Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

learn more… | top users | synonyms (1)

0
votes
0answers
31 views

Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
1
vote
0answers
10 views

A $*$-closed algebra of compact operators is completely reducible

In page 13 of Lang's $SL_2$ there is a proof that for a $*$-closed algebra $\mathscr A$ of compact operators on a Hilbert space $H$, $H$ is completely reducible. The proof follows by taking the ...
1
vote
0answers
15 views

Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
1
vote
0answers
21 views

Special case of Pieri-Rule

is there an "elementary" (read: short combinatorial) proof for the rule $$ s_\lambda \cdot s_{(1)} = \sum_{\mu} s_{\mu} $$ where $\mu$ ranges over all partitions obtained from $\lambda$ by adding a ...
2
votes
0answers
88 views

Conjugacy class $A_4$

I want to find all conjugacy classes of $A_4$. So basically what I did, I took all elements of $A_4$ and calculated their conjugates. I had no problems with $$\{e\}, \{(123),(134),(142),(243)\}, \{(...
0
votes
0answers
20 views

representations of SL_n(R) or SL_n(C)

Could someone point me to a reference for the finite dimensional representation theory of $G=SL_n(F)$ where $F = \mathbb{R}$ or $\mathbb{C}$? In particular, I want to know what this "highest weight" ...
2
votes
1answer
29 views

Is $\operatorname{Stab}(\lambda)$ generated by the simple reflections it contains, for $\lambda\in A_0$?

For a finite Weyl group, the stabilizer of an element in the fundamental domain is generated by the simple reflections of the Weyl group that is contains. Does the same still hold for the closure of ...
0
votes
0answers
9 views

Irreducible representations of locally compact semigroups

What class of semigroups have finite dimensional irreducible representations. For example, If we have a compact groups $G$ then every continuous irreducible representation of $G$ is finite ...
-2
votes
0answers
39 views

Writing some algebras by generators and relation [on hold]

I want to write the following algebras by generators and relations: $$\mathbb{C}; \mathbb{C}^{2}; M_{2}(\mathbb{C}); L^{\infty}( O(2)/C_{k}) \;\text{and}\; L^{\infty}( O(2)/D_{k})$$ where $C_{k}$ and $...
1
vote
3answers
450 views

Problem 1.24, Introduction to representation theory, Etingof

Let $k$ be a field and $n$ and $N$ be two nonnegative integers. Let $A = k[x_1, \ldots, x_n]$, and let $I \neq A$ be any ideal in $A$ containing all homogeneous polynomials of degree $\geq N$. Show ...
1
vote
1answer
45 views

In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?

In this book I read Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear ...
1
vote
0answers
37 views

Reference Request: Lie Theory For Quantum Field Theory

I have encountered the section on non-Abelian gauge theories in Peskin and Schroeder's QFT textbook, and although I am comfortable with the derivation of the Yang-Mills Lagrangian they present, the ...
1
vote
0answers
22 views

(B,N) pair and Steinberg idempotent

Let $q=p^f$ where $p$ is prime and $G$ be a finite group with a $(B,N)−$pair ($T=B\cap N$ and $W=N/T$), and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Define $$e=\dfrac{1}{[G:U]}\...
0
votes
1answer
26 views

(B,N) pair and normal subgroup

I am trying to prove the following: Let $G$ be a finite group with a $(B,N)-$pair and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Let $\widetilde{G}\triangleleft G$ such that $U\le \...
0
votes
0answers
11 views

Tannaka Krein duality for finite groups, explicit

Tannaka-Krein duality theory says that the natural mapping $G\rightarrow Aut^{\otimes}(F)$ (see http://mathoverflow.net/questions/155743/can-one-explain-tannaka-krein-duality-for-a-finite-group-to-a-...
5
votes
3answers
733 views

Two dimensional complex group representations

Michael Artin's Algebra, chapter 10 (both unstarred, and complex representations) M.8 Prove that a finite simple group that is not of prime order has no nontrivial representation of dimension 2. ...
2
votes
2answers
52 views

what does $\ltimes$ in the context of representation theory mean?

I am considering the following sentence wich is part of a theorem: '' Let $V$ be a finite dimensional unitary representation of $H=\mathbb{Z}^{2} \rtimes $ SL$_2(\mathbb{Z})$." I have no background ...
5
votes
3answers
310 views

Why is this paragraph so short?

$G$ is a connected, reductive linear algebraic group.The reference is Springer, Linear Algebraic Groups. I am having trouble making sense out of anything in this paragraph. Proposition 7.31(ii) ...
2
votes
0answers
16 views

Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
0
votes
0answers
23 views

Irreducible complex continuous unitary finite dimensional representations of SO(2)

I have to find all continuous finite dimensional complex Irreducible and unitary representations of $SO(2)$. I know that every element of $SO (2)$ can be written as $exp(J \theta ) $, where $\theta$ ...
1
vote
0answers
43 views

Determinant of a character

let two characters $\chi$ and $\vartheta$ of a finite group $G$ (assumed to be non-null). Let $\mathfrak{X}$ and $\mathfrak{Y}$ be representations of $G$ affording respectively $\chi$ and $\vartheta$ ...
2
votes
0answers
28 views

Permutations associated to a transversal and Cayley theorem

Let $G$ be a finite group with $H\le G$ and $T$ a right transversal of $H$ in $G$. $G$ acts on itself by left multiplication and so we can consider $G\le \mathfrak{S}_G$. Let $g\in G$. The permutation ...
1
vote
0answers
11 views

How to show that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding?

