algebra.

I am reading the book Elements of Representation Theory of Associative Algebras, Volume 1. I have some questions about the connectivity of the ordinary quiver of a connected algebra. On page 61, ...
0
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1answer
53 views

Topic for presentation on Group Representations, Young Tableaux, Symmetric Group

I need to do a presentation relating to group representations/Young tableaux/symmetric group; however, for all my searching, I cannot find a cool topic that I find personally interesting (and that is ...
0
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2answers
41 views

Mapping from symmetric power to a lower symmetric power

This may be a dumb question, but what are the surjective maps $$f_n:\operatorname{Sym}^n(V)\to \operatorname{Sym}^{n-2}(V),$$ where Sym$^n$ denotes the $n$-th symmetric power of $V$? Wouldn't it just ...
0
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0answers
30 views

Subalgebra generated by Cartan subalgebra and root spaces

Let $\alpha_1,...,\alpha_k$ be the roots of a semisimple Lie algebra $\mathfrak{g}$ and $\mathfrak{g}_{\alpha_i}\subset \mathfrak{g}$ the corresponding root spaces. Then the subalgebra of ...
7
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2answers
263 views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebra, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. Edit: ...
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0answers
34 views

Eigenvectors of algebraic group representation

In paper http://arxiv.org/abs/math/9905053 of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an ...
0
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1answer
21 views

How can we show that $X_A \cong T_e(Y_B)$?

I am reading the book Elements of representation theory of associative algebras 1. On Line 7 of page 37 (I attached this page below), it is said that $X_A \cong T_e(Y_B)$ since the diagram above line ...
3
votes
1answer
51 views

How to show that $Pe\otimes_B eA \cong P$?

Let $e$ be an idempotent of $A$, where $A$ is an algebra. Let $B=eAe$ and $P$ be a projective right $A$-module. How to show that $Pe\otimes_B eA \cong P$? I think that $Pe\otimes_B eAe \cong P$. But ...
1
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1answer
46 views

Is cyclic modules $=$ simple modules?

Let $A$ be an algebra with identity $1$ and $N$ be a right module of $A$ generated by $n_1 \in N$. That is $N=n_1A$. Is $N$ a simple module? I think that maybe this is not true. Let $N=A$ and suppose ...
0
votes
1answer
21 views

Is $eA$ simple?

Let $A$ be an algebra and $e$ be a primitive idempotent of $A$. We know that $eA$ is indecomposable as a right $A$-module. Is $eA$ a simple right $A$-module? Thank you very much.
0
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1answer
17 views

Simple modules over $K\times K \times \cdots \times K$.

Let $K$ be an algebraic closed field. Let $M$ be a simple module over $K\times K \times \cdots \times K$ ($n$ copies of $K$). If $n=1$, then $M \cong K$ and $\dim M =1$. If $n\geq 2$, is $\dim M =1$? ...
8
votes
1answer
69 views

Wedge pure product

Let $V$ be a vector space of $\dim n$ over $K$. Let $P$ be the set of all pure products of the form $v_1 \bigwedge v_2$. How to prove that there is a one-one correspondence between the one dimensional ...

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