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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.

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Borel condition vs strong continuity in unitary representations of finite dimensional Lie groups

I am trying to find a completely rigorous treatment of strongly continuous projective unitary representations of (analytic) Lie groups on separable Hilbert spaces in full generality. Up to my ...
ProphetX's user avatar
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39 views

Serre's Representation Theory exercise 7.3(d)

I am trying to solve Exercise 7.3(d) in Serre's Linear Representation of Finite Groups. I have solved all other parts. The important points of where I am stuck at boils down to the following facts. (I ...
Mohit Kumar's user avatar
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43 views

Mapping between vectors in irreducible Sp-representation

Let $V$ be the standard Sp-representation with symplectic basis $\{ a_i, b_i \}$. I believe the vector $(b_1 \wedge b_2) \otimes (b_1 \wedge b_2 \wedge a_3 \wedge a_4)$ lies inside the irreducible ...
Chase's user avatar
  • 326
2 votes
1 answer
46 views

Does the formal character determine the representation?

Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$. Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
Minkowski's user avatar
  • 1,562
1 vote
1 answer
72 views

"Linear independency" of Lie Brackets

I was watching this eigenchris video. At 21:49, he says: $$[g_i, g_j]=\sum_k {f_{ij}}^{k}g_k$$ for $\mathfrak{so}(3)$. Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What about ...
Cro's user avatar
  • 113
-1 votes
0 answers
37 views

Complexification of complex Lie algebras like $\mathfrak{su}(2)$

I'm reading Brian Hall's book on Lie theory. He defines the complexification $V_{\mathbb{C}}$ of a real vector space $V$ as the linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. Next, he proceeds ...
Gabriela Martins's user avatar
2 votes
0 answers
38 views

Prime ideals dividing the Artin conductor

Let $L/K$ be a Galois extension of number fields, and let $(\phi,V)$ be a representation of $\operatorname{Gal}(L/K)$. Let $\mathfrak{f}_\phi$ be the Artin conductor of this representation, which is ...
Sardines's user avatar
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2 votes
1 answer
53 views

Real Commutant Algebra of a Set of Matrices

Suppose I have a collection of $N\times N$ real, symmetric matrices $R_1, R_2, \dots$ and I want to find their orthogonal commutant---that is, the group of real, orthogonal matrices that commute with ...
Matt Mitchell's user avatar
1 vote
1 answer
33 views

Number of left ideals in the simple components of groups algebras

Let $G$ be a finite group and $K$ a field with characteristic zero. Suppose $G$ has $m$ irreducible $K$-representation $W_i$ with character $\chi_i$. $KG$ is semisimple algebra, and $$KG=KGe_1 \times \...
khashayar's user avatar
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1 vote
1 answer
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Generating a conjugate representation of an irreducible self-conjugate representation of $S_n$

Suppose we have a complex matrix representation $\Gamma_{ij}^\sigma \in \mathbb{C}^{d \times d}$ of dimension $d$ for the permutations $\sigma$ of the group $S_n$ of permutations of $n$ objects. ...
creillyucla's user avatar
1 vote
0 answers
20 views

Projective representations that are compatible with a group automorphism

We know that projective representations of a group $G$, $\rho(g) \rho(h) = e^{i\omega(g,h)} \rho(gh)$, is classified by the second group cohomology $H^2(G,U(1))$. Let us now specify a group ...
Learner's user avatar
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1 answer
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Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants [closed]

Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants (line with equation $y = x$), and let $b$ be the reflection of the same plane over the bisector of the even ...
Markus's user avatar
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2 answers
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The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

SETUP. It is a standard result that $\text{GL}(n,\Bbb{R})/O(n)$ is isomorphic to the set $P'$ of positive definite $n\times n$ matrices, as manifolds: the basic idea is that $\text{GL}(n,\Bbb{R})$ ...
SomeCallMeTim's user avatar
5 votes
1 answer
168 views

Working with character tables

I am currently a bit stumped by an old exam question, which gives a character table and wants me to deduce properties of the group: What is the order of $g$? Show that $g \notin C_G(G)$ Show there ...
Very Interesting's user avatar
2 votes
3 answers
115 views

Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?

