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

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Theorem $2.15$ of Quiver Representations by Ralf Schiffler.

I'm reading the proof of theorem $2.15$ of Quiver Representations by Karl Schiffler. The author states the following: Let $Q$ be a finite acyclic quiver, $M=(\left\{M_i\right\}_{i\in Q_0}, \left\{\...
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The representation of $SU(2)$ as a polynomial function on $\mathbb C^2$

Let $A$ element of $SU(2)$ and $p$ a polynomial function of fixed degree $l$ on $\mathbb C^2$ (in other words, $p \in P_l(\mathbb C^2)$), then the polynomial representation of $A$ in $P_l(\mathbb C^2)$...
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Relation between integrable representations and highest weight representations.

Let $g$ be a simple Lie algebra and $U_q(g)$ the corresponding quantum group. What are the relation between integrable representations and highest weight representations of $U_q(g)$? Are all highest ...
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209 views

restriction of $SL(2,R)$ representation to $SO(2)$

Let $d \geq 1$ and let $V$ be a $2d+1$ irreducible representation of $SL(2,\mathbb{R})$. We know the irreducible representations of $SO(2)$ are the $2$ dimensional spaces $V_k$ with the map $$\rho_k : ...
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Proposition 2.3 of Quiver Representations by Ralf Schiffler

I'm trying to prove proposition 2.3 of Quiver Representations by Ralf Schiffler. To any vertex $i$ finite acyclic quiver $Q$ we can associate the indecomposable projective $P(i)$, this proposition ...
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32 views

Matrices over integer fields to solve complex polynomials.

Inspired by the fruitful answer to this question regarding numerically solving polynomial equations in terms of simpler fields (in that case representing real numbers as fractions of integers), I ...
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30 views

Self conjugate representations

I am trying to understand proof of Prop. 5.1(page 64) from Fulton and Harris representation theory. I am unable to prove one statement in the proof. Which is Let $V$ be an irreducible ...
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9 views

scotts theorem and other representation theorems for order aggreeing qualitative quantitative probability measures

Scotts theorem and other theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, finite cancellation axioms ...
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Extended inner-product on complexified spaces

I am reading the book "The Index Formula for Dirac Operators: an Introduction” (via this link http://www.impa.br/opencms/pt/biblioteca/pm/PM_10.pdf) I am having trouble understanding the middle of ...
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29 views

Orthogonal basis of a Cartan of a Lie algebra with respect to Killing form.

I am trying to understand orthogonal basis of a Cartan of a Lie algebra with respect to Killing form. For example, let $g=sl_2 = \text{Span}\{h, E, F\}$. Then a Cartan of $g$ is $\mathfrak{h} = \...
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Finding a polytope in the Cartan Subalgebra

The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can. Some examples of ...
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33 views

Why must certain functions on a group take commutators to the identity?

My question is about the identity pictured above. Can't figure out why it is true.
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Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
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Action of one element of order $n$ and the $n$-th roots of unity

Let $G$ be a finite group and $\rho : G\to GL(V)$ a representation of $G$ on the vector space $V$. If $g\in G$ is one element of order $n$ we have $g^n=1$, so that $$\rho(g^n)=\rho(g)^n=I,$$ where $...
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24 views

Mock gamma factors

I wonder if analogues of gamma factors as used to defined a complete L-function of the form $\displaystyle{\prod_{j=1}^{\infty}\Gamma(\lambda_{j}s+\mu_{j})}$ with a possibly non integral value of $d=\...
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37 views

derived equivalences and finite representation type

Given two finite dimensional connected quiver algebras $A$ and $B$. Is there an easy example such that $A$ and $B$ are derived equivalent and $A$ has finite representation type but $B$ has infinite ...
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What is the dimension of this space?

