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

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Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
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Irreducible complex continuous unitary finite dimensional representations of SO(2)

I have to find all continuous finite dimensional complex Irreducible and unitary representations of $SO(2)$. I know that every element of $SO (2)$ can be written as $exp(J \theta ) $, where $\theta$ ...
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Permutations associated to a transversal and Cayley theorem

Let $G$ be a finite group with $H\le G$ and $T$ a right transversal of $H$ in $G$. $G$ acts on itself by left multiplication and so we can consider $G\le \mathfrak{S}_G$. Let $g\in G$. The permutation ...
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How to show that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding?

Let $H$ be a bialgebra and ${}_H^H YD$ the category of Yetter-Drinfeld modules over $H$. It is said that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding. ...
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Let $G=AB$ where $(|A|,|B|)=1$ and $V$ be an $\mathbb{F}[G]$ module.

Under these assumptions it is a well-known fact that if $V_A$ and $V_B$ are faithful ($V_A$ denotes $V$ as an $\mathbb{F}[A]$-module) then $V$ is also faithful. Clearly if $V_A$ and $V_B$ is ...
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Intersection of the kernel of the irreducible characters determinants

Let $G$ be a finite group. It is easy to show that $G'\le \bigcap_{\chi\in Irr(G)}Kerdet\chi$. Is there equality ? This question arises from the remarkable equalities $\bigcap_{\chi\in Irr(G)}Ker\...
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Determinant of a character

let two characters $\chi$ and $\vartheta$ of a finite group $G$ (assumed to be non-null). Let $\mathfrak{X}$ and $\mathfrak{Y}$ be representations of $G$ affording respectively $\chi$ and $\vartheta$ ...
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Finite group representation on endomorphism ring

Let $\rho:G\to\mbox{GL}(V)$ be a finite dimensional representation of a finite group $G$. We can assume the base field is $\mathbb{Q}$, but it doesn't really matter. Then we also obtain a ...
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Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
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The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
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Irreducible tensor for fundamental representation of SU(N=3)

I am trying to calculate the singlets of the tensor product $N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^*$. I know that $N_c \otimes N_c^*=1\oplus (...
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38 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
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54 views

what does $\ltimes$ in the context of representation theory mean?

I am considering the following sentence wich is part of a theorem: '' Let $V$ be a finite dimensional unitary representation of $H=\mathbb{Z}^{2} \rtimes $ SL$_2(\mathbb{Z})$." I have no background ...
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What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
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48 views

The inverse of the braiding $c: V \otimes W \to W \otimes V$.

In the article. It is said that the inverse of the map $$ {\displaystyle c_{V,W}:V\otimes W\to W\otimes V}, \\ {\displaystyle c(v\otimes w):=v_{(-1)}{\boldsymbol {.}}w\otimes v_{(0)},} $$ is $$ {\...
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Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. $$V=\underbrace{\mathbb KG\oplus\cdots\oplus \...
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Character group and lattices

Let $\Lambda$ be the complex n-th dimensional lattice over Eisenstein integers ($\mathbb{Z}[\omega]$)). The map $R: \mathbb{C} \mapsto \mathbb{R}$ is defined as following: $R(z)=R(z_{a}+\omega z_b)=...
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Basis of weight lattice in terms of root lattice

Let $\Lambda_R = \bigoplus_{i \in I} \mathbb{Z} \cdot \alpha_i$ be the root lattice of a root system $\Phi$ with simple roots $\alpha_i$ and let $\Lambda_W$ denote the corresponding weight lattice. ...
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If a finite group has only 1D irreducible representations, is it abelian? [duplicate]

I know abelian groups have only 1D representations. Is the converse proposition true? i.e. If a finite group has only 1D irreducible representations, is it abelian?
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Any theory that relates a group's representation to its subgroup's representation?

I have encountered a problem concerning a finite group $G$ and its subgroup $H$. $G$ has one more generator than $H$. I calculated dimensions of irreducible representations of $G$ and $H$ and want to ...
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1answer
41 views

Irreps of products between dihedral group and any finite group

Let $D_n$ be the dihedral group with order $2 n$. The total number of irreducible representations for $D_n$ is as follows. When $n$ is even, the total number is $\frac{n-2}{2} + 4 = \frac{n}{2} + 3$. ...
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Symmetric decompositions of $SU(2)$ representations.

Let us consider the representation theory of $SU(2)$. There is a unique irreducible representation of dimension $n$ for each $n \ge 1$, which we will denote $\mathbf{n}$, with the defining $2$-...
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Representation of diedral group $D_8$, why $\rho(a)^2=1$ if $a$ is the rotation?

I recall that $D_8=<a,b\mid a^4=b^2=1, bab=a^3>$. I have to determine all representation $\rho:D_8\longrightarrow \mathbb C^*$ of degree 1 of $D_8$. In my course it's written that since $\rho(a)^...
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61 views

Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
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Expressing $\mathbb{C}^3$ as a direct sum

$S_3$ acts on $\lbrace 1,2,3 \rbrace$, so this affords a homomorphism $S_3\to GL_3(\mathbb{C})$ (acting on $\mathbb{C}^3$). I showed the only vector fixed by the action of $S_3$ is zero. Find two ...
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Looking for examples of groups with complex representations realizable over $\mathbb{Q}$

I'm looking for examples of finite groups $G$ such that all the complex irreducible representations of $G$ are realizable over i.e the representing matrices can be chosen to have rational entries. One ...
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319 views

Why is this paragraph so short?