Suppose $G$ is a finite group, and $V$ is a complex vector space. It is often said that linear representations $\rho:G\to\DeclareMathOperator{\GL}{GL}\GL(V)$ correspond to modules over $\mathbb C[G]$. ...
Joe's user avatar
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3 votes
0 answers
45 views

What is the intuition for representations of the symmetric group?

What is the (physical) intuition for representations of the symmetric group? In particular, matrix coefficients of the Fourier coefficients corresponding to a representation. For the cyclic group $C_n$...
Jackson Walters's user avatar
3 votes
1 answer
65 views

Grothendieck ring of $Rep(\mathfrak{sl}_2)$

The Grothendieck ring of the abelian category $Rep(\mathfrak{sl}_2)$ of finite-dimensional representations of $\mathfrak{sl}_2$ is, according to Bakalov-Kirillov's Lecture notes on tensor categories ...
Minkowski's user avatar
  • 1,562
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0 answers
22 views

If $f$ is $K$-finite, then $Xf$ is also $K$-finite

This problem comes form bump, Automorphic forms and representations, p,300. It's about the ''K-finiteness'' conditions in the definition of the automorphic forms of $GL(2,A)$, where A is the adele ...
wwwwww's user avatar
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0 answers
39 views

Group of Characters of $\text{SL}_{n+1}$

We work over $\mathbb{C}$. Let $G=\text{SL}_{n+1}$ be the algebraic group of matrices with determinant $1$. Let $\text{M}(G)$ be the group of characters of $G$, that is the group of morphisms of ...
konoa's user avatar
  • 384
2 votes
1 answer
77 views

Irreps of $SU(3)/\mathbb{Z}_3$ from irreps of $SU(3)$

I'm reading Ernest Vinberg's Linear Representations of Groups. He has a nice example linking $SU(2)$ and $SO(3)$ by considering the action of $SU(2)$ by conjugation on the set of $2$ by $2$ Hermitian ...
user196574's user avatar
  • 1,846
0 votes
1 answer
30 views

On Frobenius–Schur indicator of real/complex representations

Let $G$ be a finite group with complex irreps $W_i$. Let $V$ be a real irrep of $G$. Denote $\chi_{W_i}$ and $\chi_{V}$ the corresponding characters. Each $V$ has three possibilities: Case 1: $\dim_{\...
khashayar's user avatar
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2 votes
1 answer
66 views

For two irreducible modules $V$ and $W$, $f : V\to W$ is a $G$-module isomorphism $\iff \text{span}(f)\subset V^*\otimes W$ is trivial

I 'm self-studying some representation theory and I was trying to solve a problem that says Show that if $V$ and $W$ are irreducible G-modules, then $f : V\to W$ is a $G$-module isomorphism if and ...
Fung San Gaan's user avatar
-1 votes
1 answer
56 views

Can the sum of a nonlinear irreducible character's values on $Z(\chi)$ be zero? [closed]

I need a lemma for a research problem. Suppose that I sum the values of a nonlinear irreducible character $\chi$ of a finite group over the center of that character $Z(\chi)$. Is it possible for the ...
Hanklin's user avatar
  • 15
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0 answers
22 views

conductors of representations coming from jacobians of curves

Let $C$ be a curve defined over $\mathbb{Q}$, and we denote by $J:=Jac(C)$ its Jacobian. For a prime $l$, we define by $V_l(J)=T_l(J)\otimes \mathbb{Q}_l$. There is a natural action of the absolute ...
did's user avatar
  • 323
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0 answers
71 views

Quotient of parabolic subgroup generated by a simple reflection with Borel is isomorphic to $\mathbb{P}^1$

In the context of the Demazure resolution, (Bott-Samelson varieties) I have seen the following fact; Let $G$ be a reductive group over $\mathbb{C}$. Fix a Borel subgroup $B$. Associated to a simple ...
Pambra iskra's user avatar
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0 answers
38 views

Showing irreducibility of representations and modules

Let $\rho:S_5 \rightarrow GL_5(\mathbb{C})$ be the representation of the permutation matrix $(e_{\sigma(1)},...,e_{\sigma(5)})$. I am stuck on the following: Why is $U= \{a_1e_1+...+a_5e_5|a_1+...+...
Very Interesting's user avatar
1 vote
1 answer
47 views

