Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
6
votes
1answer
35 views
Specific projective dimension of a module over bound quiver
Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver
$$\require{AMScd}
\begin{CD}
1 @>>> 2\\
@V{}VV @V{}VV \\
3 @>>> 4 @>>> 5
...
2
votes
1answer
34 views
Does an irreducible $\mathbb CG$-module have a basis of the form $u,ug_1,\dots,ug_n$?
Suppose that $U$ is an irreducible $\mathbb CG$-module and $u\in U$. Let $\operatorname{span}(u_1,\dots,u_k)$ denotes the linear span of vectors $u_1,\dots,u_k\in U$.
I was thinking along these ...
0
votes
0answers
12 views
Complex representation and Dual representation notation
Let's say we have a representation $\rho$ of $G$ on a vector space $V$. Wikipedia refers to the dual representation as $V^*$, but the dual vector space as $\overline{V}$. It does the opposite for the ...
4
votes
1answer
52 views
How to compute Nakayama functor explicitly?
I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
2
votes
2answers
37 views
On the eigenvalues of a linear transformation $\tau$ such that $\tau^3 = \mathrm{id}$
I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph.
[..] we will start our analysis of an arbitrary representation $W$ of $S_3$ by looking ...
0
votes
0answers
16 views
Symmetry of Plancherel measure (for $S_n$)
For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$:
$$
\begin{pmatrix}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 ...
1
vote
0answers
19 views
How to write down the maximal subgroups of $GL(9, \mathbb{C})$
I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
2
votes
2answers
38 views
On the proof of Schur's lemma in Fulton & Harris
I'm reading the book on representation theory by Fulton and Harris. I'm stuck with the proof of Schur's Lemma 1.7:
Schur's lemma 1.7 If $V$ and $W$ are irreducible representation of $G$ and ...
0
votes
1answer
35 views
Are all of the irreducible representations of (any) symmetric group over $\mathbb{C}$ also irreducible over a finite splitting field.
This is probably an incredibly stupid question, but I'm a novice to representation theory and finite field theory, as I've just been introduced to these concepts, so all I really need is confirmation ...
3
votes
3answers
59 views
Book recommendation for associative algebras
Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment.
...
3
votes
2answers
51 views
$\mathbb{C}[G]$-module homomorphism on finite dimensional modules and finite groups
Nice to meet you folks! I'm currently a grad student reviewing some representation theory of finite groups for prelims next year, and I'm stuck proving a simple statement. Translating the question ...
1
vote
1answer
18 views
write representation as sum of irreducible representations
Given the representation $\rho: \mathbb{Z}/3\mathbb{Z} \rightarrow GL_2(\mathbb{C})$ by $1\rightarrow \left( \begin{array}{ccc}
-1 & -1 \\
1 & 0\\
\end{array} \right)$. I have to write this ...
1
vote
0answers
28 views
Representation of Homogeneous vectorbundle = Induced representation
Hello friends of mathematics :)
I have a question about the induced representation. Suppose $G$ is a group and $H$ a subgroup of $G$. Suppose $\rho$ is a representation of $H$ on the vectorspace $V$, ...
1
vote
2answers
36 views
Group action on vector space of all functions G to $\mathbb{C}$
I have a simple question about this following action:
Let $L(G)$ be the vector space of all functions from $G$ to $\mathbb{C}$. Define an action of $G$ on $L(G)$ by
$$(\sigma f)(\tau) = f(\sigma ...
1
vote
0answers
24 views
Sum of squares of the degrees of irreducible representations equals order of group (positive characteristic case) [duplicate]
Suppose $K$ is a splitting field for a finite group $G$ such that $p = \mathrm{char} K >0$ and $p \nmid |G|$. Let $\{\rho_1, \ldots, \rho_s\}$ be the set of all irreducible representations (up to ...
11
votes
1answer
91 views
Introduction to the trace formula for people outside number theory
I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula.
I am aware that there are "standard references" on the ...
4
votes
0answers
50 views
Galois representations and normal bases
I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
1
vote
0answers
45 views
What to take from representation of $S_d$?
I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
2
votes
1answer
45 views
Submodules of tensor representations
Let $V$ be a finite dimensional vector space over a field and $T$ the tensor algebra $T=\bigoplus_{n\geq 0} T_n,$ where $T_n=V^{\otimes n}$. It's easy to see that $T$ can be viewed as a ...