Let $H$ be a bialgebra and ${}_H^H YD$ the category of Yetter-Drinfeld modules over $H$. It is said that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding. ...
0
votes
0answers
16 views

Let $G=AB$ where $(|A|,|B|)=1$ and $V$ be an $\mathbb{F}[G]$ module.

Under these assumptions it is a well-known fact that if $V_A$ and $V_B$ are faithful ($V_A$ denotes $V$ as an $\mathbb{F}[A]$-module) then $V$ is also faithful. Clearly if $V_A$ and $V_B$ is ...
1
vote
0answers
16 views

Intersection of the kernel of the irreducible characters determinants

Let $G$ be a finite group. It is easy to show that $G'\le \bigcap_{\chi\in Irr(G)}Kerdet\chi$. Is there equality ? This question arises from the remarkable equalities $\bigcap_{\chi\in Irr(G)}Ker\...
2
votes
0answers
120 views
+50

representation of a group and its center

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \...
2
votes
0answers
35 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
1
vote
1answer
26 views

Finite group representation on endomorphism ring

Let $\rho:G\to\mbox{GL}(V)$ be a finite dimensional representation of a finite group $G$. We can assume the base field is $\mathbb{Q}$, but it doesn't really matter. Then we also obtain a ...
1
vote
0answers
16 views

Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
1
vote
0answers
31 views

The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
0
votes
0answers
8 views

Irreducible tensor for fundamental representation of SU(N=3)

I am trying to calculate the singlets of the tensor product $N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^*$. I know that $N_c \otimes N_c^*=1\oplus (...
12
votes
5answers
451 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 ...
3
votes
0answers
33 views

What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
1
vote
1answer
42 views

The inverse of the braiding $c: V \otimes W \to W \otimes V$.

In the article. It is said that the inverse of the map $$ {\displaystyle c_{V,W}:V\otimes W\to W\otimes V}, \\ {\displaystyle c(v\otimes w):=v_{(-1)}{\boldsymbol {.}}w\otimes v_{(0)},} $$ is $$ {\...
1
vote
1answer
70 views

Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. $$V=\underbrace{\mathbb KG\oplus\cdots\oplus \...
2
votes
1answer
44 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
0
votes
0answers
38 views

Character group and lattices

Let $\Lambda$ be the complex n-th dimensional lattice over Eisenstein integers ($\mathbb{Z}[\omega]$)). The map $R: \mathbb{C} \mapsto \mathbb{R}$ is defined as following: $R(z)=R(z_{a}+\omega z_b)=...
0
votes
0answers
26 views

Basis of weight lattice in terms of root lattice

Let $\Lambda_R = \bigoplus_{i \in I} \mathbb{Z} \cdot \alpha_i$ be the root lattice of a root system $\Phi$ with simple roots $\alpha_i$ and let $\Lambda_W$ denote the corresponding weight lattice. ...
0
votes
0answers
30 views

Symmetric power bunde of half spinor representation

Can anyone give me a reference for understanding $$\Lambda^2_{+c}\cong S^2V_+$$ where $\Lambda^2_{+c}$ is the complexified bundle of self-dual two forms and $S^2V_+$ is the symmetric power bundle of ...
1
vote
1answer
25 views

Faithful monomial representation induced from faithful character

Let $\rho: G \rightarrow GL_n(\mathbb{C})$ be a faithful irreducible representation such that $\rho = Ind_N^G \phi$ for some 1-dimensional representation $\phi$ and normal subgroup $N$. Does $\phi$ ...
1
vote
2answers
30 views

If a finite group has only 1D irreducible representations, is it abelian? [duplicate]

I know abelian groups have only 1D representations. Is the converse proposition true? i.e. If a finite group has only 1D irreducible representations, is it abelian?
0
votes
0answers
24 views

Any theory that relates a group's representation to its subgroup's representation?

I have encountered a problem concerning a finite group $G$ and its subgroup $H$. $G$ has one more generator than $H$. I calculated dimensions of irreducible representations of $G$ and $H$ and want to ...
3
votes
1answer
50 views

Symmetric decompositions of $SU(2)$ representations.

Let us consider the representation theory of $SU(2)$. There is a unique irreducible representation of dimension $n$ for each $n \ge 1$, which we will denote $\mathbf{n}$, with the defining $2$-...
1
vote
1answer
39 views

Irreps of products between dihedral group and any finite group

Let $D_n$ be the dihedral group with order $2 n$. The total number of irreducible representations for $D_n$ is as follows. When $n$ is even, the total number is $\frac{n-2}{2} + 4 = \frac{n}{2} + 3$. ...
1
vote
1answer
58 views

Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
2
votes
0answers
59 views

Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
2
votes
1answer
22 views

Representation of diedral group $D_8$, why $\rho(a)^2=1$ if $a$ is the rotation?

I recall that $D_8=<a,b\mid a^4=b^2=1, bab=a^3>$. I have to determine all representation $\rho:D_8\longrightarrow \mathbb C^*$ of degree 1 of $D_8$. In my course it's written that since $\rho(a)^...
20
votes
5answers
359 views

Show that : $\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2= 2 n!$

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
2
votes
1answer
59 views

A question on Auslander-Bridger transpose

I am learning Auslander-Reiten Theory. When I read the book Frobenius Algebras I. Basic Representation Theory, I have some problems. On page 236-237, there is the following Proposition 4.5. Let $...
1
vote
1answer
41 views

Expressing $\mathbb{C}^3$ as a direct sum

$S_3$ acts on $\lbrace 1,2,3 \rbrace$, so this affords a homomorphism $S_3\to GL_3(\mathbb{C})$ (acting on $\mathbb{C}^3$). I showed the only vector fixed by the action of $S_3$ is zero. Find two ...