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

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2
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1answer
48 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
4
votes
1answer
30 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
22 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
2answers
46 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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
53 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 ...
9
votes
1answer
156 views

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
2
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1answer
35 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 ...
5
votes
1answer
64 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 ...
0
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0answers
27 views

Scalar Products on the Rational Function Field

Let $\mathbb{R}(t)$ be the rational function fields over $\mathbb{R}$. Are there scalar products $\langle -,- \rangle$ on $\mathbb{R}(t)$ such that multiplication with $t$ is selfadjoint, i.e. ...
-1
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1answer
39 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 ...
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0answers
20 views

Is supercuspidal representation the same as cuspidal representation?

I found that both supercuspidal representation and cuspidal representation are defined as representations which are not subrepresentations of induced representations. Is supercuspidal representation ...
1
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0answers
18 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
56 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 ...
1
vote
1answer
47 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 ...
2
votes
1answer
52 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 ...
8
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0answers
136 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 ...
2
votes
1answer
29 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
23 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
17 views

A property of simple three-dimensional Lie algebras

I am reading a solution of a problem to classify $3$-dimensional simple Lie algebras. First they prove that there exists $H$ such that $[H,X]=\alpha X$ for some $X\ne 0$ and $\alpha\ne 0$. Then they ...
3
votes
1answer
119 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
52 views

Proof in Serre/Fulton's rep. theory of Artin-Wedderburn for $\mathbb C[G]$

I have figured out a proof myself for the following theorem, but in both Serre's "Linear Representation of Finite Groups" and Fulton's "Representation Theory" books, I don't understand their comments ...
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2answers
62 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 ...
2
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0answers
17 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 ...
1
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1answer
22 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 ...
1
vote
2answers
36 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
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1answer
31 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
44 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?
0
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0answers
24 views

non degenerate representation

Let π:G →L(H) be a unitary representation Then the map π1: $\mathbb L^1(G)$→L(H) is nondegenerate. where ’nondegenerate’ is meant in the sense that For every non-zero $\xi \in \mathcal{H}$ ...
3
votes
1answer
53 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 ...
2
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1answer
84 views

Quasi-hereditary algebras

Can anyone please recommend a reference (I prefer a book chapter) on Quasi-hereditary algebras? and if it is possible tell me how to prove that Schur algebras are Quasi-hereditary (or any other ...
3
votes
0answers
65 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
61 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?
1
vote
1answer
42 views

Slodowy slices for non ADE type Lie algebras

In the first answer to: http://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c the correspondence between transverse slices of subregular elements of the nilpotent cone for a simple Lie ...
2
votes
1answer
62 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
41 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 ...
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0answers
15 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. ...
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0answers
20 views

Show that V is irreducible if and only if <$X_V$, $X_V$>=1.

Show that V is irreducible if and only if <$X_V$, $X_V$>=1. This question is on all of my past papers for 6 marks, so it's quite certain that it will come up in my exam. However we are given a ...
1
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0answers
20 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 ...
2
votes
1answer
19 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 ...
5
votes
2answers
59 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. $H=\begin{pmatrix} 1 & 0\\ 0 &-1\end{pmatrix}$, $X=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$, ...
0
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0answers
30 views

representation of $D4$

Let the representations of $D4$ : $ T: D4 \to GL(2,C)$ , $T(a)=\begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix}$ , $ T(b)=\begin{bmatrix} 0 & 1 \\ ...
0
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2answers
16 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
34 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!
0
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0answers
38 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 ...
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0answers
34 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 ...
2
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0answers
58 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 ...
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0answers
28 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
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1answer
21 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 ...
2
votes
1answer
34 views

Find radical of a quiver representation

How to find the radical of an acyclic quiver and without relations? what is the recipe? For example suppose $Q$ is a quiver with two vertices and two arrows from vertex $2$ to vertex $1$. Now suppose ...