10
votes
2answers
112 views
Path Algebra for Categories
For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
1
vote
1answer
39 views
Endomorphisms of Simple A-modules where A is a Complex algebra
Suppose $\underset{=}{\phi} \in End_A S$ is an isomorphism and $S$ is a simple (finite-dimensional?) $A$-module and $A$ is a simple $\mathbb C$-algebra. Then... must we have ...
5
votes
1answer
36 views
Connection between $\mathbb{Q}_p[G]$ and $\mathbb{Z}_p[G]$
In this post there was the comment, that having $\mathbb{Q}_p[G]$ modules, it is possible to construct $\mathbb{Z}_p[G]$ modules. How is it possible to find out when there is a bijection between ...
0
votes
1answer
38 views
Martin Isaacs's exercise 3.7 (character theory of finite groups)
I would need some help with this exercise:
Let $\chi\in{Irr(G)}$ be faithful, and suppose $\chi(1)=p^a$ for some prime p.
Let $P\in{Syl_{p}(G)}$, and suppose that $C_{G}(P)\nsubseteq{P}$. Show that ...
0
votes
0answers
42 views
Martin Isaacs's exercise 3.4 (character theory of finite groups)
I need some help with this:
Let $G$ be a simple group and suppose $\chi\in{Irr(G)}$ with $\chi(1)=p$, a prime.
Show that a Sylow $p-$subgroup of G has order p.
Thanks a lot in advance.
8
votes
0answers
55 views
Expression of basis vectors of permutation modules in different bases.
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
1
vote
1answer
39 views
Martin Isaacs's exercise 3.6 (character theory of finite groups)
I'm trying to solve this exercise, can anyone help me?
Let $G$ be a p-group, and suppose $\chi\in{Irr(G)}$. Show that $\chi(1)^2$ divides $|G:Z(\chi)|$
Thanks a lot.
1
vote
1answer
45 views
Martin Isaacs's exercise 3.5 (character theory of finite groups)
I need some help with this exercise:
Suppose $A\subseteq{G}$ is abelian, and $|G:A|$ is a prime power. Show that $G'\lt{G}$
Thank you very much in advance.
0
votes
1answer
45 views
Specific question on Sn modules
Let $L_{-1}$ denote the 1-dimensional sign-representation of the symmetric
group $S_n$ and V the standard $(n - 1)$-dimensional module for $S_n$. How to prove that V and $V \otimes L_{-1}$
are not ...
3
votes
3answers
52 views
Finding all submodules of G-modules
Let V; W be irreducible G-modules that are not isomorphic to each other.
How to prove that the only G-submodules of M:= $V \oplus W$, other than $0$ and M itself, are $V =
V \oplus 0$
and $W =
0 ...
3
votes
1answer
18 views
multiplicity of irreducible components of S3 modules
Let V denote the 2 dimensional irreducible standard module for $S_3$. I want to find multiplicity of each of irreducible components of $V^{\otimes ^{10}}$ , by writing the character for $V^{\otimes ...
10
votes
5answers
185 views
Applications of Character Theory
Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
4
votes
0answers
25 views
Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$
If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
2
votes
1answer
58 views
Question about minimal projective presentations of a module.
I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 .
On page 108, line 11-14, there is a claim:
If $P_0^{t}\to P_1^{t} \to TrM \to 0$ is not a minimal ...
3
votes
1answer
37 views
Does the projectively stable category have projective modules?
I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . On page 109, the projectively stable category is defined by $$ \underline{mod} A = mod A/\mathcal{P}. $$ ...
1
vote
1answer
34 views
Image of the projection map onto an irreducible module
Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
-2
votes
0answers
22 views
dimension of Irreducible G modules
How to show that dim(W) divides order(G), where W is an irreducible G module. Let d and n be dimension of W and order(G) respectively. I want to show that n/d satisfies a monic polynomial over Z( ...
2
votes
2answers
36 views
Group ring is not isomorphic to 2 by 2 matrices
Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the
k-algebra of 2 by 2 matrices over k, is not isomorphic to the group ring of any finite group G
over k.
2
votes
2answers
91 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.
Suppose $G$ is an abelian group and $a\in G$ and
$$f:\left<a \right>\to\Bbb T$$
is a homomorphism. Can $f$ be extended to a homomorphism on $G$:
$$g:G\to \Bbb T$$
?
$\Bbb T$ is the circle ...
7
votes
2answers
101 views
Algebraic geometry in representation theory?
I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
2
votes
0answers
29 views
Character of half-spin representation
Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum
$$\sum x_1^{\pm ...
1
vote
0answers
32 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?
7
votes
1answer
241 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 ...
2
votes
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 ...
0
votes
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?
...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
36 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
37 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
113 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 ...
