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)

2
votes
0answers
40 views

Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
2
votes
1answer
28 views

Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
1
vote
1answer
30 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
4
votes
2answers
63 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
5
votes
1answer
37 views

Recognizing action of semidirect product

I've been looking at some texts in representation theory and I see instances where the symmetric group $S_n$ and some other group, e.g., $GL(V_1) \times \ldots \times GL(V_n)$, act on a space. The ...
4
votes
1answer
35 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
1
vote
0answers
34 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
1
vote
1answer
37 views

In general, how do you construct a nontrivial representation of a group?

This is my first time studying representations. I'm not sure how to go about constructing a nontrivial representation of a group. Do I construct a function that satisfies the definition? Could you ...
0
votes
2answers
36 views

How do I construct a nontrivial linear representation of the group $G$=S' in $R^3$

$S'$ is defined as the unit circle. The product is defined as the sum of angles. How do I construct a linear representation in general? I don't know how to begin with this problem. Some ...
1
vote
1answer
29 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...
4
votes
0answers
34 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
2
votes
0answers
63 views

Fields of Research in Algebra [on hold]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
2
votes
0answers
42 views

Dual space isomorphism and the dual representation

Let $V$ be a complex finite-dimensional vector space. Then there always exists an isomorphism $V \simeq V^*$, where $V^*$ is the dual space. The isomorphism can be fixed by choosing a non-degenerate ...
4
votes
1answer
62 views

The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
-3
votes
0answers
32 views

How to prove the theorem on group algebra of permutation group [on hold]

How to prove the following theorem: If $t$ is a vector in group algebra of permutation group, then $\cal{y}t\cal{y}=\lambda_t \cal{y}$, where $\cal{}y$ is the Young operator of permutation group and ...
1
vote
0answers
18 views

Every representation of compact group is a direct sum of irreducible

Recently I asked about (references to) some results concerning representation theory of compact topological groups: here is the discussion Representation theory of locally compact groups In ...
0
votes
0answers
9 views

Stabilizer subgroup in adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
6
votes
1answer
46 views

Is there a systematic way to determine the irreducible representations of a finite group?

I was reading through fulton and harris's book on representation theory. I'm in the middle of chapter 3 and noticed their approach to finding irreducible representations of groups is pretty ...
1
vote
2answers
27 views

About the characters of representations of groups

I want to ask a question about the characters of representations of groups. we all know that the equivalent representations have the same character, and the character is a class function, so what ...
0
votes
0answers
7 views

Character degrees of 2B2(q^2)

Let $S \cong {}^2B_2(q^2), q^2 \ne 2$, and $\chi$ is the Steinberg character for $S$. What is the $\chi(1)$?
3
votes
1answer
31 views

Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...
8
votes
2answers
75 views

1-dim representations of the affine Hecke algebra for $G = SL_2$

I want to count the number of (isomoprhism clases) of one-dimensional representations of the affine Hecke algebra for $G = SL_2$. I'm doing it in two ways: (1) by explicitly looking at generators and ...
0
votes
0answers
17 views

Uniformly continuous unitary representations.

Let $U$ be a unitary rep. of $\mathbb{R}^d$ on a separable Hilbert space $H$, and $H\cong\oplus L^2_{\mu_v}(\mathbb{R}^d)$ be the spectral decomposition (according to the spectral theorem for these ...
2
votes
2answers
46 views

Realizing the oscillator algebra as a matrix Lie algebra

In Hilgert's & Neeb's Structure and Geometry of Lie Groups, they introduce a Lie algebra, which they call the "oscillator algebra," as an extension of the Heisenberg algebra. They give a basis ...
3
votes
1answer
14 views

Mappings between tensor products of group representations

All of my representations are on finite dimensional complex vector spaces. Let $G$ and $H$ be finite groups. If $V$ is a representation of $G$ and $W$ is a representation of $H$, then $V\otimes W$ is ...
4
votes
1answer
31 views

Unramified Hecke character

I'm looking for a reality check here: Let $\chi$ be a character of $(F^\times\backslash \mathbb A_F^\times)^1$ where $F$ is a number field. Call $\chi$ unramified at a place $v$ if ...
1
vote
1answer
37 views

Proof check - infinite-dimensional $\mathfrak{sl}(2, \mathbb{F})$-module

Let $L=\mathfrak{sl}(2,\mathbb{F})$ with the usual basis $(x, \ y, \ h)$ and $\text{char}\,\mathbb{F}=0$. Let $Z(\lambda)$, $\lambda\in\mathbb{F}$ the infinite-dimensional $L$-module spanned by ...
1
vote
0answers
15 views

Branching rules without previous knowledge of the projection matrix?

Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups: $$ R= R_1 + R_2 + \ldots$$ This is often called a ...
1
vote
0answers
14 views

Inner Automorphism of Lie algebras in Terms of Roots and Weights?

