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

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“Powers” of injective representations “contain” all irreducibles

Let $G$ be a finite group and let $\rho : G \to GL(V)$ be an injective representation. I need to prove that each irreducible representation of $G$ is contained in $\otimes_{i=1}^{n} \rho$ for some $n ...
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Is highest weight of a module necessarilly dominant?

Let $K$ be compact Lie group (not necessarilly connected), $\mathfrak k_0$ its Lie algebra, and $\mathfrak k$ its complexification. Choose Cartan subalgebra $\mathfrak t_0 \subseteq \mathfrak k_0$, ...
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27 views

Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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35 views

Question for recommending a good textbook in representation of quivers

I am taking representation of quivers, and the lecture notes seems not enough. So could you recommend a good textbook for this course. There is a new book "Quiver Representations, by Ralf Schiffler" ...
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isomorphism classes of representations of a quiver

Classify all isomorphism classes of representations of dimension vector 1 and 2 of the following quiver The professor briefly did the solution, but I could not understand what was going on. What he ...
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Irreducible Representations and Direct Sums

I am learning about representation theory, and my professor stated the following as a remark: Let $A$ be a $k$-algebra. Every finite dimensional representation of $A$ is a direct sum of ...
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20 views

Reference request for properties of harmonic polynomials

I am reading this paper, and on page $4$ it takes a non-degenerate quadratic form $q$ on a finite dimensional complex vector space $V$, defines the laplacian assosciated to $q$, $\Delta$, acting on ...
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33 views

left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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Sum of an irreducible character over all $G$

Let $\phi: G\to GL_n(\mathbb C)$ be an irreducible representation of a finite group $G$. Let $\chi: G\to \mathbb C$ be the character of $G$. Prove that: $$\displaystyle \sum_{g\in G}{\chi(g)}=0 $$ I ...
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36 views

Does the induced $K^G$-comodule correspond to the induced $KG$-module?

Let $G$ be a finite group with group multiplication $m\colon G\times G \to G$ and $K$ a field. Then $K^G$ (the set-maps from $G$ to $K$) is a commutative algebra with pointwise multiplication. Because ...
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38 views

Failure of the Krull-Schmidt Theorem?

Theorem 1.19 of Representation Theory of Finite Groups: Algebra and Arithmetic is the Krull-Schmidt theorem, which I screenshotted and uploaded it here, I don't have any problem with this theorem and ...
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18 views

Lattices in Lie Algebras

I am having a little confusion with the different types of lattices involved with Lie algebras. Root system: represented as euclidian vector arrows. However I have seen the same arrangement with ...
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Prove that if $\rho,\phi$ are irreducible representations then so is this representation

Let $G_1$ and $G_2$ be finite groups and let $G = G_1 \times G_2$. Suppose $\rho: G_1 \to GL_m(\mathbb{C})$ and $\phi: G_2 \to GL_n(\mathbb{C})$ are representations. Let $V =M_{mn}(\mathbb{C})$. ...
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regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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Intuition behind PBW

The PBW theorem states: $\omega:\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. Where $\mathfrak {S} $ is the symmetric tensor algebra of a Lie algebra $ L $. Where $\mathfrak ...
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Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschke's theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
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Orbits of $Sp(n,R)$ under action of $Gl(2n,R)$ by conjugation

These questions arose from a question related to K-theory, I am hoping for (big) results from the theory of linear algebraic groups to be helpful. Maybe somebody with a better background there can ...
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Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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Sum of degrees of irreducible complex characters for certain groups

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra. I have ...
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The Heisenberg group over $\mathbb{Z}/2\mathbb{Z}$

This is inspired by a problem from from Dummit and Foote. It asked me to calculate the order of every element in the Heisenberg group over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, which is defined as ...
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112 views

Verify that two linear representations are equivalent

I've a problem in verifying that two linear representations are equivalent. First of all, I have two permutation representations of the group $G=\langle\alpha ,\beta ,\gamma\rangle$ on the set ...
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Matrix representation associated to a permutation representation

I have just begun studying group representation theory. I don't understand how I can find the matrix representation associated to a permutation representation. Should I identify each permutation ...
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Why do we study representations of groups but not fields?