Unique extension of $*$-representation into an abstract multiplier algebra

I'm trying to find a proof of the following fact: Let $A,B$ be $C^{*}$-algebras and $\pi: A \longrightarrow M(B)$ be a non-degenerate homomorphism in the sense that $\pi(A)B$ densely spans $B$. Then ...
Isochron's user avatar
  • 1,399
2 votes
2 answers
51 views

Endomorphisms over direct sum of irreducible representations equal to direct sum of endomorphisms over irreducible representations

So for a group G according to Maschke's theorem we know $V=\bigoplus_{i=1}^{d} V_i^{d_i}$ where V is the regular representation, $V_i$ are distinct irreducible representations. We also know. $d_i=dim(...
Josef.K's user avatar
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1 vote
0 answers
56 views

A question about congruence classes of Lie group representations

Let $\omega_1$, ..., $\omega_{n-1}$ be the fundamental weights of the group $G=\mathrm{SU}(n)$. The restriction of the irreducible representation $\lambda=a_1\omega_1+\cdots+a_{n-1}\omega_{n-1}$ with ...
Anatoliy Malyarenko's user avatar
5 votes
0 answers
54 views

Compute the character of an irreducible representation of $S_{20}$

Here is the problem: Assume $(V,\rho)$ is an irreducible representation of $S_{20}$ with $\dim V=2*3^8*13*17*19$. Let $g$ be a transposition of $S_{20}$. Calcuate $\vert \mathrm{tr}\rho(g) \vert $. ...
fusheng's user avatar
  • 1,159
2 votes
1 answer
64 views

An infinite cyclic group has infinitely many irreducible real representations.

I'm trying to show that an infinite cyclic group $G=\langle g\rangle$ has infinitely many non-equivalent irreducible representations over $\mathbb{R}$. I have in mind the following argument, for which ...
F. Salviati's user avatar
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0 answers
16 views

Primitive non-central idempotents of a group algebra

Let $W_i;\ 1 \le i \le m$ be irreducible $\mathbb{C}$-representations of a finite group $G$ with $\mathbb{C}$-characters $\chi_i$. Let $(V,\rho$) be a $\mathbb{C}$-representation of $G$ with isotypic ...
khashayar's user avatar
  • 2,307
2 votes
0 answers
58 views

When does a Lie algebra's outer automorphism group 'inherit' a representation?

In the following I am considering finite dimensional representations of semi-simple Lie algebras over fields of characteristic $0$. Examples should illuminate what I am getting at. Consider $\mathfrak{...
Craig's user avatar
  • 821
2 votes
0 answers
28 views

Brauer characters/ modular characters are not well defined

In chapter $15$ of Isaacs' "Character theory of finite groups", he defines a field $F$ of characteristic $p$, isomorphic to the algebraic closure $\overline{\mathbb{F}_p}$ of the prime field ...
GC.'s user avatar
  • 115
0 votes
1 answer
27 views

meaning of $IBr(X | Q)$

In the paper 1, there is a notation used without specifying the meaning. It is $IBr(X | Q)$ in Definition $4.1$. What it means? Irreducible Brauer characters of the group X from a block with defect ...
scsnm's user avatar
  • 1,303
6 votes
1 answer
79 views

Schur’s lemma over $\mathbb{F}_p$

I’m studying modular representation theory, and I got really stuck with the seemingly innocent statement. Consider $\mathrm{GL}_{2}(\mathbb{F}_{p})$ and its center $Z$, which is just a set of all ...
Matthew Willow's user avatar
0 votes
1 answer
49 views

irreducible representations of Group generate left ideal in Groupring CG.

Let W be the vectorspace of an irreducible representation of a group G.$char_W(g)$ denotes the character of the representation W. With $$e:=dim W \cdot \frac{1}{|G|}\sum\limits_{g \in G} \overline {...
VeryGenericUsername's user avatar
0 votes
0 answers
48 views

Bruhat decomposition of $SL_3$

I would like to know the Bruhat decomposition of $SL_3$. In Chevalley Group Theory, we have the following theorem: Let $G$ be the Chevalley group, and $G=\langle\mathfrak{X}_\alpha | \alpha \;\text{...
Tommk's user avatar
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3 votes
1 answer
61 views

Can an irrep become reducible on multiplication by a 1 dimensional irrep?

