Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
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Image of the projection map onto an irreducible module
Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
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dimension of Irreducible G modules
How to show that dim(W) divides order(G), where W is an irreducible G module. Let d and n be dimension of W and order(G) respectively. I want to show that n/d satisfies a monic polynomial over Z( ...
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2answers
37 views
Group ring is not isomorphic to 2 by 2 matrices
Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the
k-algebra of 2 by 2 matrices over k, is not isomorphic to the group ring of any finite group G
over k.
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2answers
101 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.
Suppose $G$ is an abelian group and $a\in G$ and
$$f:\left<a \right>\to\Bbb T$$
is a homomorphism. Can $f$ be extended to a homomorphism on $G$:
$$g:G\to \Bbb T$$
?
$\Bbb T$ is the circle ...
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2answers
106 views
Algebraic geometry in representation theory?
I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
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0answers
29 views
Character of half-spin representation
Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum
$$\sum x_1^{\pm ...
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0answers
33 views
Computing Invariant Subspaces of Matrix Groups
Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?
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1answer
256 views
Do these two sets of matrices form groups?
Stimulated by some Physics backgrounds, consider the following two sets of matrices.
Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
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1answer
32 views
properties of Sym^2 vector subspace/properties of tensor products
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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0answers
22 views
Laplacian on Reductive coset spaces
Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$) and let $X_i$ denote the vector fields which generate rotations about the $x_i$-axis. My questions are:
(a) Is it true that ...
3
votes
0answers
18 views
Harish-Chandra modules of $PSL_2(\mathbb{R})$
Let $G=PSL_2(\mathbb{R})$ and $K$ a maximal torus.
Is the category of Harish-Chandra modules of $(G,K)$ equivalent to the Category of Harish-Chandra Modules of $SL_2(\mathbb{R})$ with even $K$-types?
...
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1answer
14 views
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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1answer
31 views
$\operatorname{Res}(V+W)=\operatorname{Res}(V)+\operatorname{Res}(W)$?
if there are two $R[G]$ Modules $V,W$ and $R$ some ring, $S$ subgroup of $G$. Is the formula $$\operatorname{Res}_S (V \oplus W) = \operatorname{Res}_S (V) \oplus \operatorname{Res}_S (W) $$
true? I ...
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1answer
41 views
Representations - Tensor Product prove properties of tensor product
I have a problem:
Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
3
votes
1answer
39 views
Braid Group of a Weyl Group
I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman.
The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of ...
3
votes
3answers
114 views
Are Clifford groups very *non-commutative*?
Clifford groups seem to be very non-commutative by the relation \begin{equation}
\gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}.
\end{equation} But is it really so? Can we put this degree of ...
4
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0answers
56 views
Deciding whether or not a class of modules is “big enough”
For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
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0answers
19 views
Computing a restriction of a representation
It is known (Fulton and Harris p.427 among other papers) that the restriction of $\mathrm{GL}_n$ to $\mathrm{O}_n$ yields the following branching rule
$$
...
3
votes
0answers
35 views
Duality of $Z(G)$ and $[G,G]$ in representation?
This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group.
I was thinking about its manifestation in group ...
3
votes
2answers
53 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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1answer
40 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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1answer
23 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} ...
3
votes
1answer
33 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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vote
1answer
46 views
Weights versus roots
I am not sure of the difference between weights and roots. Am I correct in thinking that the weights are the eigenvalues of the action of the maximal torus on a given representation, and the roots are ...
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votes
0answers
33 views
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$ ...
1
vote
2answers
27 views
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, ...
1
vote
1answer
22 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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1answer
53 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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0answers
23 views
Branching rule restriction to $\mathrm{O}_9 \Bbb C$ from $\mathrm{GL}_9 \Bbb C$
On page 427 of Fulton and Harris's Representation Theory, the authors give the branching rule for the above restriction as
$$
\mathrm{Res}_{\mathrm O_m \Bbb C}^{\mathrm{GL}_m \Bbb C} (\Gamma_\lambda) ...
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votes
2answers
106 views
Why are (representations of ) quivers such a big deal?
Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
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0answers
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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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30 views
$Sym^n(V\otimes U) = \oplus_{\rho \vdash n}V(\rho)\otimes U(\rho)$
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 ...
2
votes
1answer
42 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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1answer
33 views
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 ...
4
votes
1answer
59 views
What are all representations of the quiver $Q$?
Let a quiver be $Q=(Q_0, Q_1, s, t)$, where $Q_0$ is $\{1, 2\}$. The quiver has only one arrow: $\alpha: 2 \to 2$. What are all representations of $Q$? Thank you very much.
In my original ...
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votes
1answer
81 views
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 ...
3
votes
1answer
29 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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votes
0answers
18 views
Are $K$-finite vectors dense in irreducible Banach representations?
Let $\pi$ be a continuous irreducible representation of $G:=\mathrm{SL}(2,\mathbb{R})$ in a Banach space $H$, and $\pi^1$ the representation of $\mathcal{C}_c(G)$ induced by $\pi$. Suppose ...
3
votes
1answer
27 views
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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Give a bijection between unitary, degree one representations of Z and elements of T.
Definition: My book defines $\mathbb{T}$ as the unit circle in $\mathbb{C}$, i.e. $\mathbb{T}=\{z \in \mathbb{C} : |z| = 1\}$
I'm trying to answer this question: "Give a bijection between unitary, ...
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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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1answer
189 views
What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?
The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
3
votes
1answer
61 views
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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votes
0answers
12 views
Symmetry-adapted normal vibrations (the hard way)
It is my daily work as chemist to compute the number of normal vibrations of e.g. a C5H5- ring in its appropriate symmetry (here: D5h). I won't pester you with such trivialities, my professor even ...
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1answer
46 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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2answers
72 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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0answers
65 views
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 ...
2
votes
2answers
72 views
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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1answer
49 views
For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?
I'm reading the Atiyah-MacDonald book on Commutative Algebra.
At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is:
$G$ = finite group, ...
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1answer
29 views
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 ?
