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

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Submodules and $p$-adic numbers

I am a little bit confused about the terminology of simple $\mathbb{Q}_p[G]$ module. E.j.: If one take an $\mathbb{Z}_p[G]$ module $M$, then $pM$ is a submodule, so one can just look for ...
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186 views

Representations - Tensor Product prove properties of tensor product

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
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155 views

Frobenius reciprocity

I would like to ask a question on Theorem 8.6 on page 246 in this book. There is the claim that the multiplicity of $F$ in $E^G$ is equal to the multiplicity of $E$ in $F_H$. Why is this just ...
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139 views

Braid Group of a Weyl Group

I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman. The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of ...
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330 views

Definition of a “root” of a Lie Algebra

I am using the notation that $g$ is the Lie algebra of the Lie group $G$ and $T$ is the maximal torus of $G$ and $t$ is the Lie algebra of $T$ (and hence $t$ is the Cartan subalgebra of $g$). A ...
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Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
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47 views

How to compute the weights of $\Gamma_{3,1}$ the irrep of $\mathfrak{sl}_3\Bbb C$

I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest ...
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91 views

Show that for a finite field $F$ and finite group $G$, $F$ is a splitting field for $FG$

Let $F=\mathbb{F}_q$ be the field of $q$ elements and $G$ be a finite group. I'm trying to show that for an irreducible $FG$-module $V$, we have $\mathrm{End}_{FG}(V)=F \cdot 1$, i.e. that $F$ is a ...
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27 views

Question about the top of a bound representation of a bound quiver.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1. I have a on page 77. In (d) of Lemma 2.2 on Page 77, it is said that $$ L_a=\sum_{\alpha: a\to b} ...
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Exercise on representations

I am stuck on an exercise in Serre, Abelian $\ell$-adic representations (first exercise of chapter 1). Let $V$ be a vector space of dimension $2$, and $H$ a subgroup of $GL(V)$ such that ...
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Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$

Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...
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Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
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1answer
35 views

Questions about maximal submodules.

Let $A$ be a $K$-algebra and $M$ a right $A$-module, where $K$ is a field. Suppose that $M=C\oplus D$, where $C, D$ are right $A$-modules. If $C', D'$ are maximal right $A$-submodules of $C, D$ ...
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79 views

Questions about representation theory of associative algebras.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1. I have two questions on page 85. On Line 18 of Page 85, it is said that $\ker p_i \subseteq ...
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57 views

Characters of elements under every representation equal implies conjugacy

If $G$ is a group, suppose that for every $G$-module $V$ we have $$\chi_V(g_1)=\chi_V(g_2).$$ How can I be sure $g_1$ and $g_2$ are conjugate in $G$? Its easy to the reverse implication; ...
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Checking that $\rho$ is a representation

Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
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559 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
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102 views

On a sum taken over all irreducible characters: is $\sum\limits_{i=1}^t\chi_{V_i}(g)^2$ nonzero for all $g\in G$?

Suppose $G$ is a finite group and $\mathbb{k}$ an algebraically closed field of characteristic zero. Denote the irreducible representations of $G$ over $\mathbb{k}$ by $V_1,\ldots, V_t$. Is it ...
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Constructing $\text{Sym}^2(V\oplus U)\cong \text{Sym}^2(U)\oplus \text{Sym}^2(V)\oplus (V\otimes U)$ and the same for $\text{Alt}^2$.

I need to be able to show that if there exist representations $\phi_1:G\to GL(V)$, $\phi_2:G\to GL(U)$ that $\operatorname{Sym}^2(V\oplus U)$ is isomorphic to $\operatorname{Sym}^2(U)\oplus ...
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$Sym^n(V\otimes U) = \oplus_{\rho \vdash n}V(\rho)\otimes U(\rho)$

As in the title, I would like to prove that, given vector spaces $V$ and $U$ that $$Sym^n(V\otimes U) = \oplus_{\rho \vdash n}V(\rho)\otimes U(\rho)$$ Where $\rho \vdash n ...
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124 views

How does the holonomy act on the tangent space at a point?

Suppose $(X,h)$ is a compact $n$-dimensional Hermitian manifold, with holonomy group $H$. Now we know,since $X$ is a complex manifold, that $H\subset U(n)$, and that there is a representation of $H$ ...
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How is a vector a representation?

I am working on a homework problem that gives the character table for the octahedral group O, and then asks to ``decompose the vector (x,y,z) into irreps of O''. What does this mean? How can a vector ...
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345 views

Is the Hodge star an $SO(n)$-equivariant isomorphism to the dual representation?

Let $V$ be an oriented inner product space of dimension $n$. The Hodge star operator maps $\Lambda^k V\to \Lambda^{n-k}V$. In particular it maps $V\to \Lambda^{n-1}V.$ $V$ carries a representation of ...
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elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
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92 views

Induced representation is isotypical?

