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 inclusion map always a module homomorphism?

Suppose $R$ is some commutative ring and $G$ a finite group so that $R[G]$ is the usual group ring. If $M$ is some $R[G]$-module, then we can inject $M\hookrightarrow M\oplus R[G]^n$ for some ...
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Irreducible Subrepresentations of representation of $\operatorname{GL}_{3}(\mathbb{F}_q)$

For a character $\zeta$ of $\mathbb{F}_q^*$, we can construct the representation $\zeta \otimes \zeta \otimes \zeta$ of the diagonal subgroup $L$ of $\operatorname{GL}_{3}(\mathbb{F}_q)$, in the ...
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Is there a correspondence between normal subgroups and representations of a (finite) group?

I read this short introductory paper about representation theory: http://quantum.phys.cmu.edu/qm2/qmc151.pdf The dihedral group $D_3$ is described as having three representations: One faithful ...
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1answer
14 views

Induced coaction on a vector space.

Let $H$ be a Hopf algebra and let $V$ be an $H$-module. Then we have an action $H \times V \to V$ given by $(h, v) \mapsto h.v$. This action induces an $H$-action on the dual vector space $V^*$ of $V$ ...
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Infinite dimensional representation such that every subrepresentation is reducible

Let $V$ be a nonzero finite dimensional representation of an algebra $A$. a) Show that it has an irreducible subrepresentation. b) Show by example that this does not always hold for ...
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Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
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21 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
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Element of Grothendieck group is eigenvector of operator

Let $K_\mathbb{C}(G)$ be the Grothendieck group (over $\mathbb{C}$) of finite dimensional representations of a finite group $G$. Associated with any such representation $V$, there is a linear ...
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Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
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7 views

Does complexification make a self-conjugate representation non-self-conjugate?

I recently learned that a non-self-conjugate representation is not the same as a complex representation. Given a real representation $\pi$, with highest weight $\mu$ $$\pi : \mathfrak{g} \rightarrow ...
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Does the “differential” of a unitary representation give continuous operators on the space of smooth vectors?

Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which ...
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Why do “the Dynkin components of a weight play the role of eigenvalues with respect to the generators $H^i$ of the Cartan subalgebra”?

In the book "Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists" by Jürgen Fuchs,Christoph Schweigert the authors write "In the description of representations, the ...
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51 views

Algebraic Peter-Weyl theorem in the case of $G=SL_2$.

The algebraic Peter-Weyl theorem says that for a linear reductive group $G$ we have $\mathbb{C}[G] = \oplus_{V} V \otimes V^* $, where $V$ runs over the set of all non-isomorphic irreducible ...
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23 views

Representation ring of circle group over complex field

Can someone please describe how to find a representation algebra of circle group over complex field ? I am reading " representation theory of compact Lie group" chapter 3 section 7. It will be great ...
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16 views

What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$? It seems that the natural action comes from the following. We have a ...
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How to compute an integral?

I am reading the lecture notes. I am trying to understand the prove of Lemma 0.0.1.1 on page 4. From line 3 to line 4 in the proof of Lemma 0.0.1.1., how to prove that $$ \int_{F^{n-1}} ...
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30 views

Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
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1answer
18 views

Is the fundamental weight basis (a.k.a Dynkin basis) an orthonormal basis?

The simple root $\alpha_i$ basis is not an orthonormal basis, as can be seen from the Cartan matrix, which encodes how much they aren't orthonormal. For simplicity, let's assume a simply-laced Lie ...
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18 views

In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?

In this book I read Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear ...
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50 views

Are the physics and math definitions of a complex representation equivalent?

I was astonished to read at Wikipedia that The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group ...
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28 views

What does it really mean to complexify the $10$-dimensional representation of $ \mathfrak{so}(10)$?

A commonly used "trick" in $SO(10)$ Grand Unified Theories is to use a "complex" instead of a "real" $10$-dimensional representation for the Higgs fields. My problem is understanding what this ...
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Stanford math qual: Linear rep of a $p$-group over $\Bbb{F}_p$ fixes a line pointwise

I'm trying to solve a qual question that goes as follows. Let $H$ be a $p$-group and $V$ a finite dimensional linear representation of $H$ over $\Bbb{F}_p$. Then there is a vector $v \in V$ ...
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A question on $G$-map

I have a question that in studying representation theory and definition of $G$-maps we started with "Let $(\rho,V)$ and $(\rho',V')$ be two representation of $G$ over a field $F$. A linear map $T: V ...
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Restriction of some representation of $D_{12}$

Let $D_{12}$ be the dihedral group, $\langle x,y: x^2 = y^6 =1 , xy = y^{-1}x \rangle$ and $K = \langle xy \rangle \times \langle y^3 \rangle $ be a subgroup.Let $\xi^2$ denote a representation of ...
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26 views

What is character of a crystal

Let $B$ be a crystal. The character of $B$ is $$\text{char}(B)=\sum_{p \in B}X^{p(1)}$$ I don't understand how to use this formula and what is it measuring. Can anyone give me an example of how to ...
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11 views

Complexification of a Lie algebra representation in terms of weights?

