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1 vote

Is the regular representation of a given finite group unique?

The course syllabus defined it as a representation that contains all irreducible representations of a group. This is the wrong definition. The regular representation of a (finite, for simplicity) ...
0 votes

Symmetric square of an $\mathfrak{sl}_2$-representation

Instead of working with $Sym^2(V)$, we will work with $V\otimes V$, for notational convenience. As $V\otimes V$ contains $Sym^2(V)$ as a proper subrepresentation, if $v^2=e\cdot w$ for some $w\in Sym^...
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Interpretation of group elements in Cayley graph vs matrix representation

Historically we defined the composition $f(g(x)) = (g \circ f)(x)$ so that reading from left to right we would apply $g$ first, then $f$. However this gets confusing so the convention that $f(g(x)) = (...
2 votes
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Why $\mathscr{D}^k\mathfrak{g}$ is an ideal in $\mathfrak{g}$?

If $\mathfrak{h}$ is an ideal of $\mathfrak{g}$, then $[\mathfrak{g},D\mathfrak{h}]\subset [\mathfrak{h},[\mathfrak{g},\mathfrak{h}]]$ by the Jacobi identity, which is contained in $[\mathfrak{h},\...
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Weyl group of $F_4$ and its degree $4$ representations

It appears that while $ W(F_4) $ is a subgroup of $ O_4(\mathbb{R}) $ it is not a subgroup of $ SO_4(\mathbb{R}) $ or $ SU_4 $ or $ PU_4 $. That is, it has no faithful determinant $ 1 $, degree $ 4 $ ...
2 votes
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Coxeter Groups and representations of $ + $ type

Some stuff I scrounged up. First, an easier result: an element $g$ of a finite group $G$ is real if it is conjugate to $g^{-1}$, or equivalently if the character of $g$ is always real in every ...
-1 votes

What can be said about "splitting rings" for finite groups?

Here is a sketch of an argument. Let $Irr(G)=\{\sigma\colon G\to GL(V_\sigma)\}$ denote the set of irreducible representations of $G$ over $\mathbb C$. Suppose $R\subset\mathbb C$ is a ring such that ...
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Exercise 2.3 in Serre's Linear Representations of Finite Groups (and proof verification)

Your confusion about "right" (in the comments) is a common one. Let $\rho: G \rightarrow \mathrm{GL}(V)$ be the given representation. We define $\rho': G \rightarrow \mathrm{GL}(V')$ as ...
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3 votes
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Finite Rotation Groups

Here's how to show that the character $\chi_V$ of a representation $V$ determines its determinant. This is essentially an exercise in symmetric functions. To say it in as low-tech a way as possible, ...
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Generalized Binary Representation

Lemma. Given $m$ positive integer, there is $n\in\mathbb{N}$ such that $\dfrac{m}{2}<a_n\le m$. If $m = 1$, just take $n = 1$. Otherwise, the set $\{k\in\Bbb N: a_k\le \frac{m}{2}\}$ is not empty (...
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1 vote
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Knapp, Lie Groups: beyond an Introduction, Theorem II.2.15

Given part (a) $R_\mathfrak{h}(\mathfrak{g})$ is the complement of a finite union of hyperplanes in $\mathfrak{h}$ which makes it open (as well as dense in $R_\mathfrak{h}(\mathfrak{g})$). The fact ...
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2 dimensional faithful quaternionic irreps of a finite group

Let $ G $ be a finite subgroup of $ Sp_2 $. That is equivalent to being a finite subgroup of $ GL_2(\mathbb{H}) $ the group of $ 2 \times 2 $ invertible quaternionic matrices. The finite subgroups of $...
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Lattice in noncompact simple group is Ad-irreducible

The desired fact follows from a Borel density theorem type result. Thm 3 of https://www3.nd.edu/~andyp/notes/BorelDensity.pdf states "Let G be a connected semisimple $\mathbb{R}$-algebraic group ...
4 votes

complex vs modular representation theory

No, there cannot be such an equivalence, because the category of finite-dimensional representations of $G$ over a field $K$ is sensitive to $K$: the endomorphism ring of every irreducible ...
1 vote
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Row orthogonality vs Column orthogonality

