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

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Is the dihedral group Dn linearly primitive for n>2?

Let $D_n$ be the Dihedral group (of order $2n$). For $p>2$ a prime number, $\mathbb{Z}/2$ is a core-free maximal subgroup of $D_p$, then $D_p$ is a primitive permutation group, and so linearly ...
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34 views

I don't know Maschke's theorem in the group representation.

I have a question about the Maschke's theorem in the group representation. I know that Maschke's theorem says that "Every representation of a finite group having positive degree is completely ...
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1answer
26 views

An inequality for the minimal number of generators of a finite group II

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating exactly the left regular representation (with ...
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1answer
24 views

Representatons of dimension $1$ on $D_4$

Prove that there are $4$ distinct representations on $D_4$ with dimension $1$ (where the field is $\mathbb{C}$). We have just started learning representations. Getting this question, what ...
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1answer
22 views

Invariant complement to a $G$-module (not necessarily a vector space)

Let $G$ be a group, $R$ a ring (not necessarily a field), and $M$ an $R$-module. Assume we have a group action $\rho:G \times M \to M$. If there exists a $G$-invariant submodule $N \subseteq M$, is ...
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34 views

Abstract finite group vs. finite group

What is the difference between the definitions of abstract finite group and finite group? I have some exposure to the finite group theory. But I came to know about abstract finite group from ...
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120 views

Is there matrix representation of the line graph operator?

I had the need to calculate the adjacency matrix $L$ of the line graph of a certain planar $k$-regular graphs $G(n,e)$ ( $n$ vertices and $e=\frac k2 n$ edges) given its adjacency matrix $A_G$. Here I ...
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13 views

Interpreting the table of classification of the partitions of $n$

I am going through A NON-RECURSIVE EXPRESSION FOR THE NUMBER OF IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP $S_n$ by AMUNATEGUI. In table I, the classification of the partitions of n according ...
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104 views

What is Representation Theory?

I'm beginning a course that uses representation theory, but I do not really understand what that is about. In the text I am following, I have the following definition: A representation of the Lie ...
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1answer
38 views

linear characters on a Grothendieck ring of a modular category.

I am reading the paper "Rank-Finiteness for Modular Categories" by Bruillard,Ng, Rowell, and Wang. Let $C$ be a modular category and let $K_0(C)$ be the Grothendieck ring generated by simple objects ...
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15 views

Borel-Weil theorem, confused about statement in special case of $SL_2(\mathbb{C})$

Suppose that $g = sl_2(C)$, and denote by $V(m)$ the weight $m$ irreducible representation of $g$. Let $B$ be the Borel subgroup of $G = SL_2(\mathbb{C})$ consiting of the upper triangular matrices. ...
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1answer
51 views

Exercise on representations of $\mathfrak{sl}_2 \mathbb{C}$ (Etingof 1.55.j)

I'm working through these notes on representation theory: http://math.mit.edu/~etingof/replect.pdf. Currently I'm looking at exercise 1.55 part j. I've done all of the previous parts and most of the ...
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31 views

$G$ equivariant quasicoherent sheaves on $X$ as compatible $G$ actions on the total spaces?

Let $G$ be an algebraic group, and $X$ a scheme on which $G$ acts: i.e the $S$ points of $G \times X \to X$ is a group for each affine $S$. Let $F$ be a quasicoherent sheaf on $X$. There is a notion ...
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1answer
36 views

Particular on the structure of a weight $L$-module $M$, with $L$ semisimple Lie-algebra.

Let be $L$ a semisimple Lie-algebra with its root system $R \subset H^*$ Cartan subalgebra and root decomposition \begin{gather} L=H \oplus \bigoplus_{\alpha \in R}^n L_{\alpha} \end{gather} Let $M$ ...
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220 views

An inequality for the minimal number of generators of a finite group [migrated]

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
2
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1answer
38 views

Higher self-extension $\text{Ext}^i_{\mathcal{O}}(L(\lambda), L(\lambda))$ between two irreducible modules in BGG category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $\mathcal{O}$ be the BGG category for $\mathfrak{g}$. It is well-known that the set of irreducible ...
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32 views

A character on a subgroup could be written as a difference with some character on the whole group, help on argumentation

Let $G$ be a finite group, and $U \le G$ of odd order such that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t$ and assume that $U^g \ne U$ implies $U^g \cap U = 1$. Then I can not ...
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16 views

Compute the limit and show that uN converges weakly

full question I already know that the norm is 1, and that you can use the definition of weak convergence but that's where I get lost. Somebody told me I can use the Riesz representation theorem since ...
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31 views

More Hopf algebra confusion: Verifying an equation between matrix coefficients.

