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.
9,272
questions
4
votes
1
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21
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Alternative definition of Koszul algebra by injective resolutions?
Let $A$ be a positively graded algebra. This means that $A$ is a $k$-algebra graded non-negatively and $A_0 \cong k \times \dots \times k$ such that each degree is finite-dimensional.
From here, we ...
- 53
0
votes
0
answers
15
views
Necklace Lie Bracket for Species
I am currently reading the paper "Calabi-Yau Algebras and Superpotential" by Van Den Bergh, but I find it highly confusing. In Section 10 he mention that bracket
{$-,-$}$\space_{\omega_\eta}$...
1
vote
0
answers
31
views
Are there any other examples of simple not absolutely simple lie algebra besides $so(3,1)$?
I just learned the distinction between simple and absolutely simple Lie algebras and was wondering if there are other well known examples other than the Lorentz algebra.
- 184
1
vote
0
answers
55
views
Simple Lie algebras invariants
Do all simple Lie algebras have just one quadratic Casimir invariant as the Harish-Chandra isomorphism seems to imply or are there counter-examples?
- 184
1
vote
1
answer
53
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Correspondence Lie group/Torus and Lie algebra/CSA
I'm having trouble understanding the relation between Lie groups and Lie algebras.
I'm studying on this article of Kostant, Section 2.1.
So $K$ is compact connected simply connected simple Lie group, $...
- 81
3
votes
1
answer
38
views
How does this definition of Fourier transform in Fulton and Harris 3.32 relate to the usual notion of Fourier transform?
This is exercise 3.32 in Fulton and Harris' Representation Theory: A First Course.
It defines Fourier transform in a form unfamiliar to me, and I could not find any definition of Fourier transform ...
- 41
1
vote
1
answer
56
views
Eigenbundles of a vector bundle
Let $ X $ be a scheme, $ \mathcal{E} $ a locally free sheaf on $ X $ and $ G $ a finite group acting on $ \mathcal{E} $ as endomorphisms (which are necessarily automorphisms) of it. For any character $...
- 2,394
1
vote
1
answer
86
views
Is There a Rotationally Invariant Order of Points on the Sphere $S^2$
Question: Let $S^2$ denote the 2-sphere embedded in $\mathbb{R}^3$. Consider the group $SO(3)$ of rotations acting on $S^2$. Is there a strict total order $<$ on the point set $P\subseteq S^2, \...
- 321
1
vote
1
answer
50
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Trace formula of symmetric representation in Fulton and Harris Representation Theory
I am reading Fulton and Harris' "Representation Theory" and I have
some problems to understand a construction in Chapter §6.1 on pg. 77:
Let $\rho: G \to \text{GL}(V)$ a complex finite ...
- 127
1
vote
0
answers
35
views
Facts about Weyl algebra
I am trying to prove a few things about the first Weyl algebra, $W = k[x,y]/(xy-yx-1)$ over an algebraically closed field $k$ with $char(k)=p>0$. In particular, I am interested in nilpotent ...
- 725
1
vote
1
answer
61
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A PID is a semisimple ring iff it is a field
I am trying to prove that a PID $R$ is a semisimple ring iff it is a field. Clearly any field is semisimple. I am not sure about the converse. By Artin-Wedderburn, $R$ is a product of matrix rings ...
- 1,075
6
votes
0
answers
79
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Non abelian group with some nice properties
Are there examples of finite non-abelian groups of order $k$ where the number of conjugacy classes is $O(\log{k})$? Or is there any reason this is not obviously possible? I know that there lower ...
- 191
2
votes
0
answers
55
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semisimple category and characters
Let $A$ be an algebra and $\mathcal F$ be a category of finite-dimensional $A$-modules over $\mathbb C$ which is completely reducible. Then is it true that in this category the characters determine ...
- 312
2
votes
0
answers
65
views
Symmetric algebra on a graded vector space
Let $G$ be an abelian group and let $V=\oplus_{g \in G}V_g$ be a $G$-graded vector space over $\mathbb C$. Then what is the symmetric algebra on $V$ ? What is a canonical basis for the symmetric ...
- 312
3
votes
2
answers
189
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On a locally-compact group $h(x^{-1})$ constant a.e. $\implies h(x)$ constant a.e. (proof of uniqueness of Haar measure)
I'm trying to understand the final step in a proof of uniqueness of left Haar measures on locally-compact groups as presented in Knapp's "Advanced Real Analysis" (pg. 224-225) or this note ...
- 1,363
1
vote
3
answers
53
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Representation of the Symmetric Group $S_3$
I am reading Representation Theory A First Course by William Fulton and Joe Harris. In Section 3, Lecture 1, they gave a method to find all the irreducible representations of the symmetric group $\...
1
vote
0
answers
40
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Laplace-Beltrami on $SO(3)$
Denoting the double cover map $SU(2)\rightarrow SO(3)$ by $\phi$, we have an induced monomorphism $C^\infty(SO(3))\rightarrow C^\infty(SU(2))$, $f\mapsto f\circ\phi$. We know that the Lie-algebra of ...