$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}\newcommand{\Irr}[0]{\mathrm Irr}\newcommand{\norm}{\trianglelefteq}$Let me rewrite the column orthogonality relations as \begin{equation*} \sum_{...
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$\chi$ is an irreducible faithful character of $G$ then $Z(G)=\{g \in G: |\chi(g)|=\chi(1)\}$

$g\in Z(G)$ iff $|C_G(g)|=|G|$. We have that $|G|=\displaystyle\sum_{\chi\in Irr(G)}\chi(1)^2$, and so $|C_G(g)|=\displaystyle\sum_{\chi\in Irr(G)}|\chi(g)|^2$, by a direct consequence of the Second ...
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1 vote
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Let $X$ be compact metric space and $\mu_n$ be sequence of complex measures on it. Prove the following two representations of $C(X)$ is equivalent.

Your $U$ is onto because your $\mu_n$'s are mutually singular: let $\{X_n\}$ be a partition of $X$ by Borel sets such that $\mu_n=0$ outside $X_n$. Then, for any $g=\{g_n\}\in H$, wlog (up to equality ...
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3 votes

Real form of $\mathfrak{sl}_2 \mathbb{C}$ in Fulton-Harris Representation Theory

For the record, here is an arguably simpler argument showing that $\mathfrak g_0$ contains at least one ad-semisimple element $\neq 0$. Given $X \in \mathfrak g_0$, let $\lambda_1, \lambda_2, \...
4 votes
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Real form of $\mathfrak{sl}_2 \mathbb{C}$ in Fulton-Harris Representation Theory

I don't believe this claim about semisimplicity being a (Zariski) open condition, but a closely related statement is true and suffices here: having distinct eigenvalues is an open condition, because ...
2 votes
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Maximal Closed Subgroups of $ SO_5(\mathbb{R}) $

The final answer is that $SO(3)$ (embedded into $SO(5)$ as an irrep) is maximal. I'll first prove that $SO(3)$ is maximal among connected Lie groups, and then I'll worry about components after that. ...
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0 votes
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$ G $-invariant symmetric bilinear map and trivial subrepresentation of symmetric square

Let $b: V \times V \rightarrow \mathbb{C}$ be a $G$-invariant symmetric bilinear form on $V$. Define $f: S^2(V) \rightarrow \mathbb{C}$ by $f(vw) = b(v,w)$ for all $v, w \in V$. (Such a map exists ...
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Definition of Irreducible Constituent in Isaac's Character Theory

"Irreducible" or "simple" constituent is used often for groups, algebras and modules. For example, the Lie algebra $\mathfrak{f}_4$ of dimension $52$ is simple over a field of ...
1 vote
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Definition of Irreducible Constituent in Isaac's Character Theory

An 'irreducible constituent' is a constituent that is irreducible. This is just standard English, so no need to have a specific definition for that. Looking at the index of the book, 'constituent' is ...
1 vote
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Infinite groups in which all irreducible representations are one-dimensional.

If you restrict attention to finite-dimensional representations, then such groups exist; that is, there are infinite nonabelian groups all of whose finite-dimensional irreducible representations are $...
3 votes

The character of the alternating sum of the exterior powers of the standard representation

Let $S^1$ be the unit circle in $\mathbb C$, let $T=(S^1)^n$ be the $n$-torus, and let $$U=\{(z_1,\dots,z_n)\in T:z_1\cdots z_n=1\}.$$ This is a subspace of $T$ homeorphic to an $(n-1)$-torus. The ...
4 votes
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The character of the alternating sum of the exterior powers of the standard representation

If $V$ is a representation of a group $G$ of degree $d$, $g\in G$ and $p\geq0$, then the character of $\Lambda^pV$ evaluated at $g$ is $$\chi_{\Lambda^pV}(g)=\sigma_p(\lambda_1,\dots,\lambda_m),$$ ...
4 votes
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Understanding the 4 dimensional irrep of $ SU_2 $

In general, the $(n+1)$-dimensional irrep of $SU(2)$ is the $n^{th}$ symmetric power $S^n(\mathbb{C}^2)$, which can be described explicitly as the space of homogeneous polynomials of degree $n$ in two ...
1 vote
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Schur functors are pairwise non isomorphic

