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

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Bounding the degree of irreps of a finite group

Let $G$ be a finite group and $\mathbb{k}$ is algebraically closed with characteristic zero. Let $H$ be an Abelian subgroup of $G$. Show that the degree of any irreducible representation $V$ of $G$ ...
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38 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
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78 views

Connection between: Expand a representation and dual representation

I worked the proof for 3) in this link out, but I have problems with the last step: Let $\sigma$ be a irred. representation of a normal subgroup $H=\langle z\rangle$ of $G$ and $\sigma'$ its dual ...
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253 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
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78 views

indecomposable $K[G]$ modules -> get irreducible $K[G]$ modules

if I found indecomposable $K[G]$ modules, are there any techniques to get from this irreducible $K[G]$ modules? (e.j. for $k=\mathbb{Z}/p \mathbb{Z}$ and $G=C_p$) regards, Khanna
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examples for p-adic representations

Could anyone recommend me some literature or articles were I can read about the construction of irreducible representation of finite groups (like the symmetric group, alternating group or semidirect ...
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41 views

Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring ...
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54 views

Finite-dimensional representations of the Lie algebra of vector fields on a circle

I have just began to study infinite-dimensional Lie algebras and I am curious whether the Lie algebra $L$ spanned by the vector fields $z^n \partial/\partial z$, $n=0,1,2,3,\dots$ admits any ...
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122 views

Representations of Central Products

What is a good reference for learning about representations/characters of central products of groups? By central product, I mean the following. If $G$ and $H$ are groups, containing isomorphic ...
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106 views

Reference Request - Spaces of Smooth Vectors

I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
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365 views

All irreducible representations of Pauli group

I'm supposed to find out all irreducible representations of Pauli group, that is, the group generated by Pauli matrices $\sigma_k(k=1,2,3)$. It has 16 elements: $\pm 1, \pm i, \pm \sigma_k, \pm i ...
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111 views

Existence of a 1-dimensional invariant subspace

Show the existence of a 1-dimensional invariant subspace for any 5-dimensional complex representation of the group $A_4$, where $A_4$ is the alternating group of degree 4. Any hints?
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67 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
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238 views

Examples of decomposition representation

Here is a question in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg. (Question 3.8(2), page 25) that I need some hints from you : Give an example of ...
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66 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
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111 views

Question about a Corollary of Engel's Theorem

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for ...
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induced representation of tensors of irreducibles

Let $V_{\lambda}$ and $V_\mu$ be representations of the symmetric groups $\mathfrak{S}_d$ and $\mathfrak{S}_m$ respectively where $\lambda$ is a partition of $d$ and $\mu$ is a partition of $m$. It is ...
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143 views

How to compute the character of a matrix group operating on homogeneous polynomials?

I have a little problem in representation and/or invariant theory which I need help with. Let's assume $G \leq \mathbb{C}^{n\times n}$ is a finite complex matrix group which operates linearly via ...
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482 views

Tensor products and irreducible representations

Again something from Fulton and Harris I'm having trouble with: Exercise 2.33 (c). If $U$, $V$, and $W$ Are irreducible representations, show that $U$ appears in $V \otimes W$ if and only if $W$ ...
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82 views

Minimal embedding of exceptionnal Lie groups into special orthogonal groups

Let $G$ be a Lie group. The set of all $N$ such that $G$ is a subgroup of $SO(N)$ has a minimum $N_{\min}(G)$. (If I am not wrong, $N_{\min}(G)$ is supposed to be less or equal to $\dim(G)$) What is ...
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90 views

symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
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82 views

radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or ...
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364 views

Irreducible finite dimensional representations of $SL(2,\mathbb C)$

Is there a book that finds all the irreducibile finite dimensional representations of $SL(2,\mathbb C)$ without considering the Lie algebra $sl(2,\mathbb C)$? For example, how can I show "directly" ...
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209 views

clarification on the definition of a group C*-algebra

I've been trying to understand the definition of a group C*-algebra. Given a topological group $G$ and a C*-algebra $A$, let $u: G \to A$ define a unitary representation $U(G)$ of $G$ on $U(A)$, the ...
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47 views

notation question $Aut_{Modk}(A)$

What is $Aut_{Modk}(A)$ if A is a finite generated algebra, k is a field? Is it a group of automorphisms of A as a vector space over k or what?
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19 views

Embedding of a finite group in a compact connected Lie group

How can one embed a finite group $G$ in a compact connected Lie group? I think if we take a faithful unitary representation of G , that will do the job.But if $G= Z/n$, then what should be the ...
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39 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 ...
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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 ...
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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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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 ...
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67 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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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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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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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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Locally compact spaces that are not first-countable and continuity of functions on locally compact groups and continuity of group representation

If $X$ is a topological space that is first-countable, then a function $f: X \to Y$ into another topological space $Y$ is continuous if and only if $f$ is sequential continuous. Only the implication ...
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How to transform roots/weights from the simple root basis to the H-basis?

Often the roots and weights of some Lie algebra are written in terms of the simple root basis $$ r =(a_1,a_2,a_3,\ldots)=a_1 \alpha_1 + a_2 \alpha_2 + a_3 \alpha_3 +\ldots,$$ where $α_i$ denotes the ...
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19 views

Using Peter Weyl theorem to decompose an orbit

Let $G$ be a finite group and let $\pi$ be a unitary representation of $G$ on a Hilbert space $H$. Since $G$ is finite, we have that for every $v \in H$, the orbit of $v$, $\pi (G).v$, is of dimension ...
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Induced representation from subgroup to subgroup

I wonder why the third of the properties of the induced representation here (http://mathworld.wolfram.com/InducedRepresentation.html) holds. Does it follow from the universal property? I could not ...
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55 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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20 views

References about an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$.

Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times ...
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40 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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27 views

If two non-equivalent representations are irreducible then sum is 0

We have a finite group and two representations $D_1:G\to GL_n(\mathbb{C}),D_2:G\to GL_m(\mathbb{C})$ for some positive integers $m,n$. We define $$T=\sum_{g\in G}D_1(g)BD_2(g^{-1})$$ where $B\in ...
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38 views

Cohomology of permutation representation

Consider the action of $S_n$ over $\{1,...,n\}$ consider the associated representation with integral coefficients $X_n$. What are $H^r(S_n,X_n)$? More in general is there a nice way to predict the ...
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Permutation module $M^\lambda$ as induced module

If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module: $M^\lambda := ...
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Dimension of the restricted representation

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
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How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
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Relation between induced and coinduced spaces

Let $G$ be a compact Lie group and $H$ a closed subgroup of it. Let $X$ be a $G-$space. The induced $G-$space is defined to be $$G\times_H X$$ with the equivalence $(gh, x)=(g, hx)$, for any $g\in G, ...