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

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fulton and harris representation theory exercise 2.5 solution

Trying to solve Exercise 2.5: If $V$ is a permutation representation associated to the group $G$ on a finite set $X$, show that $\chi_\nu(g)$ is the number of elements fixed by $g$. Looking for hint ...
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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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Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
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1answer
12 views

Freedom in choosing the Cartan subalgebra?

What transformations are allowed regarding the Cartan subalgebra of a given group? Weights for every representation are labelled by the corresponding eigenvalues of the Cartan generators. Therefore ...
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33 views

Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$.

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block. Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...
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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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28 views

Determining the group generated by a set of roots?

I have a set of 45 roots and I want to know which group is generated by the corresponding generators. In the set are 5 diagonal (=Cartan) generators $$ (0, 0, 0, 0, 0, 0)_1,(0, 0, 0, 0, 0, 0)_2,(0, ...
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Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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Is the group-theoretic Grothendieck-Springer resolution Calabi-Yau?

Any cotangent bundle is Calabi-Yau (by which I mean the canonical bundle is trivial), so the Springer resolution $T^*(G/B)$ is Calabi-Yau. I think that the Grothendieck-Springer resolution ...
3
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30 views

Can a (Hausdorff) infinite group have only finitely many equivalence classes of irreducible unitary representations?

(Where two representations are equivalent iff they are unitarily equivalent.)
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Multivariable polynomial matrix representations

This is a follow-up to matrix representation of parabola and matrix representation to generate monomials. I found a method to build such matrices to implement this type of functionality for one ...
3
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3answers
181 views

matrix representations and polynomials

I just investigated the following matrix and some of its lower powers: $$M = \left[\begin{array}{cccc} 1&0&0&0\\ 1&1&0&0\\ 1&1&1&0\\ 1&1&1&1 ...
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1answer
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Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
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1answer
28 views

support compact modulo subgroup

I am studying (co)-induced representations of topological groups and I came across the following situation: $G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, ...
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31 views

A qualifying-exam problem related to Clifford theorem in representation theory

I am not sure : does this problem can be solve directly by Maschke's theorem, which states that every representation of a finite group is completely reducible. Perhaps I made some stupid ...
3
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31 views

How do roots act on weights?

In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an ...
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1answer
53 views

Matrix representations of parabola.

Continuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...
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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 ...
2
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1answer
52 views

Looking for various proofs there is no faithful representation $\rho:S_4\rightarrow GL(\mathbb{R}^2)$

I stumbled on the question when thinking about a representation $\rho:D_4\rightarrow GL(\mathbb{R}^2)$ as symmetries of the four points $\{(\pm 1, \pm 1)\}$. There didn't seem to be a good geometric ...
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1answer
13 views

Cartan integers are preserved by isomorphism?

Suppose you have two roots systems $\Phi\subset E$ and $\Phi'\subset E'$ with bases of simple roots $(\alpha_1,\dots,\alpha_\ell)$ and $(\alpha_1',\dots,\alpha_\ell')$ such that the Cartan integers ...
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18 views

One dimensional representations of $SL_2(\mathbb{Z}/n\mathbb{Z})$

Someone knows a reference or knows how to calculate the linear character of $SL_2(\mathbb{Z}/n\mathbb{Z})$, for an arbitrary $n$? Thanks
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1answer
42 views

How many projectives and injectives exist in a path algebra?

I do not know an efficient way to determine whether a quiver representation is projective or injective. The definitions and properties such as "Projectives are summands of free modules", etc do not ...
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Looking for reference to solve a problem (representation theory, I think?)

this question appeared on an algebra qualifying exam: Let $R$ be the group algebra $\mathbf{C}[S_3]$. How many nonisomorphic, irreducible, left modules does $R$ have and why? Would I be able to ...
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1answer
27 views

Explicitly decompose $\mathbb{C}^3$ into irreducible representations of $S_3$.

Consider the permutation representation of $S_3$ acting by permuting the elements of a basis of $\mathbb{C}^3$. Explicitly decompose $\mathbb{C}^3$ into irreducible representations. Can someone ...
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1answer
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Does the additive action of $kG$ distribute over a tensor product?

Suppose we have vector spaces $V$ and $W$ which are $kG$-modules for some field $k$ and group $G$. We have that the tensor product $V \otimes W$ is also a $kG$-module under the diagonal action $$ g ...
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1answer
36 views

Why is the $\text{End}_A(M)$-module $\text{Hom}_A(N,M)$ finitely generated?

Let $A$ be an Artin algebra and let $M,N$ be some finitely generated modules in mod(A). Why is then the $\text{End}_A(M)$-module $\text{Hom}_A(N,M)$ finitely generated? Thanks for the help.
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Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
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1answer
36 views

Examples of irreducible representations

Which of the following representations are irreducible? 1) The tautological representation of $D_n$ on $\mathbb{R}^2$ 2) The action of $U(1)$ on $\mathbb{C}$ by multiplication 3) The ...
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1answer
36 views

equivalence of Lie group and Lie algebra intertwiner

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
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1answer
19 views

*$G$-invariant* symmetric bilinear form & $G'=\Bbb Z_2\times\Bbb Z_2$.

