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

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harish chandra for sl(2,C)

Is it true that each irreducible sl(2,$\mathbb{C}$)-module, $P(\lambda,\mu)$ with $\lambda \in \mathbb{Z}$ appears as the harish chandra module of some $(\pi_{\chi},V_{\chi})$ And given ...
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65 views

Exactness Properties of Schur Functors

The title says it all: What are the exactness properties of Schur Functors? Thanks!
4
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223 views

how to find all simple modules for the given path algebra

Let $A = KQ$, where $Q$ is the quiver $$\begin{array}{ccc} & \alpha & \\ 1 & \rightleftarrows & 2 \\ & \beta& \end{array}$$ are there simple right $A$-modules with dimension ...
21
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419 views

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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50 views

Trying to get a character table of $S_{4}$ from a character table of $A_{4}$.

I have constructed a character table for $A_{4}$ and need to use induced representations to get a character table for $S_{4}$. I'm not very confident with the concept of induced representations, but ...
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1answer
71 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
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177 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
3
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49 views

Relating Modules and Representations

I am currently studying representation theory and am struggling with the concepts which relate modules and representations. The specific question I am looking at right now is this: but while ...
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1answer
75 views

Formula for idempotents in $\mathbb{C}G$

Let $G$ be a finite group with $|G|=n$. Label the irreps $V_1,\ldots , V_t$ of $G$ over $\mathbb{C}$; let $d_i$ denote the degree of $V_i$. By Maschke's theorem we have $\mathbb{C}G\cong ...
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1answer
58 views

Different orderings for highest weights of a representation

Recall that given a representation $\pi$ of $\mathfrak{sl}_n$, a weight $\mu$ is said to be of highest weight if its corresponding weight vector is annihilated by all the positive root spaces (1). ...
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1answer
81 views

Stationary paths

Let $Q$ be a finite quiver and denote the stationary parts of $Q$ by $e_{i}$. Suppose we have two arrows $f,g$ such that their composition $f \circ g$ is equal to $e_{i}$. Does this always implies ...
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38 views

Representations of $\mathrm{SU}(n)$

I have been given the following representation of $\mathrm{SU}(n)$: Let $V_{k,n} \leq \mathbb{C}[z_1,\dots,z_n]$ be the subspace spanned by the degree-$k$ homogeneous polynomials and define ...
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73 views

Representations of $\pi_1M$ and Heegaard Splittings

I am reading Floer's Instanton-Invariant paper, and am stuck on a sentence. To set the stage: Consider a closed connected oriented 3-manifold $M$ and the nonabelian group $SU_2$. Denote the ...
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109 views

A natural way of thinking of the definition of an Artin $L$-function?

Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 ...
3
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2answers
110 views

The character of a representation

Let $\chi$ be the character of a representation of a simple group $G$ and let $g\in G$. If $g$ has order two and $G\neq C_2$ then show that $\chi(g)\equiv \chi (e)$ modulo 4. The hint I get is to ...
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2answers
156 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
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1answer
205 views

the semisimple and local properties of path algebras

Let $Q$ be a finite quiver. Then the following hold: (a) If $KQ$ is semisimple, then $|Q_1| = 0$. If, moreover, $Q $ is connected, show that: (b)$KQ$ is local only if $|Q_0| = 1$ and $|Q_1| = 0$,
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174 views

Is $f(\operatorname{rad}A)\subseteq\operatorname{rad}B$ for $f\colon A\to B$ not necessarily surjective?

If I have two $K$-algebras $A$ and $B$ (associative, with identity) and an algebra homomorphism $f\colon A\to B$, is it true that $f(\operatorname{rad}A)\subseteq\operatorname{rad}B$, where ...
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63 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
3
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1answer
322 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
4
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531 views

Fundamental and the anti-fundamental representation of $U(n)$

I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate. ...
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97 views

finding highest weight of dual of a representation of a semisimple lie algebra

If $V$ is an irreducible representation of a semi simple lie algebra having highest weight $\lambda$ then what will be the highest weight of the corresponding irreducible representation $V^*$ (Dual of ...
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115 views

Ring of invariants for $\Sigma_3$

I've just started reading about classical invariant theory and I'm not seeing how the general pattern should work, maybe it's obvious I don't know... Let $k$ be a field with $\mathrm{char}\ k = 0$, ...
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1answer
33 views

intuitive explanation of sparsity / references

I know it is a vague question, but I am confused by why/when we actually want sparsity of a matrix. For example, interior-point methods work better when constraint matrix is sparse. Similarly, it is ...
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1answer
72 views

$\mathbb{Q}$-dimension of f. g. $\mathbb{Q}[\mathbb{Z}/p^l]$-modules.

My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} ...
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389 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
3
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2answers
327 views

Indecomposable modules

Suppose $q$ is a prime $(\neq 2)$ and $G$ a finite group, for example the cyclic group $C_p$. Is there a way to determine all the $\textbf{indecomposable}$ $\mathbb{F}_{q^n}[G]$ modules for some $n\in ...
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648 views

What are the irreducible representations of the cyclic group $C_n$ over a real vector space $V$?

