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Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?

Converting my comment into an answer, for a group $G$ (not necessarily finite) and a field $K$, the following categories are equivalent (actually I think even isomorphic but this doesn't really matter)...
Qiaochu Yuan's user avatar
2 votes
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Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants

For part (a), note that $G = \langle a, b \rangle$ indicates the subgroup generated by $a$ and $b$ (in which larger group?). So, in general, this can be rather large, since any word we write down in ...
Sammy Black's user avatar
2 votes

The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

You are making certain (very interesting !) mistakes. This is not quite my field, so I apologize if I say something imprecise. The overall issue is the question of "the tangent space of $P$ at $p$...
Fançois Gatine's user avatar
1 vote

Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?

I think I know where I went wrong, although I might find it difficult to explain how I was confused. In the direction where we start off with a $\DeclareMathOperator{\GL}{GL}\DeclareMathOperator{\End}{...
Joe's user avatar
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4 votes
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Working with character tables

Your solution to part 1 is correct. Part 2... not so much. Here is how I would approach these. Let $\chi_i$ denote the character corresponding to the $i$-th row of the table. If $g$ is central, then $...
Daniel Arreola's user avatar
1 vote

Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?

There is a bijection between representations of $G$ and a $\mathbb C$-vector space $V$ with a homomorphism $\xi\colon\mathbb C[G]\to \mathrm{End}_{\mathbb C}(V)$ of $\mathbb C$-algebras. However, as ...
Kenta S's user avatar
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3 votes
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Grothendieck ring of $Rep(\mathfrak{sl}_2)$

This is a version of the formal character. It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of ...
Qiaochu Yuan's user avatar
3 votes
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Irreps of $SU(3)/\mathbb{Z}_3$ from irreps of $SU(3)$

(All representations are complex and finite-dimensional throughout except for at a handful of points where I talk about real representations.) In general, if $V$ is an irreducible representation of a ...
Qiaochu Yuan's user avatar
2 votes
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On Frobenius–Schur indicator of real/complex representations

It means both in different contexts, as far as I know. It's unfortunate but they can be disambiguated based on whether the author is discussing real or complex representations. The Frobenius-Schur ...
Qiaochu Yuan's user avatar
1 vote
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Locally compact group whose unitary irreducible reps are one dimensional

This is always true, no assumptions concerning direct integral decomposition are needed. More precisely: if $G$ is a locally compact, Hausdorff topological group and all irreducible representations of ...
Mogget's user avatar
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2 votes
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For two irreducible modules $V$ and $W$, $f : V\to W$ is a $G$-module isomorphism $\iff \text{span}(f)\subset V^*\otimes W$ is trivial

Perhaps it is better to identify $V^* \otimes W$ with $Hom(V,W)$. The action of $G$ on $Hom(V,W)$ is then seen to be $$ g \cdot f := (g \cdot f)(v) = g \cdot f(g^{-1} \cdot v)$$ Recall now the ...
Guster's user avatar
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Fulton and Harris: Exercise 1.3 in section 1.1

Here's an answer with all the details filled in. For a general vector space $V$ (not necessarily of dimension $n$) we have a bilinear map $B:\bigwedge^kV \times \bigwedge^{n-k}V \to \bigwedge^nV \to \...
Samuel Johnston's user avatar
1 vote

Can the sum of a nonlinear irreducible character's values on $Z(\chi)$ be zero?

This sum of values is not equal to zero if and only if $\chi$ is constant over $Z(\chi)$. First, write the restriction of $\chi$ to $Z(\chi)$ as $\chi_{Z(\chi)}=\chi(1)\lambda$, where $\lambda$ is a ...
Deif's user avatar
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1 vote

Local systems are the same as modules over chains of based loop space?

Recently, I try to write a proof for this fact. Maybe you already know the proof. However, let me write down a direct argument here in case someone need it. Recall Lurie's monodromy equivalence $$\...
Bingyu Zhang's user avatar
1 vote

Unique extension of $*$-representation into an abstract multiplier algebra

I remember when you last asked this question, I thought there was a counterexample and wrote it down. It didn’t end up working, and I couldn’t figure out a solution in the end. Let me try my hand ...
David Gao's user avatar
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0 votes

Endomorphisms over direct sum of irreducible representations equal to direct sum of endomorphisms over irreducible representations

As commented 8 min ago: Schur also tells you that the only morphism between two "distinct" (i.e. non-isomorphic) irreducible representations is $0$.
Anne Bauval's user avatar
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0 votes

Endomorphisms over direct sum of irreducible representations equal to direct sum of endomorphisms over irreducible representations

Schur's lemma tells us that for two irreducible unitary representations $(\pi, V_\pi)$ and $(\tau, V_\tau)$ we have $\dim \text{Hom}_G(V_\pi, V_\tau) = 1$ iff $\pi\cong \tau$ as representations and $\...
GhostAmarth's user avatar
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1 vote

How is representation theory linked to invariant theory?

