Questions tagged [representation-theory]
For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.
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Explicit 28 Lie algebra matrix representations of $Spin(8)$: Two half-spinor representations
A simple Lie group $𝑆𝑝𝑖𝑛(8)$ has 2 half-spinor representations and 1 vector representation (coming from standard vector representation of SO(8)), all of them have dimension 8. One can compose a ...
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Trying to understand the proof of the main theorem in Tannakian categories
I am trying to understand Lemma 2.13 in Deligne and Milne's Tannakian Categories
Lemma: Let $C$ be a $k-$linear abelian category and let $\omega:C\to\text{Vec}_k$ be a $k-$linear exact faithful ...
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Show that $\phi:A_4\longrightarrow GL_3(\mathbb{C})$ is a irreducible representation
Given the representation $\phi:A_4\longrightarrow GL_3(\mathbb{C})$ with:
$$\phi_{(123)}=\begin{bmatrix}
0 & 0 & 1\\
1 & 0 & 0\\
0 & 1 & 0
\end{bmatrix}\, \text{and} \,\, \...
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Show irreducibility of polynomial representation of $SU(2)$
I am looking at this exercise for 2 days and honestly cannot make any progress, so I really appreciate any help.
Let $V_n$ ⊆ $C[x_1, x_2]$ be the space of all homogeneous polynomials of degree
n. Let $...
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Lifting projective representations from Lie algebras
Let $\mathbb{H}$ be a Hilbert space and $\mathbb{P}(\mathbb{H})$ the projective space of one dimensional linear subspaces of $\mathbb{H}$, that is,
$$\mathbb{P}(\mathbb{H}):=\mathbb{H}\backslash\{0\}/\...
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A doubt on a statement about simple algebras in Weil's Basic Number Theory
The following is from Chapter IX, Section 1, Corollary 5 of Weil's classical book mentioned in the title
In the proof of Corollary 5, he used the fact that the kernel of $F_L$ is a two-sided ideal, ...
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Line bundles on complete flag varieties independent of isogeny class
Let $G$ be a semisimple connected linear algebraic group over an algebraically closed field $k$. If $G^{sc}$ is the simply-connected cover of $G$ (i.e., the semisimple connected simply-connected ...
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Why does $\rho(n) = \rho(1 + \dots + 1) = \rho(1)^n$. where $\rho$ is a homomorphism?
I am studying representation theory, and below is an excerpt from my lecture notes.
Suppose that $G = (\Bbb Z,+)$. Then, if $\rho$ is a representation of $G$, it is completely determined by $V$ and ...
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Proving that a matrix that commutes with all other matrices is a multiple of the identity [duplicate]
I'm trying to prove that if there is $Z \in M_n(\mathbb{C})$ such that $[X, Z] = 0$ for all $X \in M_n(\mathbb{C})$, then $Z=cI$ for some complex number $c \in \mathbb{C}$.
I first noted that if $Z$ ...
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Relation between $SU(8)$, and $Spin(8)$ and $SO(8)/(\mathbf{Z}/2)$
It is east to use the special unitary group to contain the special orthogonal group
so $$SU(n) \supset SO(n) .$$ For $n=8$, we have
$$SU(8) \supset SO(8).$$
We know that the $SO(8)$ has a double cover ...
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Real / complex Lie algebra $\Rightarrow$ Real / pseudoreal/ complex representations (any correlation)?
I know that the distinction of real / complex Lie algebra.
For example,
The $su(2)=so(3)=sp(1)$ is a real Lie algebra.
The $sl(2,\mathbf{C})=so(1,3)$ is a complex Lie algebra.
Question 1: How are ...
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spin representations v.s. semi-spin representations from $\rho: \operatorname{Spin}(n, {\mathbb C})\to SL(N, {\mathbb C})$
I learned that the complex representations $$\rho: \operatorname{Spin}(n, {\mathbb C})\to SL(N, {\mathbb C})$$ of complex Spin groups $\operatorname{Spin}(n, {\mathbb C})$ helps to study the compact ...
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SU(5) Lie algebra: Derive the 10-dimensional matrix representation, from the given 5-dimensional fundamental matrix representation
My question (Abstractly)
If we know how to write the matrix representation of the fundamental representation of SU(N), could we use them to derive the matrix representation of other representations of ...
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Decomposition of $k^*$-representations
Let $k$ be an algebraically closed field of characteristic zero.
Let $V$ be an (algebraic) $k^*$-representation, not necessarily of finite dimension.
Is it true that $V$ decomposes in $k^*$-isotypical ...