Is there a theorem like this for the induced representation? Let $N$ be a normal subgroup of a finite group $G$ and $\rho$ be an irreducible linear representation over any field $k$. Then one of ...
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Representations of $\text{GL}_2(\mathbb{Q})$

Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
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207 views

Finding irreducible representations of the following group using GAP

Given the following group of order 24, $$ G = \langle a,b \mid a^2=b^3=(abab^2)^2=1\rangle$$ how can one find (all) the irreducible representations using GAP? Since I have not installed GAP yet, I ...
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Completing a Character Table for a Group of Order 18

I have the following homework question: A group of order 18 has the following partial character table, where $y=-\frac{1}{2} + xi$: \begin{array}{c | c c c c c} \hline\hline & g_1 & g_2 ...
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Different induced representations - same simples?

is the following case possible: $\pi_1, \pi_2$ two simple representations of the same subgroup over an arbitrary field. $\operatorname{Ind}(\pi_1)$ and $\operatorname{Ind}(\pi_2)$ have equal ...
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126 views

Some questions about representation theory in the modular case

I'm working on a paper which uses representation theory in order to compute some characters and deduce arithmetical statements about certain field extensions. Let $\Delta$ be a group of order prime ...
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136 views

Question about the radical of the Jacobson radical.

I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
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Subspaces stabilized by representations of $\mathrm O(9)$

I am trying to figure out what representations of maximal subgroups of $\mathrm{GL}_{n^2}$ stabilize one dimensional subspaces in $\mathrm{GL}(\mathrm{Sym}^n(\Bbb C))$. More precisely, let the setup ...
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highest weight module correspondence with irreducible representation

Let g be a simple Lie algebra. L(λ) be the irreducible g -module of highest weight λ . are all highest weight modules irreducible ?
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Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
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2answers
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Isomorphism of faithful representations

Let $G$ be a group and $f,g: G \rightarrow GL(V)$ be two faithful representations over some field $K$ with $f:x\mapsto f(x)$ and $g:x \mapsto f(x^{-1})$. I would like to find out if $f$ and $g$ are ...
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470 views

Faithful irreducible representations of cyclic and dihedral groups over finite fields

How to determine all the faithful irreducible representations of $\mathbb Z_n$ and $D_{2n}$ over $GF(p)$, where $p$ is a prime not dividing $n$?
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Some irreducible character separates elements in different conjugacy classes

Let $x$ and $y$ be elements that are not conjugate in $G$. Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$. Clearly the "irreducible" part isn't important, ...
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75 views

Representations over $\mathbb{Q}_p$

I would like to understand representations over the $p$-adic field $\mathbb{Q}_p$ and find simple $\mathbb{Q}_p[G]$ modules for a finite group $G$. Is there some famous literature like for ...
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Do we have $e(\operatorname{rad}^2 A)=\operatorname{rad}^2 (e A)$?

Let $e$ be a primitive idempotent of $A$, where $A$ is a finite dimensional algebra over an algebraically closed field $K$. Do we have $e(\operatorname{rad}^2 A)=\operatorname{rad}^2 (e A)$? Here ...
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Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
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Artinian ring with zero finitistic dimension

Let $R$ be a left artinian ring with identity. Suppose $R$ contains copies of all its simple right $R$-modules. Is it true that every left $R$-module of finite projective dimension is projective (so ...
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111 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
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Tensor product of an irreducible $G$-representation and a one-dimensional representation [duplicate]

If $G$ is a finite group, $V$ is an irreducible $G$-representation and $W$ is any 1-dimensional $G$-representation (both over an algebraically closed field of characteristic zero), show that $V ...
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Induced representation - Workout for $S_3$

I want to give an induced representation of $S_3$. Elements of $S_3$ are given by: $$e,\ c=(123),\ c^2=(132),\ r=(12),\ rc=(23),\ rc^2=(13)$$ with the following relations: $$r^2=c^3=e,\ rcr^{-1}=c^2$$ ...
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Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
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Linearizations induced by the trivial one?

Let $k$ be a field, algebraically closed for simplicity. $G=GL_{n+1}(k),$ $X=\mathbb{P}_k^n$ and consider the action $G\times X\rightarrow X$ given by $(g,x)\mapsto gx$ (thus induced by usual left ...
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Proving the 3-dimensional representation of S3 is reducible

The 3-dimensional representation of the group S3 can be constructed by introducing a vector $(a,b,c)$ and permute its component by matrix multiplication. For example, the representation for the ...
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Nonabelian group with all irreducible representations one-dimensional

All irreducible representations of an abelian group are one-dimensional. For a finite group, the coverse is also true - if all irreducible representations are one-dimensional then the group is ...
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Standard representation of $\frak S_4$

On p. 18 of Representation Theory: A First Course, Fulton and Harris write The character of the standard representation is $\chi_V = (3, 1, 0, -1, -1)$. Note that $|\chi_V| = 1$ so $V$ is ...
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Induced representation, isomorphism between vectorspaces

I want to prove the following statement: The mapping $Ind_H^G{V}\rightarrow V^m: \psi\mapsto(\psi(g_1),\ldots,\psi(g_m))$ is a isomorphism. Here is $g_1,\ldots,g_m$ a representing system of $G/H$ ...