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

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Fixed Spaces for Group Elements

what is the GAP code for finding the fixed space? A list of row vectors that form a base of the vector space $V$ such that $v M = v$ for all $v$ in $V$ and all matrices $M$ in the list $mats$.
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1answer
78 views

representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition). Define a representation $\pi_{m,\,n}: \text{SU}(n) \to ...
4
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57 views

How to check if a representation of su(2) is irreducible

I have found a representation $\rho$ of the group $G=Su(2)$. I want to show that this representation is irreducible but I don't know how. Finding all invariant subspaces seems very difficult. I ...
3
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1answer
42 views

Character of a tensor product of $\mathfrak{sl}_2$-modules

Let $V$ be a finite-dimensional $\mathfrak{sl}_2$-module. There is a standard base $\{e,f,h\}$ in $\mathfrak{sl}_2$, I use standard notation ($h$, for instance, is the diagonal matrix with $1$ and ...
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588 views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
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82 views

“Converse” Schur's lemma [duplicate]

For representations over an algebraically closed field one can formulate Schur's lemma in the following form: Every endomorphism of irreducible representation is of the form $\lambda\cdot id$ I ...
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Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
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Decomposiom of the representation of $SU(N)$

Let $T$ be the "fundamental" representation (I mean the one in which the matrices representing the group elements are simply themselves) of $SU(N)$ group. I have \begin{pmatrix} SU(N-1)& 0\\ ...
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26 views

Representatives of simple $\mathbb C[\mathbb D_3]$-modules (left modules)

Problem Let $\mathbb D_3$ be the symmetry group of the equilateral triangle. Give a complete list of the representatives of the simple left $\mathbb C[\mathbb D_3]$-modules. My attempt at a solution ...
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50 views

Is the restriction of the regular representation of a finite group always a multiple of the subgroup?

For an inclusion of groups $H \hookrightarrow G$, define the restriction $\operatorname{Res}^G_H$ of representations as precomposition with the inclusion map. Also, define the complex regular ...
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45 views

No invariant complement?

How do I show that the representation $\rho: \mathbb{Z} \to \text{GL}_2(\mathbb{C})$ with $$\rho(1) = \begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$$ has an invariant subspace with no invariant ...
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56 views

No character theory, any representation ${\bf{GL}}_N(\mathbb{C})$ is reducible, upper bound

Let $G$ be any finite group. How do I show without character theory that there is a number $N = N(G)$ so that any representation $\rho: G \to {\bf{GL}}_N(\mathbb{C})$ is reducible (and finding an ...
2
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1answer
37 views

Subalgebra condition in Engel's theorem

An equivalent version of Engel's theorem says that Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then there exists ...
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1answer
30 views

Lie subalgebra in $Der(\mathbb{C}[z])$ isomorphic to $\mathfrak{sl}_2$

I am to prove that $\{(az^2+bz+c)\frac{\partial}{\partial z}:a,b,c\in\mathbb{C}\}$ regarded as a Lie algebra is isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. I guess it is possible to build a basis ...
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45 views

Extensions of representations

I'm again confronted with an exercise from Etingof's book "Introduction to representation theory" (page 30 of http://math.mit.edu/~etingof/replect.pdf) Problem 2.22. Let ...
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108 views

random walk on finite cyclic group

Suppose that I have a random walk on the finite cyclic group of order $d > 2$, where the initial probability distribution $Q$ assigns the values $p, q, r$ to $-1, 0, 1$, respectively, where $p + q ...
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Is $SL(2, 3) $ a subgroup of $SL(2, p)$ for $ p>3$?

As the title says, I was wondering whether $SL(2,3)$ is a subgroup of $SL(2,p)$ for $p>3$. I know that it is for $p=5$ (it can be found explicitly using the quaternionic representation), and I ...
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matrix representation of free group with metric requirement

Look at this Cayley diagram of the free group generated by 2 elements, $F_2 = \langle a, b \rangle$: The 2 elements marked by green and pink are "unrelated" in the sense that they are far apart in ...
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Matrix representations of free groups?

What is the general form of faithful matrix representations of free groups? How about for the simple case of $F_2$?
4
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78 views

Exercise on Induced Representations of 1 dimensional complex representation

I'm having a hard time trying to solve the following problem coming from Eingof's book "Introduction to representation theory" (page 55 of the book in PDF format ...
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34 views

Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
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Is a semidirect product of linear groups a linear group?

It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product? Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, ...
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2answers
51 views

Why representations become functions?

I am trying to answer Problem 5 below, but why irreducible representations become functions(i.e. $f_{m}$), aren't representations homomorphisms from $G$ to $GL(V)$? (It would also be helpful if ...
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Questions for compact lie group representations.

I have two questions about representations of compact Lie group. If all irreducible representations of a compact Lie group are one-dimensinal, then G is abelian. An infinite compact Lie group has an ...
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12 in the definition of Virasoro algebra and Regge symmetry

In the definition of Virasoro algebra, there is a following condition on the generators: $[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$ Now, Regge symmetry is the following ...
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How to determine $Hom (M,M)$ for an irreducible $R$-module $M$?