I am thinking of the following situation: The lie algebra $g = sl_2(\mathbb(C))$, and $V(1)$ is the unique dimension 2 irreducible representation (the defining representation). Let $U$ be the ...
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1answer
32 views

$V(1)$ generates the tensor category of representations of $sl_2(\mathbb{C})$ - what exactly does this mean?

Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra ...
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1answer
24 views

Algebra of matrix coefficients in the dual of a hopf algebra, confusing verification

Let $H$ be a Hopf-algebra, and let $V$ be a finite dimensional $H$-module (a module for the algebra structure of $H$). For $f \in V^*$ and $v \in V$, we get $c^V_{f,v} \in H^*$ via $c^V_{f,v}(u) = ...
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1answer
31 views

On the decomposition of the group ring $\mathbb Q[G]$ over the rationals if $G$ is finite and cyclic

Let $G$ be a cyclic finite group of order $n$. I tried to determine the structure of the group ring $\mathbb Q[G]$ over the rationals $\mathbb Q$, what I got for even $n$ is $$ \mathbb Q[G] = A ...
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1answer
36 views

Module is semisimple if and only if all its finitely generated submodules are semisimple

I'm trying to show whether a module is semisimple if and only if all its finitely generated submodules are semisimple is true or not. I know that a semisimple module has semisimple submodules, ...
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3answers
43 views

Does any module with finite number of elements have a simple submodule? [closed]

Does any module with finite number of elements have a simple submodule? Not sure if this is true or not, struggling to find a counterexample
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23 views

linear span of all matrix coefficients is $C(G,\mathbb{C})$

Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...
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3answers
74 views

Is $\mathbf{Q}/\mathbf{Z}$ semisimple as a $\mathbf{Z}$-module? [closed]

Is $\mathbf{Q}/\mathbf{Z}$ semisimple as a $\mathbf{Z}$-module? I feel as though this should be obviously not true, but struggling to show why.
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34 views

Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove ...
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30 views

Easy to generate subgroups of the symmetric group $S_n$

Which subgroups of the symmetric group $S_n$ can be generated in polynomial or subexponential time?
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1answer
62 views

Regarding the representation theory of $SL_2(\mathbf{R})$.

Dear friends of mathematics, I have the following question for you. (a) According to Wikipedia there is a unique irreducible (real??) $2$-dimensional representation of $SL_2(\mathbf{R})$, which must ...
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36 views

Reference for an isomorphism

Let $A$ be a finite dimensional algebra over a field $K$ and $D:=Hom_K(-K)$ the natrual duality of mod-$A$. Let $M$ and $N$ be $A$-bimodules. Then there is an isomorphism $A$-bimodules: $Hom_A(M,D(N)) ...
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32 views

The size of the automorphism group of a graph

I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et a. It is said on page 14 that the size of the automorphism group of a graph is either $1$ or ...
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2answers
23 views

Tensor product with irreducible representation has no $G$-invariant submodules

Let $\rho: G \to GL(V)$ be a finite dimensional irreducible representation of a group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and let $R$ be a commutative ring with ...
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25 views

How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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1answer
19 views

Indecomposable representation of central elements have single eigenvalue

I was working through Etingof's notes on representation theory, but quickly ran into a stumbling block on the following problem. Let $A$ be an algebra, $z \in Z(A)$ and V an indecomposable finite ...
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18 views

“Standard representation” and their irreducibility

I think the standard representation of $SL_n(\mathbb{C})$ is irreducible, simply because for every $g\in GL_n(\mathbb{C}))$ there is a $\alpha\neq 0$ scalar with $\alpha g$ in $SL_n$. Is this a ...
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2answers
27 views

Coordinate ring of $GL_2$.

