Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
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16 views
Question about an almost split sequence.
On page 124 of the book Elements of representation theory of associative algebras, volume 1, Example 3.10, I computed the modules in this example.
$$
S(3)=0\leftarrow 0 \rightarrow K \leftarrow 0, ...
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0answers
20 views
Is $D=\hom_K(\cdot, K)$ exact?
Let $K$ be an algebraically closed field. On page 31 of the book Elements of representation theory of associative algebras, volume 1, from Theorem 5.13 (a) we see that $D=\hom_K(\cdot, K)$ is exact. ...
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0answers
13 views
Question about the decomposability of the radical.
Let $A$ be an algebra over an algebraically closed field $K$ and $M$ an indecomposable $A$-module. Suppose that $M$ is indecomposable. Can we conclude that $\operatorname{rad}M$ is indecomposable? ...
1
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0answers
15 views
Questions about socles and radicals.
Let $M$ be a module of a finite dimensional algebra $A$ over an algebraically closed field $K$. Let $N=M/\operatorname{rad}M$ be the top of $M$ and suppose that $N$ is simple. Let $D=\hom_K(\cdot, ...
3
votes
1answer
28 views
Reference request: Auslander-Reiten quivers of which types of algebras have been found?
I would like to know Auslander-Reiten quivers of which types of algebras have been found.
For quantum groups, there is a paper http://arxiv.org/abs/1202.1714 by Julian Külshammer. I search the ...
2
votes
0answers
13 views
Partial cycles in projective resolutions of square-free algebra
Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand?
I suspect not, but have not ...
1
vote
0answers
18 views
A technical problem on constructing smooth vectors in a representation
Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$, and $\pi:G\to\mathrm{GL}(H)$ a continuous representation of $G$ in a Banach space $H$. Let $\pi^1:\mathcal{C}_c(G)\to\mathrm{End}(H)$ be ...
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votes
0answers
21 views
Representation of even/odd type
I have seen in my reading in Dynkin's Maximal Subgroups of the Classical Groups that an irreducible representation $\phi$ of $A_n, D_{2k+1},$ or $E_6$ has a bilinear invariant if and only if the ...
4
votes
1answer
53 views
Why is Lie derivative smooth?
Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$. Suppose $X\in\mathfrak{g}$ and $f:G\to\mathbb{R}$ is smooth. The Lie derivative of $f$ with respect to $X$ is the function ...
4
votes
1answer
83 views
How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?
Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
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1answer
19 views
Invariant hermitian forms and irreducible representations
Let $V$ be a vector space over $\mathbb{C}$ of finite dimension $n$, $G$ is a finite group and $T:G\rightarrow GL(V)$ its irreducible representation that sends each $g$ into $T_g$.
Let $E:V^{\bigoplus ...
4
votes
1answer
45 views
Group homomorphisms into a field
Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
6
votes
1answer
105 views
+250
Decompose $P$ into the direct sum of irreducible representations.
Note: I need help with part (c).
Consider the representation $P: S_3 \rightarrow GL_3$ where $P_{\sigma}$ is the permutation matrix associated to $\sigma$.
a) Determine the character $\chi_P : S_3 ...
7
votes
1answer
41 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
42 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 ...
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votes
1answer
23 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
58 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
38 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 ...
1
vote
1answer
26 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 ...
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0answers
20 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
40 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 ...
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votes
1answer
37 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
63 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
21 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
31 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
37 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
26 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
92 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
54 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
47 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
46 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
115 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
40 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 ...
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votes
1answer
40 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
44 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
58 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
42 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
51 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.
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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
53 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
19 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
186 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
26 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
59 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
23 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.