My background is in physics, and I am learning a little bit of representation theory of finite groups (with matrices over the field of complex numbers; this will be the setting for my question). ...
user196574's user avatar
  • 1,846
3 votes
1 answer
50 views

Invariants of $2$-torsion group under involution

Let $G=\{1,\tau\}$ be the group with two elements and let $A$ be a free abelian group of finite rank on which $G$ acts (via group homomorphisms). Let $B$ be a $2$-torsion group, also with an action ...
Hans's user avatar
  • 3,615
2 votes
0 answers
44 views

Values of Characters of the representation of permutation

I am just getting into representation theory and while I feel that I understand the general concept of characters, I still struggle with actually calculating them. I have the following problem: Let $...
Very Interesting's user avatar
5 votes
0 answers
37 views

Which transitive $G$-sets appear when repeatedly inducing and restricting $G/H$, where $H\subseteq G$ is an inclusion of finite groups?

Let $G$ be a finite group and $H$ be a subgroup. Then $G/H$ is a transitive $G$-set. We can define a sequence of $G$-sets as follows: $$X_0=G/H$$ $$X_{n+1}=Ind_H^GRes^G_HX_n$$ Like every finite $G$-...
Peter Huston's user avatar
0 votes
0 answers
24 views

How to prove that submodules of $L = b(n,F)$ indecomposable

Let $F$ be a field,$L = b(n, F)$ and $V = F^n$. Let $e_1, ..., e_n$ be the standard basis of $F^n$. Define $W_r = Span\{e_1,...,e_r\}$, with $1 \leq r \leq n$. It can be proved that $W_r$ a submodule ...
Chen's user avatar
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0 votes
0 answers
42 views

What are the irreducible representations of $A_3$?

I've got some notes saying the Character table of the alternating group $A_3$ is given as in the attached image. I can't seem to figure out what the representations $\rho_1, \rho_2$ are supposed to be....
Merkel_Bot's user avatar
3 votes
0 answers
56 views

Dualizing a finitely generated bimodule

Let $k$ be a semisimple associative ring and $V$ be a $(k,k)$-bimodule. Suppose that $V$ is both left and right $k$-finite. Let $V^* := \mathrm{Hom}_k(V, k)$ be the dual of $V$ as a left $k$-module. ...
Caitlyn Kiramman's user avatar
0 votes
0 answers
24 views

Decomposition of primitive central idempotents in group algebras

Let $W$ be an irreducible $\mathbb{C}$-representation of a finite group $G$ with character $\chi_W$. A primitive central idempotents of the group algebra $CG$ is: $$e=\frac{\dim_{\mathbb{C}}(W)}{|G|}\...
khashayar's user avatar
  • 2,307
1 vote
1 answer
48 views

Eigenvectors of a matrix that commutes with subgroup of permutation matrices

I have a certain symmetric matrix $M_{ij}$, that is invariant under certain permutations of some of the indices $i=1,\dots,N$. More precisely, there exists a subgroup of permutations $G$, such that if ...
a06e's user avatar
  • 6,771
0 votes
0 answers
19 views

Why Is There No Oscillator Representation for Operators in Planar N=4 SYM Theory?

Im studying the planar N=4 Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum systems, ...
iron's user avatar
  • 45
2 votes
1 answer
62 views

Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)

I look for the reference (or proof) of the following fact which is from appendix (B $27$) of Dixmier's book on $C^*$-algebras. Claim: Let $A$ be an algebra (not necessarily commutative) over a field $...
Charles Ryder's user avatar
0 votes
1 answer
37 views

Relation between linear representation and "induced adjoint representation" of Lie algebra?

Consider a representation $\rho \colon \mathfrak{g} \mapsto \mathrm{End}(V)$ of a Lie algebra $\mathfrak{g}$ on a vector space $V$. What can we say about the induced representation on the space of ...
Another User's user avatar

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