The character can be defined for finite-dimensional representations of an arbitrary group $G$ (or more generally for finite-dimensional modules over an arbitrary algebra), and it's an invariant of ...
0 votes
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Representation of a finite group $G$ on the symmetric and exterior power of a $\mathbb C G$-module

I was trying to show that the action of any $g\in G$ on $(v\otimes w - w\otimes v)$ and on $(v\otimes v)$ always gives $0$, in which case, the action of $G$ induces an action on $Sym^2(V)$ and $\...
1 vote
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Matrices for linear transformations that operate by multiplying two other matrices from right and left (Serre's linear representations: exercise 2.3)

The explicit calculation should go like this; hope this will clarify the matter. Let $e_{i}$ be a basis of $V_{1}$, and $f_{j}$ be a basis of $V_{2}$. For a fixed $s \in G$, write $$ p_{1,s}^{-1} e_{k}...
1 vote

Quotient of HP^n by SO(3) = Aut(H)

For concreteness, I'm going to view $\mathbb{H}P^n$ as consisting of the points of the form $[z_0:...:z_n]$ with $0\neq (z_0,..,z_n)\in \mathbb{H}^{n+1}$, where $[z_0:...:z_n]$ is identified with $[...
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1 vote

A question about irreducible representations of SO(3) group

To get an understanding of the higher-dimensional representations, think of the $SO(3)$ action as a rotation of functions instead of a rotation of vectors. This is how we get higher-dimensional ...
4 votes
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Integral basis for representations from diagonal action of $ GL_n $

No. Even if $V$ is defined integrally $V^{\otimes n}$ can have complex subrepresentations that can't be defined integrally. In fact, so long as $V$ is faithful (i.e. does not factor through a ...
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0 votes

Books on representation theory of algebraic groups

Beginner: Gunter Malle, Donna Testerman: Linear algebraic groups and finite groups of Lie type. Robert Steinberg: Lectures on Chevalley groups. Armand Borel: Properties and linear representations of ...
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1 vote
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Doubly transitive action and orbits in $X\times X$

i. It is redundant to state that the action is transitive - showing that the action is transitive using the definition above is immediate by choosing $x=y', y=x'$. It is however a necessary condition ...
2 votes

Relation between eigenvalue and representation of compact Lie group

I don't know the precise details but this is how it should go in broad strokes. If $G$ is a compact semisimple Lie group it can be equipped with a canonical Riemannian metric given by translating the ...
1 vote

Particular Weyl group longest word

There is an inductive procedure: for the group $S_2$ the longest word is $s_1$. For $S_3$ it is $s_1 s_2 s_1$. For $S_4$ it is $$s_1 s_2 s_3 s_1 s_2 s_1,$$ and for $S_5$ it is $$s_1 s_2 s_3 s_4 s_1 ...
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3 votes

Decomposing $Sym^2\mathbb{C}[X]$ without using characters

Yes, this can be done as follows, and this argument generalizes to $S_n$ with no difficulty. $S^2$ of a permutation representation is itself a permutation representation: namely it's the permutation ...
4 votes
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Representation of $\mathfrak{sp}_4\mathbb{C}$

Like you write, the wedge product $\wedge : \bigwedge^2 V \otimes \bigwedge^2 V \to \bigwedge^4 V$ is symmetric, so we can regard it as a a symmetric $\bigwedge^4 V$-valued symmetric bilinear form $$\...
1 vote

Degrees of irreducible representations of an abelian group

I think I got it. Let $K$ be a field of characteristic zero and $G$ a finite abelian group. Let $V$ be a simple $KG$-module with character $\psi$. Let $\bar{K}$ be an algebraic closure of $K$. Assume ...
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1 vote
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Exercise 2.1 in Serre's Linear Representations of Finite Groups

As pointed out in the comments by Captain Lama my approach is quite complicated. Instead one could use the Proposition 3 (and not the formulas used in the proof of said proposition which I tried above)...
1 vote
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Rotations and Hyperspherical Harmonics

If $V_\ell\subset L^2(S^{n-1})$ consists of the homogeneous degree $\ell$ functions with basis $\mathcal{B}=\{Y_\ell^m\mid 1\le m\le N_\ell\}$ then $\mathcal{B}'=\{Y_\ell^m\circ R\mid 1\le m\le N_\ell\...
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