Let $GL_2$ be the group of all 2 by 2 invertible matrices over a field $K$. Let $x_{ij}$ be the function on $GL_2$ such that $x_{ij}(a) = a_{ij}$ for $a = (a_{ij}) \in GL_2$. Is the coordinate ring ...
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4answers
28 views

Class function and character of $S_3$ representation

I particularly need help with question 2. The $\textit{character table}$ for $S_3$ is given as follows:$$\begin{array}{c|c|c|} & \text{1} & \text{(12)} & \text{(123)} \\ \hline ...
3
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1answer
47 views

(Infinite) Non-abelian group with only linear characters

If $G$ is an abelian group, then every irreducible character has dimension one (i.e. is linear), for finite group we also have a converse. Do we have a converse for infinite groups? Or: Does there ...
3
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1answer
37 views

A direct proof that for finite $G$ we have “$G$ abelian iff all irreducible characters are linear”

A finite group is abelian iff all its irreducible characters have dimension one (hence are linear). A common proof uses that the number of irreducible representations equals the number of conjugacy ...
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0answers
26 views

Constructing character table of subgroup from character table of whole group

If $\psi : G \to \mathbb C$ is a character and $U \le G$, then $\psi_{|U} : U \to \mathbb C$ is a character, but I guess this might not be irreducible with respect to $U$. But is it possible to ...
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1answer
33 views

How to see that the decomposition of representation into a direct sum of irreductibles is unique?

Suppose I have a representation $V$ of a finite group $G$, and decompose $V$ into $V = V_1 \oplus V_2$ where $V_1$ and $V_2$ are irreducible representations. If I understand correctly, it is known ...
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24 views

How to construct a character table? (E.g Klein 4 group)

Could someone explain to me how you make a character table? Say I wanted to give the character table for the Klein $4$ group, $K$. $K$ is isomorphic to the product $\mathbb{Z}/2\mathbb{Z} \times ...
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Resources/tips to improve intuition for Representation Theory (general advice)

I find Representation Theory, despite having worked through solutions and theory, still something I do not understand Could anyone provide any advice or maybe some good videos, texts that help to ...
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15 views

Decompose $(\mathrm{Sym}^2 \mathbb{C}^2) \otimes (\mathrm{Sym}^2 \mathbb{C}^2)$ into irreducible representations of $\mathrm{SL}_2 \mathbb{C}$

Question: Let $V=\mathbb{C}^2$ be the standard representation of $\mathrm{SL}_{2}\mathbb{C}$. Decompose $(\mathrm{Sym}^2 V)\otimes (\mathrm{Sym}^2 V)$ into irreducible representations $\mathrm{SL}_2 ...
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1answer
54 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma}$?

In quantum mechanics we learn about the Pauli spin matrices: $$ \sigma_1 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( \begin{array}{cc} i ...
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0answers
43 views

Center of the group algebra of the symmetric group

How to prove that the center of the group algebra of the symmetric group is generated by 1-cycle conjugacy classes? I mean, that the center (consisting on class functions) is multiplicatively ...
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41 views

Showing that the right and left regular representations are equivalent

Let $G$ be a finite group. Let $C(G,\mathbb{C})$ be the complex vector space of all functions from $G$ to $\mathbb{C}$. We define two representations of G on $C(G,\mathbb{C})$: the left regular ...
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1answer
14 views

Are “Orbit Subspaces” Disjoint?

The orbits of a group action partition the set. Does a linear group action on a vector space break it into a direct sum? I.e. do the subspaces $V_w = \text{span}\left\{g\cdot w: g\in G\right\}$ ...
2
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0answers
48 views

Embeddings into symmetric structures

In the recent months I've come across a phenomenon which seems to come up in several areas of algebra making me wonder if there's a larger concept behind it, which I just fail to grasp. Namely, ...
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22 views

Induced representations of compact groups

I am taking a seminar that follows Serre's book "Linear Representations of Finite Groups", and I am preparing a talk on Chapter 7 on induced representations (Frobenius reciprocity, Mackey's formula ...