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

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3
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Why is the arrow ideal $R_Q$ of a finite, connected, acyclic quiver $Q$ equal to the Jacobson radical?

If $Q$ is a finite, connected, acyclic quiver, why does the arrow ideal $R_Q$ equal the Jacboson radical $J$ of the quiver algebra $k(Q)$? It comes up in showing that the quotient $k(Q)/R_Q$ is a ...
1
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0answers
12 views

Relation between compact Lie group and Lie algebra representation

Currently I'm studying representation theory for compact Lie groups and I don't know how to link representations of Lie algebra to representations of corresponding Lie group, ie. suppose I have a ...
1
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1answer
26 views

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 ...
3
votes
1answer
17 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 $V$ ...
4
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1answer
63 views

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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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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9 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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24 views

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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27 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 ...
0
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1answer
56 views

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 ...
6
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1answer
76 views
+50

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 ...
3
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2answers
29 views

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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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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33 views
+50

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 = ...
0
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1answer
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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0answers
19 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 ...
2
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1answer
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 ...
0
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1answer
43 views

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 ...
0
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0answers
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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0answers
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 ...
3
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2answers
52 views

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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0answers
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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0answers
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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21 views

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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0answers
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 ...
2
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2answers
47 views

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 ...
2
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1answer
19 views

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 ...
2
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1answer
62 views

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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0answers
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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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33 views

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 ...
6
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1answer
100 views

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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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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0answers
46 views

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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37 views

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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48 views

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 ...
2
votes
1answer
12 views

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 ...
3
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1answer
32 views

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.
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1answer
33 views

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 ...
2
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1answer
37 views

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 ...
2
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39 views

Homological Conjectures

Let The strong Nakayama conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,R)=0$ for $i \geq 0$, then $M$ is zero. The generalized Nakayama conjecture If $S$ is a simple module and ...
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What is an algebra and what is it's representation?

Heyho, i've kind of got an understanding problem what exactely an algebra and especially it's representation is. In my case it is said, that the relation $R_{12}(u-v) (L(u) \otimes \mathrm{I}) \; ...
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1answer
41 views

Understanding a step in this proof

I have it tagged as representation theory because it's out of my rep theory book but I'm really just misunderstanding a group theory aspect here. So the statement to prove is: if $G$ has odd order ...
2
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1answer
14 views

Abelian semi-simplification representation

Let $K$ be a number field with $G=\mathrm{Gal}(\bar K/K)$. Consider the representation of $G$ with value in a $\mathbb{F}_p$-vector space $V$ of dimension 2 $$ \rho : G \longrightarrow ...
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Let G be an abelian group. Let V be an irreducible faithful CG-module. Prove that dimV = 1 and G is cyclic.

I was wondering if I could get some help with the following problem. I know how to prove it with Schur's Lemma but I'm having problems without it. Let G be an abelian group. Let V be an irreducible ...
0
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1answer
90 views

How do modern algebraists think about diagonal matrices?

Let $\mathbb{K}$ denote a field and $A$ denote a $\mathbb{K}$-algebra. Then given a $\mathbb{K}$-subalgebra $\Delta$ of $A$, I suppose it make sense to declare that $m \in A$ is ...
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51 views

Do we have $\delta(ab)=\delta(a)\delta(b)$ implies $\Delta(cd)=\Delta(c)\Delta(d)$?

Assume that $B$ is an algebra which is also a coalgebras (we do not assume that $B$ is a bialgebra: we do not assume $\Delta(cd)=\Delta(c)\Delta(d)$). Assume that $A$ is $B$-comodule algebra. Then we ...
2
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1answer
30 views

Two questions about Schubert calculus and Schur functions.

I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify ...
2
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34 views

Schubert calculus and number of lines satisfying some properties.

I am reading the file. I have a question on pae 18. It is said that: Given a line in $\mathbb{R}^3$, the family of lines intersecting it can be interpreted in $G(2, 4)$ as the Schubert variety $$ ...