An automorphism is a homomorphism of a group $G$ onto itself. For Lie groups this induces a Lie algebra $g$ automorphism, i.e. a map of the Lie alegbra onto itself that preserves the Lie bracket. An ...
0
votes
2answers
21 views

s3 representation in 2d in matrix form

Trying to arrive at how matrix form of standard representation of Symmetric Group S3 has the form $((-1,1),(0,1))$ for permutation $(1,2)$. Please let me know details. - Thanks
1
vote
1answer
33 views

Do we have $\mathbb{C}[SL_n] = \oplus_{\lambda, \text{ht}(\lambda)\leq n} V_{\lambda} $?

The coordinate algebra $\mathbb{C}[SL_n]=\mathbb{C}[x_{ij}: i, j \in \{1, \ldots, n\}]/(\det(x_{ij}) - 1)$ is a representation of $SL_n$: $(g'.f)(g)=f(g'^T g)$. Let $V_{\lambda} = \langle e_T : T ...
1
vote
1answer
40 views

How to understand that minors are matrix elements in fundamental representations of $SL_n$?

In the video, Lecture 3 of June 14, 49:00-53:00, it is said that "minors are matrix elements in fundamental representations of $SL_n$". What are fundamental representations of $SL_n$? How to ...
6
votes
1answer
72 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
1
vote
0answers
66 views

Orbit of a Weight Vector?

Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$. We can write every element of a given ...
1
vote
0answers
23 views

Equivalence of continuity conditions of a group representation on an infinite-dimensional space

Let $V$ be an (infinite-dimensional) Banach space and $G$ a locally compact topological group (with a countable basis of neighbourhoods of $1$, and which is a countable union of compact subsets). I ...
0
votes
1answer
41 views

Representations of the form $\varphi: G \rightarrow GL(V)$ vs $\phi: G \rightarrow Aut(A)$

Standard representation theory studies homomorphisms of the form $\varphi: G \rightarrow GL(V)$ where $V$ is a vector space. How much does the focus of representation theory change if one considers ...
1
vote
0answers
13 views

What is the Wedderburn decomposition of $\mathbb{R}[D_{2n}]$?

Hi so I have been looking everywhere and can't seem to find a general formula for the Wedderburn decomposition of the real group ring of the dihedral group ring of order $2n$, $\mathbb{R}[D_{2n}]$. ...
1
vote
2answers
54 views

Representation of $sl(2,R)$.

I am interested in the unique (up to isomorphism) $5$-dimensional representation of the Lie algebra $sl(2,R)$. I understand that one can choose the module $V_4 = ...
2
votes
0answers
24 views

List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
3
votes
0answers
28 views

Question concerning a correspondence between basis elements of the Schur algebra and some matrices

I have the following question: Let $k$ be an infinite field and let $S_k(n,r):={A_k(n,r)}^{∗}=\text{Hom}_k(A_k(n,r),k)$ and $A:=A_k(n):=\text{polynomial functions on}\ \Gamma:=\text{GL}_n(k)$ and ...
1
vote
1answer
27 views

Explicit Representation of the SU(N) Simple Roots in with redundant coefficents?

Commonly the simple roots for $SU(n)$ groups are given as $n$ dimensional vectors, although root-space is $n-1$ dimensional. The $SU(n)$ Wikipedia article explains: Here, we use n redundant ...
3
votes
0answers
54 views

Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
4
votes
1answer
61 views

Is this parabolic induction?

In one of my previous questions, @PL. explained the idea of parabolic induction. In my bachelor thesis I used a technique quite like this, to find all irreducible representations of a certain group. I ...
1
vote
0answers
37 views

General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where $$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$ or in other words, M commutes with all members of the the ...
5
votes
1answer
71 views

Question concerning a property of polynomial functions on $\Gamma:=\text{GL}_n(K)$ and the Schur algebra

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.4b) part (i) on page 14: Consider the map $e : ...
2
votes
0answers
35 views

How to prove that the Schur algebra is isomorphic to a certain endomorphism ring?

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.6c) on page 18: Let $K$ be an infinite field ...
2
votes
0answers
53 views

Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
3
votes
0answers
45 views

Dominant dimension $\geq 2$ implies a certain double centralizer property

let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: $\bullet$ $Ae$-dom.dim.$(A)\geq 2$ ...
2
votes
1answer
25 views

Question concerning a faithful module over an Artinian ring

Let $A$ be an Artinian ring and $M$ in $\operatorname{\mathfrak{mod}} A$. Is it true that $M$ is faithful if and only if there is an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for ...
2
votes
1answer
45 views

Representation theory of locally compact groups

My knowledge about representation theory of locally compact groups is rather scattered. As I got more interested with this subject, I would like to know some good references, where I could learn the ...