There is representation theory pretty much everywhere you look here, but to keep this answer reasonably short, I will only consider the example of $V = \mathbb C^2$. Then the polynomial functions $\...
krm2233's user avatar
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1 vote

How is representation theory linked to invariant theory?

Before we consider the group action, first, the ring of polynomials $\mathbb{C}[x_1, \dots, x_n]$ is isomorphic to the symmetric algebra $S(V)$ where $V$ is an $n$-dimensional vector space whose basis ...
Chris Jones's user avatar
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2 votes
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An infinite cyclic group has infinitely many irreducible real representations.

This is all fine but you are also working much too hard; you skipped the case of degree $1$ polynomials for some reason even though it is easier. $1$-dimensional representations correspond to ...
Qiaochu Yuan's user avatar
0 votes

Equivalence between parabolic 1-cocyles and right representations of modular group (from a paper of Zaiger)

For your first question, writing $f(S)|_{1+U+U^2}$ explicitly, using the definition of cocycle, we get \begin{equation} f(S)|_{1+U+U^2}=f(S)+f(SU)+f(SU^2)-f(1)-f(U)-f(U^2) \end{equation} AS you stated,...
Ashar Tafhim's user avatar
1 vote

If ${\text{char}(k) = p}$, and ${p \nmid |G|}$, are two representations of $G$ over $k$ isomorphic iff they have the same Brauer character?

The whole purpose of Brauer characters is to get rid of the positive characteristic of the field we're working with. A fundamental result in Brauer theory is that for every prime $p$ $($even if $p$ ...
GC.'s user avatar
  • 115
1 vote

irreducible representations of Group generate left ideal in Groupring CG.

The part of the text you're quoting is confusingly written, but actually it does not claim that $I = (e)$ is a minimal left ideal, which is good because it's not true. This is what it says (Fulton and ...
Qiaochu Yuan's user avatar
0 votes
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meaning of $IBr(X | Q)$

The meaning of $\operatorname{IBr}(X\mid Q)$ is given by the definition (Definition 4.1) where the notation is first used. It gives a long list of conditions that constitute what it means for the ...
Jeremy Rickard's user avatar
5 votes
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Schur’s lemma over $\mathbb{F}_p$

Since we’re over a finite field, $\mathbb{End}_{\mathbb{F}_p[G]}(V)$ is also a field, but I am not sure that it is exactly $\mathbb{F}_p$, why couldn’t it be a field $\mathbb{F}_{p^n}$? In general, ...
Qiaochu Yuan's user avatar
3 votes
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Can an irrep become reducible on multiplication by a 1 dimensional irrep?

The tensor product $V \otimes I$ of an irreducible representation and a $1$-dimensional representation is always again irreducible. This is because tensoring with $I$ establishes a bijection between ...
Qiaochu Yuan's user avatar
3 votes
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Invariants of $2$-torsion group under involution

Nope. Let $A = \mathbb{Z}^2$ and let $\tau : A \to A$ be given by the matrix $$\tau = \left[ \begin{array}{cc} 1 & 0 \\ 2 & - 1 \end{array} \right].$$ Writing $e_1, e_2$ for the standard basis,...
Qiaochu Yuan's user avatar
1 vote

Eigenvectors of a matrix that commutes with subgroup of permutation matrices

This can be understood in terms of representation theory of finite groups which you can learn about from many sources. Let $V$ be the vector space of column vectors on which these matrices are acting. ...
Qiaochu Yuan's user avatar
1 vote
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When are Idempotents elements of a semisimple algebra primitive

Based on the comments, a primitive central idempotent is a central idempotent that cannot be written as a sum of two central orthogonal idempotents. If we define that a primitive idempotent is an ...
khashayar's user avatar
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1 vote
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Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)

Since $\pi \oplus \pi'$ is finite-dimensional, the intersection $I = \ker \pi \cap \ker \pi'$ has finite codimension, so both representations factor through the finite-dimensional quotient $A/I$. ...
Qiaochu Yuan's user avatar
0 votes

Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

Everyone seems to be working too hard: By the Jordan-Holder theorem for modules, any simple module has a well-defined multiplicity in $F[G]$ (viewed as a left module over itself), and so there are at ...
krm2233's user avatar
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0 votes

Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

This is just an appendix to @Qiaochu Yuan 's nice answer that I hope could be useful (please correct me if I'm wrong). In the notation above, since $A=F[G]$ is Artinian (as finitely generated algebra),...
F. Salviati's user avatar
1 vote
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Relation between linear representation and "induced adjoint representation" of Lie algebra?

