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

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Existence of weights of a finite dimensional representation of a semisimple Lie algebra

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. I want to show that every finite dimensional irreducible representation of $\mathfrak{g}$ is a weight module, and I need the existence of at ...
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55 views

Proving that $Hom_G (V,W)$ is 1-dimensional when $V,W$ are irreducible

Question: Let $G$ be a group. For any two representations $V,V'$ of $G$ over $\mathbb C$, let $Hom_G (V,V')$ denote the space of all linear maps $h: V\rightarrow V'$ such that $h\rho'_g = \rho_g ...
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143 views

Problem 5.15, I. Martin Isaacs' Character Theory

Isaac's Character theory of finite groups book, Problem 5.15: Let $H \subseteq G$ and suppose $\phi$ is a character of $H$ with $det(\phi)=1_{H}$. Let $\chi={\phi}^{G}$ and show ...
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160 views

bests book of representation theory for algebraic number theorists

I am looking for some of the best books on representation theory for an algebraic number theorists> I would prefer a book that is more number theoretical (e.g, galois representations, p adic ...
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131 views

Isomorphic representations of $\mathbb Z$

I've read a statement in my notes that I am confused about: Representations $\rho, \rho' : \mathbb Z \to \mathrm{GL}(V)$ are isomorphic iff we may choose bases such that $\rho(1)$ and ...
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Is a surjective maps from a projective module to another projective module bijection?

Let $M$ and $N$ be projective $A$-modules. If we know that $f: M \to N$ is surjective and $g: N \to M$ is surjective, can we conclude that $M$ is isomorphic to $N$? More generally, if $M$ and $N$ are ...
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121 views

Radicals of modules using linear algebra

Let $\mathcal{A}$ be an associative algebra over a field $F$ with generators $a_1,a_2,\ldots,a_n$ and let $M$ be an $\mathcal{A}$-module. We can specify $M$ with the following data: a ...
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Usefulness of the concept of equivalent representations

Definition: Let $G$ be a group, $\rho : G\rightarrow GL(V)$ and $\rho' : G\rightarrow GL(V')$ be two representations of G. We say that $\rho$ and $\rho'$ are $equivalent$ (or isomorphic) if $\exists ...
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349 views

Question regarding the definition of direct sum decomposition of a representation

Please bear with me. I am trying to learn representation theory of finite groups from J.P. Serre's book by myself. Here, the author has used the word 'representation' for the homomorphism $\rho : ...
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120 views

When is $R(G\times H) = R(G) \otimes R(H)$?

Suppose $G$ and $H$ are discrete groups. If $\rho_G$ and $\rho_H$ are reps of $G$ and $H$ on $V_G$ and $V_H$, respectively, then we get a rep of $G\times H$ on $V_G\otimes V_H$ by sending $(g,h)$ to ...
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Exercise 2.13, I. Martin Isaacs' Character Theory

I am trying to solve the exercise 2.13 in Isaacs' Character Theory Book. However I met some difficulties, let me sketch out what I am thinking so that you may tell me a hint. The problem 2.13 is ...
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135 views

Questions about the Space of Matrix Coefficients

Apologies in advance for the basic question: In reading up on representation theory, I came across a confusing definition for the $M(\rho)$, the space of matrix coefficients of a representation $(G, ...
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287 views

Representation problem from Serre's book

I asked this question yesterday on the setting of an exercise problem (Ex 2.8) from Serre's book "Linear representations of Finite Groups" (I'm teaching myself representation theory...) Now that that ...
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427 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
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124 views

Representation problem: I don't understand the setting of the question! (From Serre's book)

Ex 2.8 of Serre's book "Linear Representations of Finite Groups" says: Let $\rho:G\to V$ be a representation ($G$ finite and $V$ is complex, finite dimensional) and $V=W_1\oplus W_1 \oplus \dotsb ...
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Representations of $SO_3(\mathbb{R})$ from $SU_2(\mathbb{C})$

Define $V_n$ as the linear space of all homogeneous polynomials of degree $n$ in two variables $x$ and $y$. Define also the representation $\rho_n$ of $SL_2(\Bbb{C})$ on $V_n$ by: ...
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277 views

Irreducible characters form orthonormal basis of set of class functions

I am reading Serre's book (Linear Representations of Finite Groups). Theorem 6 in chapter 2 says that the irreducible characters $\chi_1,\dotsc,\chi_h$ of a finite group $G$ form an orthonormal basis ...
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84 views

Bounds on Young Tableau Element locations

I'm having trouble finding some elementary results on the following. Let $Y$ be a standard Young Tableau of shape $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$ with $N:=\sum_{i=1}^n\lambda_i$. My ...
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167 views

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does ...
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366 views

Degrees of faithful irreducible representations of $\mathbb{Z}_n$ over finite fields

I came across the following theorem while studying representation theory over finite fields. A cyclic group $\mathbb{Z}_n$ has a faithful irreducible representation of degree $d$ over $\mathbb{F}_p$ ...
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164 views

When a group algebra (semigroup algebra) is an Artinian algebra?

When a group algebra (semigroup algebra) is an Artinian algebra? We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group ...
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144 views

Standard representation of $O_h$ in $\mathbb{R}^3$

I want to give the standard representation of the complete octaedergroup $O_h$ in $\mathbb{R}^3$. To which group is $O_h$ isomorphic, and how to obtain a standard representation of the group? What ...
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125 views

Extending and inducing irreducible representations

I sense this may be a simple question, but it is one I haven't been able to find an answer for, possibly due to the use of different terminology. Referring to this question: Faithful irreducible ...
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62 views

What is the smallest dimension possible for a representation of $D_8 \times Q_8$ which is faithful over $F$?