The function $π(v)$ interchanges the coordinates of the vector $v$ randomly. For example: $v = (1,3,7,9), π(v) = (7,3,1,9)$. Fix some vector $v ∈ \mathbb{R}^n$ and construct a linear hull of all ...
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Understanding some notation and concepts regarding Universal Enveloping Algebras

I'm learning about Universal Enveloping Algebras, and I just want to clear up my understanding of a bit of notation and confirm my understanding of a bit of the concept. If we have a lie algebra $\...
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Correspondence between $\mathfrak{sl}(n,\mathbb{C})$ and $\mathfrak{gl}(n,\mathbb{C})$ representations

There is a one-to-one correspondence between irreducible $\mathfrak{gl}(n,\mathbb{C})-$representations and $\mathbb{C} \times \{ \text{irreducible } \mathfrak{sl}(n,\mathbb{C}) \text{ representations}...
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Question about Tensor Product as a representation of $\mathbb{C}[G]$-module

I am learning about representations of finite groups, and trying to clear up some my understanding of Symmetric and Alternating squares. If $V$ is a vector space, let $P: V \otimes V \rightarrow V \...
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Determine whether $f(Z(G))\subset SL_n(F)$, where $f$ is a representation.

Let $G$ be a finite group, $F$ a field, and $f:G\to GL_n(F)$ a representation. Determine whether $f(Z(G))\subset SL_n(F)$, where $Z(G)$ denotes the center of $G$. I am a beginner in representation ...
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Character of an Induced Representation

In Fulton's book, page 34, it is stated that, To compute the character of $V = Ind\space W$, note that $g \in G$ maps $\sigma W $ to $g \sigma W$, so the trace is calculated from those cosets $\...
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14 views

Isotypic component of a vector bundle

Suppose that $E \to B$ is a vector bundle with fibre $F$. Let $G$ be a finite group and assume that $F$ has the structure of a $G$-module. Let $\chi$ be another $G$-module. I have come across a ...
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Induced Representations from Tensor Product of Modules

In Chapter 7 of J P Serre's book on Representation Theory, he defines a $\mathbb{C}[G]$-module as $W'= \mathbb{C}[G]\otimes_{\mathbb{C}[H]}W$ and claims it to be the $\mathbb{C}[G]$-module obtained ...
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Computing representations of $SL_2(\mathbb F_3) \ltimes C_3^2$ explicitly

I am trying to explicitly construct the complex representations of $SL_2(\mathbb F_3) \ltimes C_3^2$, with the natural action of matrices on two-dimensional vectors by multiplication. Here are the ...
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21 views

Irreducible group representation of subgroup

Does exist a way such that given $G$ a finite group, $H\leq G$ and $GL(V)$ build a group representation over $G$ (let's call it) $\rho\colon G \rightarrow GL(V)$ such that $H$ has an irreducible ...
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Computing $\operatorname{ad}_x$ where $\operatorname{ad}$ is the adjoint representation [duplicate]

Let $\operatorname{ad}_x:L\rightarrow GL(L)$ be the adjoint representation. In Humphreys "Introduction the Lie-Algebras and representation theory" one can find this example where $x,y$ and $h$ are ...
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29 views

References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
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27 views

Find all the invariant invariant subspaces for the regular representation of $S_3$.

Using a little program I can build the regular representation of $S_3$ (the permutation group of three elements). The textbook that I am reading only tells that the regular representation contains all ...
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On representations of a nonabelian group of order $pq$

Let $p,q$ primes number s.t. $p>q$ and let $G$ a non abelian group of order $pq$. 1) Determine all degree of irreducible representation 2) Show that $|[G,G]|=p$ (where $[G,G]=\left<ghg^...
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Methods for finding infinite subset of set of poset representations

This is based on the David Arnold book on Abelian Groups and Representations of finite posets. Let $F$ be a field. Let's suppose we have a vector space $U_0$ over $F$ and four subspaces of it, $U_1,...
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A survey on Lie-Kolchin Theorem

While, I was reading something in Lam's book "A first course in Noncommutative Rings" stumbled upon a Theorem called "Lie-Kolchin Theorem" and I am bit confused with it. Firstly, if I understand ...
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Motivation for introducing $A^{\prime} := \frac{1}{\left|G \right|} \sum_{g \in G} \Delta(g^{-1}) A \Delta(g)$ in representation theory proof

Assume $\left|G \right|= \left| G \right| \cdot 1_F$ is invertible in $F$. Let $\Delta:F \to GL_n(F)$ be a representation and $U \subseteq F^n$ be an $F$-subspace that is $\Delta$-invariant. Then ...
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Solvability and representation of finite groups

Let $G$ be a finite solvable group, and let $G=G^{(0)}\unrhd G^{(1)}\unrhd...G^{(n)}=1$ be its derived series. Is it true that any irreducible representation of $G$ has dimension at most $n$?
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Why all irreducible representations appear in the regular representation?