If you are looking at the semisimple case it should be clear to see that $\operatorname{End}(W)$ will be a subrepresentation if $W$ is. Indeed if we split $V$ into irreducible representations $W_i$, ...
Callum's user avatar
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0 votes

Exterior algebra and space of spinors

Now I think I understand how to connect the two approaches. Firstly let us remind how people construct the matrices in the first approach. The basic observation is that if we define $$b_a^+ = \frac{1}{...
Gold's user avatar
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2 votes

Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

Here's another proof, which is basically the same, but doesn't rely on the Artin-Wedderburn theory. Let $R$ be an Artinian ring, and let $\{I_\alpha\}$ be its set of maximal right ideals. By the ...
Steve D's user avatar
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4 votes
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Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

No specific property of the group algebra is needed beyond that it is finite-dimensional: Proposition: If $A$ is Artinian then there are finitely many isomorphism classes of simple $A$-modules. ...
Qiaochu Yuan's user avatar
0 votes

Exterior algebra and space of spinors

$ \newcommand\lcontr{\mathbin\rfloor}\newcommand\lintr{\mathbin\lrcorner} \newcommand\form[1]{\langle#1\rangle} \newcommand\Ext{\mathop{\textstyle\bigwedge}} $They seem to be the same (assuming you're ...
Nicholas Todoroff's user avatar
0 votes

Reference for representation theory over the reals

I have finally found the book that contains the central primitive idempotent of the group algebra $KG$ for any field $K$ such that $|G| \in K^{\times}$ with the detailed proof. Here is the book and ...
khashayar's user avatar
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2 votes
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Exercise 2.19 in Isaacs's book on character theory

You have already done the hard part, which is establishing the hint. You can now finish the problem in one line, using a previous problem from chapter 2. (As mentioned in the comments below, this is 2....
Steve D's user avatar
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0 votes

Bilinear form on a finite dimensional algebra over a field

Your notation is a little confusing; I assume you mean "over $K$" and "over $E$" respectively and that $K \hookrightarrow E$ is a field extension, but as written those could be ...
Qiaochu Yuan's user avatar
2 votes
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Problem 2.14 from Isaacs's Character Theory of Finite Group

I will show that $n\leq\chi(1)$. Since $n$ is a $p$-power for some prime $p$, if $H\cap\ker\chi\not=1$ for every $\chi\in\text{Irr}(G)$ then it would follow that $K\subseteq H\cap\ker\chi$ for the ...
Deif's user avatar
  • 1,176
2 votes
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Does the monoid of non-zero representations with the tensor product admit unique factorization?

Counterexamples are much easier to produce than this and exist already when $G = C_2$ and with two $2$-dimensional representations, see here. Abstractly the problem is that the representation ring is ...
Qiaochu Yuan's user avatar
2 votes

Does the monoid of non-zero representations with the tensor product admit unique factorization?

A counterexample is given by Nate at https://math.stackexchange.com/a/4436073/491450 : Taking $G = A_5$, we have $$ V_4 \otimes V_5 \otimes V_3 \cong V_4 \otimes V_5 \otimes {V_3}' $$ where the ...
Smiley1000's user avatar
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2 votes
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Question on the Proof of Proposition 6.6 in the Book Introduction to Lie Algebras by Karin Erdmann and Mark J. Wildon

From Lemma 5.5, I can see that if $z \in L$, then $a(zv) = z(av) = \lambda(a)zv$, thus $zv$ is also a eigenvector for $a$, where $a \in A$. I don't see the connection of it with eigenvector of $z$ in $...
Matthew Towers's user avatar
1 vote

What is the difference between a group representation and an isomorphism to GL(n,R)?

I would say that the most common definition of a (linear) representation of a finite group $G$ on a finite-dimensional complex vector space $V$ is that it is a homomorphism $\rho:G\to\...
Joe's user avatar
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2 votes
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Group representation determined up to isomorphism by image of homomorphism in $GL(V)$?

This is not true. The problem is that the group structure of the image does not tell you how it acts on the vector space. As a counterexample, consider two 1-dimensional representations $\mathbb{Z}/4\...
Sverre's user avatar
  • 594
2 votes
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Partial order on power set & set of partial orders

No. Suppose $X$ has at least $2$ elements; $a,b\in X$, $a\ne b$. Extend the partial order $\subseteq$ to a partial order $\preceq$ on $2^X$ so that $\{a\}\preceq\{b\}$. Suppose there is a set $\...
user14111's user avatar
  • 1,647
2 votes

What is the difference between a group representation and an isomorphism to GL(n,R)?

Let $G$ be a finite group and $V$ a finite-dimensional complex vector space. Then a representation of $G$ on $V$ is a group homomorphism $\rho: G \to \operatorname{GL}(V)$. (Note that $\rho$ can never ...
Smiley1000's user avatar
  • 1,647
7 votes

What is the difference between a group representation and an isomorphism to GL(n,R)?

There are two ways one can define a representation of a group $G$ on a vector space $\mathbb{C}^n$ (The field does not have to be $\mathbb{C}$, nor does one need to specify a collection of 'standard' ...
Coherent Sheaf's user avatar
1 vote
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What is the connection between double commutant theorem and group representation theory?

They are not the same result. The von Neumann double commutant theorem applies to infinite-dimensional Hilbert spaces and infinite-dimensional operator algebras acting on them, which are not ...
Qiaochu Yuan's user avatar

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