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

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GNS construction and representations

I am currently reading about C* from the following notes (math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf). In the proof of GNS construction theorem 1.12.4 page 50 there is something I don't ...
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25 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 ...
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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. ...
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39 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 ...
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56 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 ...
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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 ...
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17 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 ...
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7 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 ...
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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 ...
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2answers
26 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 ...
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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 $\chi(1)$?
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30 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 ...
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63 views
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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 ...
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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 ...
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2answers
43 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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13 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 ...
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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
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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 ...
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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 ...
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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 ...
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65 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 ...
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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 ...
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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 ...
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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 $2n$, $\mathbb{R}[D_{2n}]$. ...
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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 = ...
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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 ...
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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 ...
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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 ...
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53 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 ...
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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 ...
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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 ...
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70 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 : ...
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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 ...
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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 ...
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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.$(A)\geq 2$ ...
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1answer
24 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
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1answer
43 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 ...
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15 views

Questions about distributions on $l$-spaces.

I am reading the paper. I have some about distributions on $l$-spaces. On page 7, Section 1.7. Let $X$ be an $l$-space. Locally constant complex-valued functions on $X$ with compact support are ...
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Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi ...
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Matrix representation of function concatenation using other basis than polynomials.

I have now familiarized myself with the Carleman-matrices which represent function composition of polynomials (actually taylor series terms) and built some of my own. I noticed that for any finite ...
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$SO(n)$ algebra relations in the vector rep

The $\mathfrak{so}(n)$ algebra has some relations between generators always indicated as $$\left[T_{ij}, T_{kl}\right] = \delta_{ik}T_{jl} - \delta_{jk}T_{li} - \delta_{jl} T_{ik} + ...
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1answer
24 views

Function composition as representable by matrices?

I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and ...
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1answer
21 views

Root space $L_\alpha$ is completely contained in simple ideal?

I'm having trouble understanding a section in Humphrey's Lie algebras on page 74. Suppose $L$ is a semisimple Lie algebra which decomposes as a direct sum of simple ideals $L_1\oplus\cdots\oplus ...
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1answer
56 views

How can I show that the characters in sense of irreducible representations are the same as the character maps from the burnside matrices?

My Task is: Let G be a finite group. 1. Let $C_1 = \{e\}, C_2,..., C_k$ be the conjugacy classes, and let $v_1,..., v_k$ be the normalised eigenvectors of the Burnside matrices of G, then for all s ...
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Weights system corresponding to reflected Dynkin diagram?

Given a set of weights corresponding to the $SO(10) Dynkin diagram How can I transform these weights into weights that correspond to the Dynkin diagram ?
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1answer
24 views

Dihederal Group $D_{2n}$ Where $n$ is even/odd

I know that the group presentation of $D_{2n}$ is the following $$D_{2n} = \big<a,b: a^n=b^2=1,b^{-1}ab =a^{-1} \big>$$ Now if we consider the case where $n$ is even and we write $n =2m$ for ...