Let $G$ be a finite group and $R$ the regular representation. That is, as a vector space $R = F(G)$ is the free vector space with basis $G$. If the basis is $\{e_g : g \in G\}$ the action is defined ...
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Weyl's theorem confusion

Weyl's theorem states that given a semisimple Lie algebra $\mathfrak{g}$, any $\mathfrak{g}$-module $V$ is completely reducible. If we consider the case of $\mathfrak{g}= \mathfrak{gl}(1)$, then ...
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Decomposition of a representation into a direct sum of irreducible ones

I'm studying representation theory and in the book (Fulton and Harris) the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite ...
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Equivalent group representations commutative diagram

I am confused by the commutative diagram displayed above. Why is $\varphi:V \to V$ and not $G \to GL(V)$? Analogous question for the mapping $\psi: W \to W$.
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Representations of an abelian group

Let $V$ be an $F$-vector space, and let $f:G\to GL(V)$, where $G$ is a group. For $g\in G$, how can we show that if $G$ is abelian then the eigenspace of $f(g)$ is a $G$-invariant space? Moreover, ...
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Constraint for matrix representation for general irreducible permutation group.

Say I have a matrix $\bf P$ for which is ensured that $P_{ij} \in \{0,1\}$. Then consider this requirement: $$\sum_{k=0}^{n-1}{\bf P}^k[1,0,\dots,0]^T = [1,1,\dots,1]$$ Should this be enough to make ...
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Constructing an explicit isomorphism between automorphism group of bijective $F$-linear mappings and group of intertible $n \times n$ matrices

I'm going over some class notes: In the literature, sometimes a representation of $G$ over $F$ is defined as a pair $(V, \rho)$ where $V$ is a finite-dimensional $F$-vector space and $\rho: G \to \...
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Non-rational G-modules

Let me recall the definition of a rational $G$-module from M. Brions notes Introduction to actions of algebraic groups (Def. 1.6) Let $G$ be an affine group scheme over $\mathbb{C}$. A rational $G$-...
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Anti-involution on universal enveloping algebra of a Lie algebra.

Let $\mathfrak{g}$ finite dimentional semisimple Lie algebra and $\sigma$ the usual chevalley anti-involution that fixes the Cartan subalgebra $\mathfrak{h}$ sends the weight space $\mathfrak{g}_\...
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How to find a basis of weight vectors

I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal ...
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Problem with Molien's formula for covariants

If $E$ and $H$ are finite-dimensional faithful representations (over $\mathbb{C}$) of a finite group $G$, with $H$ irreducible. The Molien formula describer the Poincaré series of the covariants as $$ ...
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algebras without identity

This problem is an exercise from Drozd-Kirichenko's book Finite Dimensional Algebras, page 29. Let $k$ be a field. Let $A$ be a $k$-algebra not necessarily with identity. Let $\overline A$ be the ...
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Reduction of a representation of the Symmetric Group $S_3$

I have this representation of $S_3$ obtained in the usual way $$\varrho\left(\sigma\right)e_i=e_{\sigma_i}$$. Being more explicit the representation is this one: $$\varrho\left(e\right)=\left(\begin{...
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Differences between realizations and representation of a group

I am studying an introduction to group representation theory on my relativity class' lecture notes. I've previously learned in other classes and also on the Wikipedia article that a representation $T$ ...
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Computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$

How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$? Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 ...
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$G$-invariant symmetric, nondegenerate form is unique up to scalar

Let $V$ be a f.d. representation of a finite group $G$ over a field $F$. A standard argument shows there is a $G$-invariant, symmetric, nondegenerate bilinear form on $V$. If $(-,-)$ is any such ...