Groups are great objects to work with as we all know. With surprisingly little structure, we can say fairly general things. However groups can be difficult to manage and so we look to representations ...
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Representation of a group, and finite index subspaces

Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints. Let $G$ be a group, not necessarily finite. $V=\mathbb C [G] $ a vector space with basis $(e_g, ...
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25 views

Semi-simple Irreducible Representations

I am studying the Representation Theory of Lie Algebras and came across this dilema. When can the representations of semi-simple Lie algebras be irreducible? I thought Weyl's theorem said this ...
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Describing all $\rho$-invariant inner products

Let $z$ satisfying the equation $z^3=1$ be a generator of the cyclic group $\mathbb{Z}_3= \{ 1 , z,z^2 \}$. You are given that $\rho : \mathbb{Z}_3 \to GL(\mathbb{C}^2)$ defined by $$\rho(z) = ...
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Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
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PBW proof proposal

One version of the PBW theorem states: $\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. I am curious if this is a possible proof for the PBW theorem, part is taken ...
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The comultiplication on $\mathbb{C} Q $ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
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The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
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129 views

Extensions and Kazhdan's Property (T)

Is Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?
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Jacobson radical of the integral group ring

I am trying to prove that the Jacobson radical of the integral group ring $\mathbb{Z}G$ for a finite group is zero. Most of what I find on semisimplicity, Jacobson semisimplicity, has to do with ...
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61 views

Weight spaces of Verma modules

Let $\mathfrak g$ be a semisimple Lie algebra generated by $x_i^+,x_i^-$, $1\leq i\leq n$, via the Chevalley-Serre relations and let $V(\mu)$ be a Verma module with highest-weight $\mu$. I gather that ...
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Multiplicity of irreducible $\mathbb{C}S_n$-modules

A known result in the representation theory of the symmetric group $S_n$ says: "Let $T_{\lambda}$ be a Young tableaux corresponding to a $\lambda \vdash n$, and let $M=M_{1} \oplus M_{2} \oplus ...
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Is the Transpose of a Representation an Equivalent Representation?

Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two ...
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Explicit formula for invariant inner product of the standard representation of $S_3$

Let $V$ be a representation of a group $G$ over $\mathbb{C}$. Given the standard Hermmitian inner product $\langle\cdot,\cdot\rangle$ on $V$ we can always define a $G$-invariant inner product by ...
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History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
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51 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
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Compute the isotropy representation

Suppose $SU(1,1)$ acts on the open unit disc $\mathbb{D}$ in the natural way, by linear fractional transformations. The isotropy group is $U(1),$ since it stabilizes the point $0.$ I am trying to ...
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One-dimensional representations of S5

The only one-dimensional representations of $S_5$ are the trivial representation and the sign representation. Why are these the only ones? Here's what I've got so far: the image of any ...
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Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
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Slodowy slices for non ADE type Lie algebras

In the first answer to: http://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c the correspondence between transverse slices of subregular elements of the nilpotent cone for a simple Lie ...
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Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$?

Let $B$ be a Borel subgroup of $GL_n$ and $W$ the Weyl group of $GL_n$. Let $w_1, w_2 \in W$. Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$? If this is true. How to prove it? Thank you very much. Edit: If $l(w_1 ...
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Dynkin diagram automorphisms and weights

Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
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Why $\rho(t)^{-1}(H-\frac{\partial}{\partial h_{\rho^{\vee}}}) \rho(t) = H - \frac{1}{2}(\rho^{\vee}, \rho^{\vee})$?

I am reading the paper. On page 17, line 15, why $$ \rho(t)^{-1}(H-\frac{\partial}{\partial h_{\rho^{\vee}}}) \rho(t) = H - \frac{1}{2}(\rho^{\vee}, \rho^{\vee}) $$? Here $$ H = \frac{1}{2} \sum_{i\in ...
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Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
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Does every Lie algebra come from commutator of some associative product operation?

Suppose $\mathfrak{g}$ is an Lie algebra. Is it possible to define an associative product operation $\star$ on $\mathfrak{g}$ such that $[A,B]=A\star B - B \star A$ ? If it is not possible to do so ...
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Examples of the local Langlands correspondence

I'm trying to compile some examples of the local Langlands correspondence, with the aim of motivating the statement and also just giving some concrete traction on how it works. I would especially like ...
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matrix of the dual representation: inverse of the transpose

I have a doubt concerning the dual representation. Can someone check that what I wrote is correct please? Let $A: V \longrightarrow V$ be linear, the dual map $A^T : V^* \longrightarrow V^*$ is ...
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Reference Request: Characters of Finite General Linear Groups

I've been looking at J.A. Green's article The Characters of Finite General Linear Groups and it seems that Green in this article comes up with a way of calculating all irreducible characters of a ...