$G$ is a connected, reductive linear algebraic group.The reference is Springer, Linear Algebraic Groups. I am having trouble making sense out of anything in this paragraph. Proposition 7.31(ii) ...
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Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
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33 views

Schur test and it's relation to representation theory

I was told by analyst who doesn't know about such things that Schur's test relating to boundedness of integral operators is somehow a version of Schur's lemma on irreducible representations, the group ...
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61 views

Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
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1answer
29 views

Faithful monomial representation induced from faithful character

Let $\rho: G \rightarrow GL_n(\mathbb{C})$ be a faithful irreducible representation such that $\rho = Ind_N^G \phi$ for some 1-dimensional representation $\phi$ and normal subgroup $N$. Does $\phi$ ...
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Correlation among a given representation of a finite group and a representation given by the composition with an automorphism

Assume that $G$ is a finite group, and $p : G \rightarrow GL_n(\mathbb{F})$ is a linear represenation. Furthermore assume that the image of $G$ lies inside a subgroup of $GL_n(\mathbb{F})$, say for ...
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Tensor invariants constructed from identity tensor

It is evident that tensors constructed from copies of the identity tensor (and scalars) eg $t^{ij}_{kl} = 2 \delta^i_k \delta^j_l - \delta^i_l \delta^j_k$ are invariant under any matrix group, and ...
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Showing a rep of $sl(2,\mathbb{K})$ is irreducible

Let $V$ be a $m+1$-dim $K$-vector space with char$K=0$. Let $(v_0,v_1,\dots,v_m)$ be a basis of $V(m)$. Now suppose I construct a representation of $sl(2,K)$ on this representation. How do I show ...
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Restricted linear representations of abelian groups

If $G$ is a group (say finite for simplicity though the question applies to infinite groups as well), what can one say about the subgroup $G^*_n = \text{Hom}(G, \mu_n)$ of the group of all linear ...
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Which groups have only real representations?

An irreducible representation $\rho$ (with character $\chi$) of a finite group is called a "real" representation if its Frobenius-Schur indicator is 1: $$\frac{1}{\lvert G \rvert} \sum_{g \in G} \chi\...
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101 views

Basic Algebras: Definition

A $k$-algebra $A$ is called basic if for every set of primitive orthogonal idempotents $\left\{e_1, \dots , e_n\right\}$ such that $1=\sum_{i=1}^ne_i$ we have that $$e_iA\cong e_jA\Leftrightarrow i=j.$...
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Proving Schur's lemma

Schur's Lemma: Let $(\Pi_i,V_i)$, $i=1, 2$ be two irreducible representations of a group $G$, and let $\phi : V_1 \to V_2$ be an intertwiner. Then either $\phi = 0$ or $\phi$ is a vector space ...
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38 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
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Inner product of Induced permutation representation and an irrep $\langle {\chi \uparrow^{S_n}_{D_n}}_{\mathbf{ 1}_{D_n}} , \chi_\rho \rangle_{S_n}$

I am trying to compute the inner product of the characters of the induced permutation representation from the trivial representation of a dihedral group $D_n$ of order $2 n$ to $S_n$ and an irrep $\...
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Odd order subgroups of $PSL(2.q)$

Let q be an odd prime power. Is it true that every odd order subgroup of $PSL(2,q)$ is abelian ? If yes, how can it be proven ?
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Centralizer of a Sylow $2-$subgroup of $PSL(2,q)$

Let q be an odd prime power. By a classic result, a Sylow 2−subgroup $P$ of $SL(2,q) $ is generalized quaternion. It is an irreducible subgroup of $GL(2,q)$ (since otherwise its natural ...
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Can we do representation theory for algebras with >2 operations?

Suppose I define an algebra with 3 or more operations, perhaps using universal algebra. Would it be meaningful to talk about the representation theory of this algebra? In particular, I am interested ...
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The complex numbers $\mathbb{C}$ are a Frobenius algebra over $\mathbb{R}$

I try to show that there is a functional lineal $f:\mathbb{C}\longrightarrow\mathbb{R} $ whose Kernel contains no nonzero left ideals. Define $f:\mathbb{C}\longrightarrow\mathbb{R}$ by $f(a+bi)=a$ ...
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Characters of $d$-dimensional representations have size at most $d$.

Let $\chi$ be the character of a $d$-dimensional complex representation $\rho$ of a finite group $G$. Then $\vert \chi (g)\vert \le d$ for all $g$ and if we have equality (for all $g$) then $\rho(g)=\...
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Factorization of Schur polynomials

For a weakly decreasing sequence of non negative integers $\lambda = (\lambda_1, ... , \lambda_n)$ the Schur polynomial $S_\lambda$ is defined as $S_\lambda(x_1,x_2,...x_n) = \sum_T x_1^{t_1}x_2^{...
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A Sylow $2-$subgroup of SL(2,q) is irreducible

Let $q$ an odd prime power. By a classic result, a Sylow $2-$subgroup of $SL(2,q)$ is generalized quaternion. How can I show that it is an (absolutely) irreducible subgroup of $GL(2,q)$ ? I have ...
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1answer
38 views

Definition of induced representation

Definition. Suppose that $H$ is a subgroup of the finite group $G$ and $\sigma \colon H \to \operatorname{GL}(W)$ is a representation of $H$. Then the induced representation from $H$ up to $G$ denoted ...
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All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
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Stable equivalences preserving injective dimension

Let $A,B$ be two finite dimensional connected algebras and $F$ be a stable equivalence between their stable module categories (module category modulo projectives). Are there some natural conditions ...