1
vote
0
answers
20
views
1-dimensional representations and anti-automorphisms.
Suppose we are in the following setting. Let $A$ be an associative algebra and over a field $\mathbf{K}$, $B\subset A$ a subalgebra and $M$ be a finite dimensional left module of $A$. Let us ...
- 569
7
votes
2
answers
104
views
Does $S_n$ always embed into $GL_{n-1} (\mathbb{F}_p$)?
$S_n$ is the symmetry group of the standard $n-1$-simplex, which is the convex hull of the standard basis vectors in $\mathbb{R}^n$. One can orthogonally project this shape onto the plane $x_1 +...+ ...
- 173
0
votes
1
answer
49
views
Suppose $G$ is a solvable group and $|G|=n$, show that $G$ is isomorphic to a subgroup of group of upper triangular complex invertible matrices. [closed]
I've proved that all subgroups of upper triangular complex invertible matrices is solvable, but I find it too hard to show the inverse proposition.
- 76
1
vote
1
answer
61
views
Modules over the ring $\mathbb{F}_p[C_p]\cong \mathbb{F}_p[X]/(x^p-1)$
I would like to understand the category of modules over the group algebra $\mathbb{F}_p[C_p]\cong\mathbb{F}_p[X]/(x^p-1)$.
I am interested in computing the group cohomology of $C_p$ with coefficient ...
- 1,805
4
votes
0
answers
59
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Exponentiating a representation and Baker–Campbell–Hausdorff formula
Let $G = \{\rho \in \mathrm{Aut}(\mathbb{C}[[t]]) \,|\, \rho(t) \in t + t^2 \mathbb{C}[[t]] \}$ be a subgroup of continuos $\mathbb{C}-$automorphisms of $\mathbb{C}[[t]]$ and $\mathfrak{g} = t^2 \...
- 883
0
votes
2
answers
65
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Definition unramified local representation
We can define unramified representation of $GL(2,\mathbb Q_p$) as a representation $(\pi , V) $ such that $V^K\neq \{0\}$ for $K$ the maximal compact subgroup i.e. $GL(2,\mathbb Z_p)$.
However, I do ...
- 2,865
5
votes
1
answer
87
views
Universality of Hecke algebra of a finite group
I am solving an assignment problem on the Hecke algebra of a finite group, and looking for an idea that might help find a right direction.
Given a pair of finite groups $G\geq K$, the Hecke algebra $\...
- 766
1
vote
1
answer
72
views
Jordan-Holder theorem for group algebras
I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite ...
0
votes
1
answer
32
views
What do Kassel, Rosso, and Turaev mean by "duality"?
In their book "Quantum Groups and Knot Invariants", Kassel, Rosso, and Turaev prove that $U_q\mathfrak{sl}(N+1)$ has a PBW basis. I'm having trouble following the last step, though.
In ...
- 303
0
votes
1
answer
55
views
Can an irrep be a tensor product of representations
Consider a unitary irrep of a compact Lie Group $G$ onto a Hilbert space $\mathcal H$, $\pi\colon G\to\mathcal U(\mathcal H)$. Now assume that $\mathcal H$ can be decomposed into a tensor product, $\...
- 29
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0
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29
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Are there recommended series of online videos or books that are directly related or indirectly related to the textbook? [closed]
I want to learn Group representation theory by self-study and the textbook that I will use is "The symmetric group Representations, Combinatorial, Algorithms and symmetric function" by Bruce ...
- 39
-2
votes
0
answers
41
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general structure theory of associative algebra [closed]
Recently reading literature, there is a very difficult to understand:"$F\left[ ad_{x}\right] $ is a semisimple, finite $F$ algebra and owing to general structure theory,$G$ is a completely ...
- 35
1
vote
0
answers
49
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Reading off module properties from the companion matrix
Let $P\in \mathbb{F}[x]$ be a monic polynomial of degree $n$ over a field $\mathbb{F}$, and $M_P$ its companion matrix. The matrix $M_P$ gives a module of $\mathbb{F}[x]$ on $\mathbb{F}^n$, by letting ...
- 1,075
0
votes
1
answer
33
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Solution to system of nonlinear equations
I am trying to comple the character table of a finite group with seven conjugacy classes $c_1, \cdots, c_7$:
Character table
If I use the orthogonality of the column vectors of table I get a system of ...
1
vote
1
answer
66
views
Why is $\pi(\sigma)$ a well defined mapping $V^{\otimes n} \to V^{\otimes n}$?
Given an $R$-module V, where $R$ is a commutative ring (we even assume $\mathbb{Q}$ is a subring, though I don't think that's required for my particular question), we wish to view the wedge product $V^...
- 163
0
votes
0
answers
44
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Homomorphism of Differentials
Let $D\mathcal{F}_{\alpha}: \text{T}G\vert_{e} \to \text{T}G\vert_{\alpha}$ be the differential of a local diffeomorphism $\mathcal{F}$ of the Lie group $G$, where $\alpha \in G$ and $g \in G$. ...