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
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Tensor product of representations of a product group?

Given some group $G$ that can be written as product of two other groups $$G = G_1 \times G_2 $$ and some representation of this group written in terms of representations of $G_1$ and $G_2$ $$R = ...
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Solving the Character table for $A_4$ and derived algebra characteristics.

Say I am given a group with 4 conjugacy classes $C_1, C_2, C_3, C_4$ with orders 1,3,4,4 respectively. I will call the conjugacy class characters $\chi_1,\chi_2,\chi_3,\chi_4$ respectively. I am ...
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Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & ...
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1answer
38 views

Fulton and Harris: Exercise 1.3 in section 1.1

This is exercise 1.3 on page 5 of Fulton and Harris Representation Theory: A First Course. Exercise: Let $G$ be a finite group, let $V$ be an $n$-dimensional $\mathbb C$-vector space and let $\rho: ...
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Stone representation theorem and right(or left-) one-sided ideals in a ring

Consider Marshall Stone's representation theorem: https://en.wikipedia.org/wiki/Stone's_representation_theorem_for_Boolean_algebras I would like to know in whichspecific way, if any, it is connected ...
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2answers
30 views

Find character table for symmetric group $S_3$

This group contains all permutations of 3 elements, so it has order 3!=6. Its three congruency classes are {1}, {(1,2),(1,3),(2,3)}, {(123),(132)}. As we know that the number of congruency classes ...
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1answer
20 views

Every base of a root system arises as indecomposable positive roots of a regular element?

I'm confused about a line in the Theorem p48 in Humphrey's's book on Lie Algebras. He's proving that every base $\Delta$ of a root system $\Phi$ arises as the set of $\Delta(\gamma)$ of ...
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Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
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2answers
83 views

Irreducible representations of $\operatorname{GL}_3(\mathbb{F}_q)$

I am trying to find all irreducible representations of $G = \operatorname{GL}_3(\mathbb{F}_q)$. I know that the order of $G$ is $(q^3-1)(q^3 - q)(q^3 - q^2)$ and the number of conjugacy classes is ...
2
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1answer
27 views

Existence of groups corresponding to dimensions of irreducible representations

If there are $r$ irreducible representations of a group $G$, we know that $|G| = \sum_{i=1}^{r}d_{i}^2$ and $d_i$ divides $|G|$. Suppose we have a decomposition of $N$ such that $N = ...
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1answer
40 views

Algebraic Peter-Weyl theorem in the case of $G=SL_2$.

The algebraic Peter-Weyl theorem says that for a linear reductive group $G$ we have $\mathbb{C}[G] = \oplus_{V} V \otimes V^* $, where $V$ runs over the set of all non-isomorphic irreducible ...
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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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1answer
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What would be the “action” in functional analysis?

I am reading Simmons' "Topology and Modern Analysis". He keeps bringing up the idea of studying the set of all structure preserving mappings to obtain info regarding the structure of a certain normed ...
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Equality on pg. 40 of Humphreys's Lie Algebras, $\kappa(t_\lambda,t_\mu)=\sum_{\alpha\in\Phi}\alpha(t_\lambda)\alpha(t_\mu)$?

I don't understand part of an equality on page 40 of Humphreys's book on Lie Algebras. Suppose $L$ is a semi-simple Lie algebra over an algebraically closed field of characteristic $0$, and $H$ a ...
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3answers
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All simple modules are projective $\Rightarrow$ semisimple [duplicate]

Let $A$ be a finite dimensional algebra over a field $K$. It is clear that if $A$ is semisimple, then every simple module is projective. Does the converse hold ? It seems false, but I can't find a ...
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1answer
42 views

Nonlinear representation of SU(2) and SU(4)

Consider a nonlinear representation of a group acting on the space of real vector $\phi_i$ in the form: \begin{equation} \phi_i\rightarrow \sum_jM_{ij}\phi_j+\delta\phi_i \end{equation} where $M$ is ...
4
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1answer
52 views

Right modules Vs Left modules.

I have been reading Frobenius Algebras, Volume 1 By Andrzej Skowroński, Kunio Yamagata. On page 18 I came across the following paragraph, and I founded interesting, I will quote it and then ask my ...
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1answer
38 views

Questions about fundamental representations of $SL_3/U$.

Consider the group $SL_3$. Let $U$ be the subgroup of $SL_3$ consisting of all upper triangular unipotent matrices. Then the algebra $\mathbb{C}[SL_3/U]$ is generated by $a_{11}, a_{21}, a_{31}, ...
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2answers
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Proving that an $FG$-homomorphism is surjective

Assume that $V$ is an $FG$-module.Prove that the subset $$V_0 = \{v \in V : vg = v \space \forall \space g \in G \}$$ is an $FG$- submodule of $V$. Also show that the function $$\phi: v \to ...
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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 ...