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Polynomials invariant under the action of SO(3)
Let $SU(2)$ act on $V_2=\{ax^2+bxy+cy^2:a,b,c\in\mathbb C\}$ by $$\begin{pmatrix}a&b\\c&d\end{pmatrix}f(x,y)=f(ax+cy,bx+dy).$$ Then $SU(2)$ also acts on the $n$-th symmetric power $S^n V_2$. ...
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Representation theory of matrix groups over a finite field
Let $G$ be the multiplicative group of $n\times n$ invertible matrices over the finite field $\mathbb{F}_2$ of two elements. Consider the representation $\sigma(M):e_x\mapsto e_{Mx}$ acting on the ...
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If the left-regular representation is restricted to a closed subgroup, is it weakly equivalent to the left-regular representation of that subgroup?
Let $G$ be a second-countable locally compact group. We consider its left-regular representation $\lambda^G: G\to\mathcal{U}(L^2(G))$ given via $\lambda^G_g(\xi)(h)=\xi(g^{-1}h)$.
Recall: Given any ...
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Embeddings of several Lie groups and their geometry embedding
The question concerns some problems about the Lie groups and representations. And the geometry embedding of the several Lie groups.
We start from a fixed common special unitary group SU(2), with the ...
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Showing that $k[G]$ is a self-injective module
Let $k$ be a field, $G$ a finite group, and $k[G]$ the group ring. I'm trying to show that $k[G]$ is self-injective, meaning that it is injective as a (left) module over itself.
One possible approach ...
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Determine a basis of $Z(\mathbb{C}S_3)$
Determine a basis of $Z(\mathbb{C}S_3)$ explicitly as matrices in the group algebra when the group algebra is written as a direct sum of matrixalgebras.
I know that the dimension of $Z(\mathbb{C}S_3)$...
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What is meant by a defining representation?
I'm aware that there are similar questions, but none of them are really something that I can approach with my current knowledge of groups and representations (which is the first 3 chapters of a book ...
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Find a certain decomposition of $\mathbb{C}S_3$
I need to find a decomposition of $\mathbb{C}S_3$ in the following way:
$\mathbb{C}S_3=\mathbb{C}S_3e_1\oplus\mathbb{C}S_3e_2\oplus\mathbb{C}S_3e_3$
with $e_i=|G|^{-1}\sum\limits_{g\in G}\chi_i(\text{...
- 991
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Branching rule for $S_n$ proof by James
Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
It can be found in "The Representation Theory of the Symmetric ...
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Can every group be represented by a group of matrices?
Can every group be represented by a group of matrices?
Or are there any counterexamples? Is it possible to prove this from the group axioms?
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A group of order $pq$
I want to prove that if a group of order $pq$ (where $p,q$ are primes) with $p>q$ is not abelian, then $p \equiv1$ mod $q$. I don't know if this is correct but I think I have a proof using ...
CommunityBot
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Hall-Littlewood polynomials and elementary symmetric functions-- Chapter III (2.8) in Macdonald's "Symmetric Functions and Hall Polynomials"
I'm confused about the proof of Chapter III (2.8), page 209 in Macdonald's book, see
proof of (2.8).
Here is the background. Let $\Lambda_n$ be the ring of symmetric polynomials in $r$ variables, i.e. ...
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Question about a proof in Fulton and Harris book (Representation theory: a first course)
I would like to understand the proof of proposition $1.5$ from the book "Representation theory: a first course" of Fulton and Harris.
$\textbf{Proposition}$: If $W$ is a subrepresentation of ...
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Must the norm of a character be an integer?
I am working on a problem and I was very tempted to take for granted that for any complex character $\chi$ of a finite group $G$, the norm
$$\langle \chi, \chi \rangle = \frac{1}{|G|}\sum_{g\in G} \...
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Counting the multiplicity of weight of a Verma module without using the Kostant partition function or the Weyl character formula.
For a semisimple Lie algebra $\mathfrak{g}$ and a Cartan subalgebra $\mathfrak{h}$, if $\mu \in \mathfrak{h}$ and $W_{\mu}$ is the associated Verma module, I want to see that the multiplicity of $\...
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Constructing a surjective intertwining map onto Verma module
I want to prove the following result:
Theorem: Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra, $\mu \in \mathfrak{h}$, and $W_{\mu} = \mathfrak{U}_{\mathfrak{g}...
- 4,007
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Cohomology ring of grassmannian and Pieri rule
I am learning Schubert variety and I came across a problem to understand a particular detail (I asked the same question on mathoverflow : https://mathoverflow.net/questions/397999/cohomology-ring-of-...
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Space of maps in induced representation
One of the definition of induced representation from a subgroup $H$ to the group $G$ comes from certain types of maps; which are termed somewhere as $H$-equivariant maps.
Let $G$ be a finite group, $H$...