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6
votes
3
answers
383
views
Has every infinite simple group a faithful irreducible representation?
Has every infinite simple group a faithful irreducible representation?
This question solves the finite case. However, the proof requires a non-trivial linear representation of a finite group. I want ...
- 389
0
votes
0
answers
28
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Group algebra and the classification of simple modules
This is a question from one of our 'ancient' qualification exams. I am not familiar with representation theory, so the second question baffles me a lot.
Let $G=\langle g: g^3=1\rangle$ be the cyclic ...
- 65
1
vote
1
answer
57
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Is the character table of $S_3$ unique?
I'm trying to construct the character table of $S_3$ group.
$n_c$
class
$1$
$\bar{1}$
$2$
1
$I$
1
1
2
3
$(12),(23),(13)$
1
a
b
2
$(123),(132)$
1
c
d
As a brute force method, I imposed every ...
- 13
1
vote
0
answers
83
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Invariants of $\mathfrak{sl}_2 \oplus \mathfrak{sl}_2$.
Let $V_d$ be a $d+1$-dimensional $k$-vector space with $\text{char} k = 0$. Suppose that $V_d$ is an irreducible representation of the Lie algebra $\mathfrak{sl}_2$ and let $k[V_d]^{\mathfrak{sl}_2}$ ...
- 7,711
-1
votes
1
answer
52
views
Confusion regarding matrix representation properties
I am studying about matrix representation of finite groups. If the group is defined as
\begin{equation}
G=\{e,a,b,c,.....\}
\end{equation}
then the matrix representation is defined by the collection ...
- 113
3
votes
0
answers
79
views
Classification of 4 dimensional real associative unital algebra
I think I have a complete list for all the commutative ones, maybe with possible repeats (I did try my best to make sure none are same up to isomorphism):
$\mathbb{R}^4 \simeq \begin{bmatrix}a&0&...
- 193
0
votes
1
answer
91
views
Block diagonal matrices
It is known that any real skew-symmetric matrix is similar to the block diagonal matrix of the form
$$M(\Lambda)=
\begin{pmatrix}
J_1 & & \\
& \ddots & \\
& & J_n
...
- 325
2
votes
0
answers
17
views
Decomposition of the row and the column preserving groups given a young diagram
I have some problems understanding a decomposition used by Fulton and Harris in their book on representation theory (and also Procesi in his Lie group's book).
Given a partition of $d$, i.e. a vector $...
- 105
3
votes
1
answer
37
views
Number of copies of irreducible representations for transitive actions
Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then $|X| \le |G|$ since $|X| \big| |G|$ by the orbit-stabilizer theorem. Let $\pi_i;\ \ 1\le i \le m$ be irreducible ...
- 1,953
2
votes
1
answer
41
views
Fulton Harris Lemma A.28
I am trying to understand the proof of the lemma A.28 in the representation theory book by Fulton and Harris. Let $x_1,\ldots, x_k$ and $y_1,\ldots, y_k$ two distinct indipendent variables. Denote $...
- 105
0
votes
0
answers
26
views
Uniqueness of characters of transitive actions
Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then, $|X| \big| |G|$ by the orbit-stabilizer theorem. Let this action induce a permutation representation $\rho: G \to ...
- 1,953
2
votes
1
answer
37
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2nd order Commutant of permutation matrices
Let $d\ge 2$ denote an integer, let $S_d$ denote the group of permutations of $d$ elements, and let $L(\mathbf{C}^d)$ denote the space of linear operators on the Hilbert space of $d$-dimensional ...
- 41
0
votes
1
answer
39
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For a compact group $G$ and any finite-dimensional complex representation $(\pi,V)$, how to show that $V$ admits an invariant inner product?
To demonstrate that an inner product $\langle\ ,\ \rangle$ on $V$ is invariant, one need to check
$$\langle\pi(g)v,\pi(g)w\rangle=\langle v,w\rangle.$$
For an arbitrary inner product $\langle\langle\ ,...
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4
votes
1
answer
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Methods for Finding A Representative of Each Element in a Quotient Set $S/G$
Question:
Let $G$ be a group and $S$ be a $G$-set. I am interested in finding a representative for each orbit in the quotient set $S/G$, especially in cases with an infinite number of orbits, i.e., ...
- 321
3
votes
2
answers
106
views
Finite-dimensional faithful representation of a matrix algebra is completely reducible
I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have somequestions about the finite representations of matrix algebras.
It was said that any finite-...
0
votes
0
answers
25
views
What is split semisimple algebra over field k
I can't find the definition of this terminology. Actually, there is a similar question 'what is the split k-algebra?'. But the notes in the answer is unavailable and I can't add comments somehow. So I ...
3
votes
1
answer
38
views
Noetherian, self-injective ring $R$ with non-torsionless $R$-module
Let $R$ be an (associative, unital) ring. By an $R$-module we mean a left $R$-module. We call an $R$-module torsionless if it can be embedded into a direct product of the regular $R$-module $R$.
I am ...
- 1,075