Consider $D_8 \times Q_8$, where $D_8$ is the dihedral group of order 8; $Q_8$ the quaternions. Let $F$ be a field of characteristic not equal to 2. What is the smallest dimension possible for a ...
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37 views

Show the $\mathbb{C} S_3$-module of dimension 2 has $S(V \otimes V)$ is not irreducible

Consider the $\mathbb{C} S_3$-module of dimension 2, call it $V$. I want to concretely show that $S(V \otimes V)$ is not irreducible. I found a representation for $S_3$ over $\mathbb{C}$ of degree ...
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482 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
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104 views

If $M \simeq N$ in ${\tt stmod}(G)$ will $M \oplus \text{(proj)} \simeq N \oplus \text{(proj)}$ in ${\tt mod}(G)$?

Let $G$ be a finite group and ${\tt stmod}(G)$ the stable module category for $G$, i.e., the category whose objects are $G$-modules and whose morphisms are $G$-module homomorphisms modulo those that ...
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170 views

The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k $ -algebras to groups that is representable ...
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283 views

Relationship between number of conjugacy classes and number of irreducible representations of a group

For a finite group G the number of irreducible representations over an algebraically closed field F is at most the number of conjugacy classes whose sizes are coprime to the characteristic of F. What ...
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127 views

If $V$ is an irreducible representation then is $S(V\otimes V)$?

Let $V$ be an irreducible $FG$-module of dimension $2$. Is $S(V\otimes V)$ irreducible? Why? $G$ is a finite group. $F$ is a field, its order is unspecified. $S(V\otimes V)=\{x \in V\otimes V : ...
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Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$. Is this because spinor representation are projective representations? If so, where does ...
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298 views

Show that a p-group has a faithful irreducible representation over $\mathbb{C}$ if it has a cyclic center

A p-group is a group of order $p^d$ where p is a prime. If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible ...
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35 views

Split 3D representation of S3 in irreducible components

I saw from this post that you can prove that the 3d representation of S3 is reducible. What if I want to split this representation in a sum of irreducible representation?
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Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
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632 views

Division algebra over a an algebraically closed field

I am reading my notes and I stumbled upon a proof that I dont fully understand, and I was hoping maybe someone could clear the details. The main goal was to show that if $k$ is algebraically closed, ...
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faithful irreducible representation of $A_{4} \times Q_{8}$

Construct a faithful irreducible representation of the group $A_4 \times Q_8$ $A_{4}$ is the alternating group $Q_{8}$ is the quaternions
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Find all irreducible representations of $A_4$ over $\mathbb{F}_p$ where p is an odd prime

I have no issue with finding representations over $\mathbb{C}$ but I'm not quite sure how to find them over finite fields, particularly when that field might not be algebraically closed or the ...
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137 views

Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$

I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$. I'm reading a paper with a result that implies ...
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Weights set spans

Definition Let $T$ be a torus and $R: G \to GL(V)$ a representation. $R(T)$ is a collection of commuting matrices and therefore can be simultaniously diagonalized. For a character $\lambda \in ...
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449 views

What is the contragredient representation?

Let $V=M_2(\Bbb C)$ be the set of all $2$x$2$-matrices. Let $G=B$x$B$ where $B$ is the group of $2$x$2$ lower triangular matrices with non-zero diagonal entries. Then G acts on $V$ by $\rho ...
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106 views

First group cohomology and composition factors

Let $G$ be a finite group. Let $k$ be a field ($\text{char}(k)=p>0$). Let $P(k)$ be the projective cover of $k$. Assume that for any nontrivial simple $kG$-module $M$ we have $H^1(G,M)=0$. Does it ...
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392 views

First Homology Group and Abelianization

On the Wolfram Mathworld article for Commutator Subgroup, it states that the first homology group is the abelianization, $$H_{1}(G) = G \big/ [G,G]$$ which totally blows my mind because I've only seen ...
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A question of the book Elements of the Representation Theory of Associative Algebras: Volume 1

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . I have a question on page 9, line -3 (see Page 9 here). It is said that $$h_1f_X = f_Y h_2.$$ I am ...
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Partial cycles in projective resolutions of square-free algebra

Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand? I suspect not, but have not ...
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Reference request: Auslander-Reiten quivers of which types of algebras have been found?

I would like to know Auslander-Reiten quivers of which types of algebras have been found. For quantum groups, there is a paper http://arxiv.org/abs/1202.1714 by Julian Külshammer. I search the ...
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Representing sums of matrix algebras as group rings

Let $A = M_{n_1}(\mathbb R) \oplus M_{n_2}(\mathbb R) \oplus ... \oplus M_{n_m}(\mathbb R)$ be a direct sum of real matrix algebras. Under what conditions does there exist a group ring $\mathbb R[G]$ ...
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Question about an almost split sequence.

On page 124 of the book Elements of representation theory of associative algebras, volume 1, Example 3.10, I computed the modules in this example. $$ S(3)=0\leftarrow 0 \rightarrow K \leftarrow 0, ...
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Question about the decomposability of the radical.

Let $A$ be an algebra over an algebraically closed field $K$ and $M$ an indecomposable $A$-module. Suppose that $M$ is indecomposable. Can we conclude that $\operatorname{rad}M$ is indecomposable? ...
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Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
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How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...