More exactly, I'm considering $R$ to be a finite dimensional $\mathbb C$-algebra. For any $R$-module $M$, the $\mathbb C$ vector space $Hom_R(M,M)$ contains scalar multiplication and hence contains ...
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1answer
23 views

Can we define unitary representations on semigroups

A representation on a semigroup $S$ is a pair $(\pi,H_\pi)$ where $\pi$ is a homomorphism from $S$ into $B(H_\pi)$ and $H_\pi$ is a Hilbert space. In the group case, a representation $\pi$ of a group ...
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462 views

Can someone tell me in dummy terms what Left and Right regular representations are.

In my book it just says that the left regular representation is the map f in the Cayley's Theorem proof, but I just don't understand what are they? Why we need to find them?
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1answer
40 views

Should Ext-quiver be a full sub-quiver of its AR-quiver for a basic hereditary algebra A over algebraic closed field K?

For a basic hereditary algebra A over algebraic closed field K, prove its Ext-quiver $\Gamma_{A}$ is a full sub-quiver of its AR-quiver $\Delta_{A}$. I have no clue for this.
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1answer
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Lie algebra representations and tensor product decompositions.

Find the weights for $V_{L_1 - 2L_3}$, where $L_1, L_2, L_3$ are the weights for the standard representation of $\mathfrak{sl}_3 \Bbb{C}$ on $V \cong \Bbb{C}^3$. In order to find these weights, ...
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105 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
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2 smooth algebraic varieties, isomorphism $\mathcal{D}_{X \times Y} \to p_1^* \mathcal{D}_X \otimes p_2^*\mathcal{D}_Y$ as sheaves of rings?

Let $X$ and $Y$ be two smooth algebraic varieties. How do I see that I have an isomorphism$$\mathcal{D}_{X \times Y} \overset{\sim}{\to} p_1^* \mathcal{D}_X \otimes p_2^*\mathcal{D}_Y$$as sheaves of ...
3
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4answers
112 views

A trigonometric integral identity

How can we prove the following identity? $$ \int_{0}^{2\pi}\cos^{n}\left(\,\theta\,\right) \sin\left(\,\left[\,n + 1\,\right]\theta\,\right)\sin\left(\,\theta\,\right) \,{\rm d}\theta ={\pi \over ...
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Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
2
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2answers
99 views

$SU(2)$ acting by conjugation, decomposition into irreducibles

I am attempting past exam questions of the Cambridge Math Tripos. I know how to do the first few parts, which involves giving the irreducible representations of $U(1)$ and $SU(2)$. But I am not sure ...
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1answer
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Decomposition into irreducibles of representations of semisimple Lie groups.

Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ it's Lie algebra. Then $\mathfrak{g}$ is semisimple. Let $V$ be a finite dimensional representation of $G$. Viewing $V$ as a ...
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1answer
41 views

Irreducible representation

I know that correspondence every pure state on a C*-algebra $A$, there is an irreducible representation of $A$. Also we have the following theorem: Let $A$ be a C*-algebras and $(\pi,H)$ be an ...
1
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1answer
23 views

Restriction of an irreducible representation

Let $A$ be a C*-algebra and $\pi:A \to B(H)$ be a irreducible representation. Could we claim $\pi_{|B}$ is an irreducible representation if $B$ is a C*-subalgebra of $A$ ?
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construction of an injective representation of $C_0(X)$

Let X be a locally compact noncompact Hausdorff space and consider the C$^*$-Algebra $C_0(X)$ of continuous functions vanishing at infinity. I want to construct an injective *-represenatation of ...
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Duals of representations of affine group schemes, in particular $\mathrm{GL}_n$

Duals of representations of affine group schemes Let $R$ be a commutative ring. If $G$ is a group and $V$ is a dualizable i.e. finitely generated projective $R$-module on which $G$ acts, then it is ...
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0answers
24 views

How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?

Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$: $$ ...
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1answer
54 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
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3answers
95 views

Infinite dimensional representation such that every subrepresentation is reducible

Let $V$ be a nonzero finite dimensional representation of an algebra $A$. a) Show that it has an irreducible subrepresentation. b) Show by example that this does not always hold for ...
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1answer
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character of irreducible representations of odd-ordered groups

I want to prove that if $G$ is a group and the order of $G$ is odd, and $\chi$ is a real-valued irreducible character of $G$, then $\chi$ must be the trivial representation, $\chi = \epsilon$. So ...
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1answer
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Difference between to Tensor products with regards to modules

What would be the difference between $$ \otimes_B $$ and $$ \otimes $$ both in the following context and in general? Let A be a ring with $$ B \subset A $$ and M a B-Module. We can construct the ...
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Irreducible representations of $S_n$ [duplicate]

I want to prove that $S_n$ has an irreducible representation of dimension $n-1$. Intuitively, I know that the $\forall n$, the trivial representation is irreducible, and there should be some ...
3
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On Schur map and tableaux

My post refers to Jerzy Weyman's "Cohomology of vector bundles and syzygies" pag. 37. Let $R$ be a ring and $E$ a free $R$-module of rank $n$. Let ${e_1, \cdots, e_n}$ a basis of $E$. Let us consider ...
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2answers
49 views

If $V$ is a $\mathbb CG$-module then we may take $\rho(g)$ as a diagonal matrix?

If $G$ is a group and $\mathbb K$ is a field let $\mathbb KG$ be the usual group ring. We know a representation $\rho:G\longrightarrow GL(V)$, where $V$ is a $\mathbb K$-vetor space, is the same as a ...
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1answer
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Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
3
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2answers
137 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...