It suffices just to consider a linear transformation $f$ such that $f^n=id$ and require $V$ to have no proper subspace invariant under $f$. But I still don't have a picture of what's going on.
3
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1answer
298 views

Representation of simple groups

Let $G$ be a finite simple group, prove that $G$ does not have a non-trivial represention of degree $1$. Remark: I intend to prove by contradiction. Let $V$ be a non-trivial $\mathbb{C}G$-module of ...
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3answers
485 views

Lie algebra action from Lie group action: coordinates

Here's the setup: I have $SL(2;\mathbb{C})$ acting on $V = \mathbb{C}[z,w] = \oplus_d V_d$, where $V_d$ is the homogeneous complex polynomials of degree $d$. The action is precomposition: ...
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1answer
764 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
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232 views

Generalizing Artin's theorem on independence of characters

Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$. Can this theorem be ...
2
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1answer
366 views

Algebra over a finite field is a field [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field Let $\mathbb{F}$ be a finite field and let $A$ be a finite-dimensional ...
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107 views

Commutators in finite groups

I get stuck with this question : Let $g$ be an element in a finite group $G$, and let $k$ be an integer coprime to $|g|$. Prove that $g$ is a commutator in $G$ if and only if $g^k$ is a commutator. ...
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1answer
348 views

Young diagram for standard representation of $S_d$

I'm working through Fulton-Harris and I'm kind of "stuck" at the following question. I'm looking for representations of $S_d$, the symmetric group on $d$ letters via Young Tableaux. The question is: ...
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3answers
79 views

Let $G$ be a group. Why is the subgroup $N$ generated by all elements in $G$ of the form $ghg^{-1}h^{-1}$ normal?

Also I need to show that every 1-dimensional representation of $G$ arise from some 1-dimensional representation of $G/N$.
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1answer
75 views

Conjecture about irreps exhaused by direct factors of reps formed from conjugacy classes

More precisely (the conditions can be tweaked to be more general as we may desire), let $G$ be a finite group and $K$ an algebraically closed field whose characteristic does not divide the order of ...
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1answer
317 views

Relation between irreps of SO(2) and SO(3)?

Is there a relation between the irreducible representations of SO(2) and SO(3)? For instance, consider an n-by-n matrix representation of SO(3), $G$, if I restrict all of my rotations to have a common ...
8
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1answer
143 views

Fulton and Harris A.23

I am reading the appendix of Fulton and Harris pg. 459 and am trying to understand the following setup. Suppose $\lambda : \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_k \geq 0$ is a partition of ...
5
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1answer
98 views

Endormorphism ring of quaternions isomorphic to the quaternion ring

I found the quoted question in this post interesting: Confusion regarding what kind of isomorphism is intended. I don't have commenting privileges just yet, and since the question already has an ...
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1answer
79 views

Confusion regarding what kind of isomorphism is intended.

For a class I'm taking this semester, I was given this question: (To guarantee clarity, I am quoting the full question even though my own question is only regarding the final sentence.) Let $G$ be ...
4
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1answer
181 views

Representation of Lie algebra of $\textrm{SU}(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, Let $U\in SU(2)$ is just a linear map on $\mathbb{C}^2$, Define a Linear Transformation $\Pi_m:V_m\rightarrow V_m$ given by ...
2
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1answer
127 views

Trace and identity are the only linear matrix invariants?

This question is obviously related to that recent question of mine, but I feel it’s sufficiently different to be posted as a separate question. Let $V$ be a finite-dimensional space. Let ${\cal L}(V)$ ...
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121 views

Bounding the degree of irreps of a finite group

Let $G$ be a finite group and $\mathbb{k}$ is algebraically closed with characteristic zero. Let $H$ be an Abelian subgroup of $G$. Show that the degree of any irreducible representation $V$ of $G$ ...
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1answer
268 views

Highest weight of dual representation of $\mathfrak{sl}_3$

Suppose I have an irreducible representation $\phi:\mathfrak{sl}_3 \to \mathfrak{gl}(V)$ of the Lie algebra $\mathfrak{sl}_3$. Now I have been asked to express the heighest weight of the corresponding ...
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2answers
109 views

uniqueness of induced representation

I am studying the book "Representation Theory" by Fulton and Harris. And I just can not understand the part where they prove the uniqueness of induced representation. If someone could explain it I'd ...
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80 views

Help with Fulton and Harris question 6.16

The Pieri formula gives a decomposition $$\textrm{Sym}^d V \otimes \textrm{Sym}^d V = \bigoplus \mathbb{S}_{(d+a, d-a)}V,$$ the sum over $0\leq a \leq d$. The left-hand side decomposes into a ...
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1answer
96 views

Some representation of $SU(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, could any one tell me how is that Linear Transformation look like explicitly?And would you please tell me how this is an ...
3
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1answer
64 views

Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$

Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that means that all proj. indecomposable $A$-modules and all inj. indecomposable $A$-modules ...
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217 views

Hom and Tensor Product of Linear Maps

I am reading Claudio Procesi's book on Lie groups and on page 105 there is something I don't understand. Let $U,V,W$ be vector spaces. Let us consider the product space $\hom(V,W) \times \hom(U,V) $ ...