# Tagged Questions

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

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### Nebentypus of contragredient representation

Let $k$ be a local non-archimedean field with ring of integers $\cal O$ and maximal ideal $\frak p$. Let $\pi$ be an irreducible admissible $\infty$-dimensional representation of $\text{GL}_2(k)$ ...
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### Degrees of irreducible complex characters of alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
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### In what way is the representation not continuous

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf pp.7-8 Following the definition of a linear representation it states 'if V (an E-vector space) is equipped with a topological structure and if ...
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### representations of Lie algebras

I am studying irreducible representations of Lie algebras when our filed is of positive characteristic, I need an explicit explanation with example (or an article) which describes the differences what ...
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### An example contradicting the validity of Maschke's theorem for infinite groups.

I am learning representation theory of finite groups and I am in doubt about a homework problem: Let $G = \mathbb{Z}$ and $V = \{(a_1 , a_2 , . . . )|a_i ∈ R\}$ be a vector space of infinite ...
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### $[L_+^m, L_y^n]$ in the $SO(3)$ Lie Algebra

Let $SO(3)$ be generated by infinitesimal rotations $L_x, L_y, L_z$ such the typical relations $[L_x, L_y] = L_z$ and similar. Let $L_\pm = L_x \pm i L_y$ be the raising and lowering operators. Is ...
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### Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
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### If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
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### action of $SU(2)$

Let $L=\mathbb C[u,v]$ (the $\mathbb C$-algebra of polynomials over two commuting variables $u,v$. For each non negative integer $n$ let $L_n$ be the linear subspace of homogeneous polynomials of ...
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### Normal basis theorem

Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is ...
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### Some questions on Langlands Classification of Irreducible Admissible Representation

I am trying to construct some representations using Langlands classification theorem. But I get confused and have some problems when constructing these representations..... i) In the classification ...
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### Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
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### Finding expansions for elements of a group algebra

Suppose $G$ is a finite group of order $\left|G\right|$, with an associated group algebra $K[G] = \left\{\sum_{g \in G} a_g g\right\}$ over, say, the complex numbers. Suppose we represent $G$, and by ...
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### Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
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### What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$?

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$? Here $k$ is a non-archimedean local field. Thank you very much.
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### What does it mean for a representation to contain a character?

I'm trying to understand the statement "The representation $\pi$ contains the trivial character of $N$ if and only if it contains an irreducible representation $\sigma$ of $B$ containing the trivial ...
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### Showing that $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real

I am trying to show that for a finite group $G$ and $g\in G$, $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real for all $\chi$ irreducible characters of $G$. I have the following: I first want ...
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### Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal $O(n)$-...
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### restriction of spin representation to block diagonal subgroup

What is the restriction of the (complex) spin representation of $so(n+m)$ to the block diagonal subalgebra $so(n)\times so(m)$? A naive guess is that it is the (complex) tensor product of the two ...
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### Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
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### Representations of Nilpotent Lie Algebras

Let $\mathfrak{g}$ be a rational, nilpotent Lie algebra. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. But this representation generally is not ...
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Let $\mathbb{A}$ be the ring of adeles. Let $N$ be a natural number. Let $\Gamma(N)=\ker(\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})))$. I saw the following embedding of $\... 0answers 101 views ### Which finite groups contain the two specific centralizers? This is a question requiring the good knowledge of group theory: (Q1) Which finite groups$G$contains some specific centralizers both of these two groups: i. the elementary group$Z_2^4$, ... 0answers 55 views ### A construction of$\mathfrak{e}_8$in Fulton and Harris In section$22.4$of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra$\mathfrak{e}_8$is constructed using a method of Freudenthal. For background, I will ... 0answers 84 views ### Lie Algebra of Reduced Heisenberg Group Identities I am having problems trying to understand a statement by Howe in his paper "On the role of the Heisenberg group in harmonic analysis". Here is the setting: Howe defined the (reduced) Heisenber group ... 0answers 40 views ### On an invertible element for equivariant K-theory Fix a positive integer$m$. Let$G = \lbrace h\in\mathbb C | h^m = 1\rbrace$and$(X,\pi)$the standard representation of$G$. Namely$X = \mathbb C$and$\pi:G \to GL(X)$is defined by$\pi(h)v=h v$... 0answers 56 views ### Learning representation theory of real reductive lie groups I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject ... 0answers 32 views ### What do diagonal matrices do in irreducible repns of SL$_2(\mathbb{Z}/N\mathbb{Z})$? Let$N \in \mathbb{N}, \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$. For every$a \in \mathbb{Z}_N^\times$put$R_a = \begin{pmatrix} a^{-1} & 0 \\ 0 & a\end{pmatrix}$and also set$T = \begin{...
Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $A$ by generators and relations helps in the study of structure of the algebra ...
Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].