EDIT: I found in this book the sentence: The weight system of a real representation of $G$ is defined to be the weight system of its complexification I think if someone can explain what this ...
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35 views

Degree of $\mathbb{Q}_p$-irreducible representations of a cyclic group of order $p^n$ for a prime p

Let $p$ be a prime and $\mathbb{Q}_p$ denotes the $p$-adic numbers. Then how to prove that the degree of the nontrivial $\mathbb{Q}_p$-irreducible representations of a cyclic group of order $p^n$ is ...
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37 views

Where to the degrees of freeedom go when a complex representation becomes a real representation of a subalgebra?

As an example consider the complex $16$-dimensional representation of $\mathfrak{so}(10)$. When $\mathfrak{so}(10)$ is reduced to the subalgebra $\mathfrak{so}(9)$, the complex $16$-dimensional ...
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Determining the minimum dimension required for embedding a finite group

Consider the groups $S_3$ and $S_4$ which are the symmetric groups on 3 and 4 elements respectively. We note that $S_3$ can be realized geometrically as the set of all rotations and reflections of a ...
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Isomorphism between colimits.

I actually need something weaker than this but my hope is that this holds in its fullest generality. Let $I$ be a small diagram and $I'$ a full subcategory of $I$. Let $F: I\to {\rm vec}$ be a functor ...
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Intuition behind the construction of Young Symmetrizer

I've been studying the representation theory of groups from Tung's "Group Theory in Physics." I understand Young symmetrizers of different Young diagrams are essentially primitive idempotents in the ...
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Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
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Space of arbitrary rotations of a cube

Suppose I have a cube $[-1,1]^3\subset\mathbb{R}^3$. I am allowed to rotate it about any angle/axis through the origin rather than just $90^\circ$ about the coordinate axes, e.g., by applying ...
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Permutation unitary in a tensor product

Given a matrix of the form $$ A = B_{1} \otimes B_{2} \otimes B_{3} \otimes... \otimes B_{n} $$ how can I find a matrix that gives me a permutation of , say, two of the elements: $$ A = B_{2} ...
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Representation of D_{12}

Let $D_{12}$ be the dihedral group, $\langle x,y: x^2 = y^6 =1 , xy = y^{-1}x \rangle$ and $H = \langle xy \rangle$ be a subgroup.Let $\xi$ denote a representation of $D_{12}$ which sends y to ...
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What shall I learn in order to understand Auslander-Reiten theory and tilting theory?

I work on cluster algebras and quivers and hence I need to understand Auslander-Reiten theory and tilting theory as soon as possible. I have read some noncommutative algebra and homological algebra ...
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319 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and ...
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27 views

What's the meaning of simple algebra?

I am reading a linear algebra material, in the book it mentioned : Every simple algebra has an exact irreducible representation Could anyone provide a proof of this claim ? And what is the more ...
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Countably many projections on more than continuos vector space with trivial commutant?

Is there such an example? An $\mathbb{F}_2$-vector space $V$ of dimension strictly more than the continuos $c=|2^{\mathbb{N}}|$, and a numerable set of commuting $\mathbb{F}_2$ projections ...
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248 views

Generating the partners in a multi-dimensional irreducible representation.

I am trying to block diagonalize a Hermitian matrix using the irreducible representations of its symmetry group. Using the group's character table, it is straightforward to generate a set of ...
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What is the Wedderburn decomposition of $\mathbb{R}[D_{2n}]$?

I have been looking everywhere and can't seem to find a general formula for the Wedderburn decomposition of the real group ring of the dihedral group ring of order $2n$, $\mathbb{R}[D_{2n}]$. Does ...
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Lie group representation and inner product

Let $G$ be a connected semisimple Lie group.Now let $\theta$ be the Cartan involution of $G$ and let $(\pi,V)$ be a finite dimensional representation of $G$. On page 22 of Analysis and geometry on ...
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Let G be an abelian group, and V be a faithful irreducible representation of G over C

This seems kind of obvious to me but I'm really having trouble thinking of what to do! Any help would be appreciated. Let G be a finite abelian group, and V be a faithful irreducible representation ...
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Representations of direct sums of matrix algebras [closed]

I'm reading Introduction to Representation Theory by Pavel Etingof et al. and I want to do most of the exercises. But I must stop by the exercise on page 25. I can prove part (a) easily by direct ...
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1answer
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Unit commutes with $H$-action.

Let $H$ be a Hopf algebra. Let $A$ be an $H$-module algebra. Then the unit map $\eta: k \to A$ commutes with the $H$-action. It is said that "$\eta: k \to A$ commutes with the $H$-action" is ...
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Let $v\in V-0$, then $\varphi _{v}: k[x]\rightarrow V : f \mapsto f.v$ is a surjective $A$-module homomorphism.

Proposition. Let $A=k[x]$ and let $(V,\rho )$ be a finite dimensional irreducible $A$-module. Let $v\in V-0$, then $\varphi _{v}: k[x]\rightarrow V : f \mapsto f.v$ is a surjective $A$-module ...
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Complex irreducible representations of the Klein 4 group

I wrote an answer to the following question. Can someone please verify it? Completely and explicitly describe, up to isomorphism, the set of all complex irreducible representations of the Klein 4 ...
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Completely and explicitly describe, up to isomorphism, the set of all complex irreducible representations of $\mathbb{Z}_n$

Can someone please verify my answer to this question? Completely and explicitly describe, up to isomorphism, the set of all complex irreducible representations of $\mathbb{Z}_n$. For each $x ...
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1answer
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About a character of group rep

We know that a character of a group representation has the same value for conjugate elements, my question is: are the elements which have the same character, conjugate?or it's not necessary?! thanks.