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Reading character tables
I am currently studying for my algebra qualifying exams and always get tripped up by character tables (though it seems they should be much more straightforward). Here is a practice problem that I am ...
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Local systems are the same as modules over chains of based loop space?
Let $M$ be a "good" topological space such as a manifold or a CW complex and assume it's connected. We use $\operatorname{Loc}(M)$ to denote the category of local systems either as a dg ...
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how to compute the Euler characters of a Grassmannian?
Let $G(n,m)$ be the Grassmannian of all n-dim subspaces of an m-dim vector space over $\mathbb{C}$. How to compute the Euler characters of $G(n,m)$? For example, $G(1, 2)$ is $\mathbb{C}P^1$ which is $...
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Decomposition of finite-dimensional representation of semisimple Lie-Algebra into irreducible subrepresentations.
This is my first post on MathSE, since I could not find a helpful answer to my question on here yet.
Let $(L,[\cdot,\cdot])$ be a finite-dimensional semisimple complex Lie-Algebra, $H\subseteq L$
a ...
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Structure of $*$-representations of matrix $C^*$-algebras
I'm trying to solve the following exercise from Paulsen's book "Completely bounded maps and operator algebras":
Here, $M_n$ is the $C^*$-algebra of complex $n \times n$ matrices.
I don't ...
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Reference for Langlands functoriality conjecture view towards classical examples
I want to know if there's any good reference on Langlands functoriality conjecture which provides connection with classical examples. What I have in my mind are followings:
Classical Rankin-Selberg (...
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Checking whether a particular group has an efficient, faithful representation as a matrix group
There are cryptographic protocols being developed for non-abelian groups. For some protocols it is necessary to know whether the group has an efficient representation as a matrix group (say, a matrix ...
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Character of the irreducible representation $ψ^λ$ : $S_4$ → $Aut_C(S^λ)$
I am struggling with these exercise from group representations and would really appreciate some steps to take or sources with similar exercises.
The task is to compute the character of the irreducible ...
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Question about group algebras and representations
Hello I am currently learning about group algebras and representations. And our professor gave us in the corona-online semester only handwritten notes and thats all..
And there was one topic I am kind ...
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Functoriality of associating to a group algebra $kG$ its center $Z(kG)$, where $G$ is finite
It's well known that associating a group $G$ to its center $Z(G)$ is not functorial (read: doesn't extend to give us a functor $\mathsf{Grp} \to \mathsf{Ab}$). A simple counterexample is given by ...
- 2,635
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Representations of symmetric (and alternating) groups over finite fields.
I want to learn representations of finite groups (actually in particular symmetric and alternating groups) over finite fields. For now, I am particularly interested in the semisimple case. Can anybody ...
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Representations $M_n(\mathbb{C}) \to B(H).$ [duplicate]
Consider the following fragment from Paulsen's book "Completely bounded maps and operator algebras". Here $M_n= M_n(\mathbb{C})$ are the complex $n \times n$ matrices.
How can we prove this?...
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Condition for a matrix to belong to the spin-$s$ irreducible representation of $\text{SU}(2)$
The spin-$s$ irreducible representation of $\text{su}(2)$ is generated by the three $2s+1$ dimensional spin matrices $iX,iY,iZ$. Since these spin matrices are anti-Hermitian, they span a subspace of $\...
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Group action on a space over a division ring $D$.
I am confused about very basic thing.
Let $V$ be a right $D$ module where $D$ is a division ring. Suppose that a group $G$ acts on $V$ (from right) linearly and faithfully.
If $D$ is a field, we can ...
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Spin groups and Clifford Algebra
Consider $\Bbb R^n$ with standard norm and its Clifford algebra $Cl(\Bbb R^n)$. It decomposes as $Cl(\Bbb R^n)=Cl_0(\Bbb R^n)\oplus Cl_1(\Bbb R^n)$ (even part and odd part). The group $\text{Spin}(n)...
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Duality between hermitian and skew-hermitian matrices
Let $H_n= \lbrace A \in M_n(\mathbb{C}), A^* = A \rbrace$ be the space of hermitian matrices, and let $iH_n= \lbrace A \in M_n(\mathbb{C}), A^* = -A \rbrace$ be the space of skew-hermitian matrices.
I'...
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Lie groups in Harish-Chandra's class
A Lie groups in Harish-Chandra's class satisfies the following:
$G$ is a real Lie group and its Lie algebra $\mathfrak g$ is reductive;
$G$ has only finitely many connected components;
$\mathrm{Ad}(G)...
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Automorphism groups of which lattices act irreducibly on the ambient Euclidean space
Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider the finite group
$$
\mathrm{Aut}(L) = \{ f \in \mathrm{O}(V) \mid f(L) = L \},
$$
...
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