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

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Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1\geq\lambda_2\geq\ldots\geq \lambda_n)$. Is there any physical ...
3
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1answer
435 views

Converse of Schur's First Lemma

Suppose $G$ is a closed subgroup of $SU(d)$, $d>1$, and let $\rho$ be a $d$-dimensional special unitary representation of $G$. Suppose that if a matrix $A$ commutes with all of $\rho(G)$ for all ...
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1answer
89 views

Schur's first lemma for finitely generated continuous groups of $SU(d)$

Suppose that a finite set $S$ of $d\times d$ special unitary matrices densely generates a representation $\rho$ of a continuous subgroup of $G$ of $SU(d)$. That is, for every $\epsilon>0$ and ...
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2answers
527 views

Schur-Weyl Duality ( Classical ) and the Double Commutant reference request

I would like to ask for any reference suggestions on the topic of Schur-Weyl Duality for GLn ( directly GLn, not through the lie algebra ) and the double commutant theorem. The section on this ...
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1answer
101 views

Monodromy Representations

Let, be $V$ a connected smooth manifold and $q_1,q_2\in V$ and $F:U\to V$ a connected covering of degree $d$. This covering induces two monodromy representations $\rho_1:\pi_1(V,q_1)\to S_d $ ...
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105 views

Representations of direct sums of matrix algebras [closed]

I'm reading Introduction to Representation Theory by Pavel Etingof et al. and I want to do most of the exercises. But I must stop by the exercise on page 25. I can prove part (a) easily by direct ...
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1answer
231 views

Some irreducible characters of the Symmetric group $S_n$

I want to have characters of some irreducible $S_n$-modules corresponding to certain partitions $\lambda$ of $n$, the computations using Frobenius formula get complicated and I am unable to find in ...
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86 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
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2answers
319 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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47 views

Product of two squares in finite groups

Let $G$ be a finite group and $g$ an element of order $n$ in $G$. Assume that $g$ is a product of two squares. Moreover, assume that $n$ and $k$ are coprime. Prove that $g^k$ is also a product of two ...
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1answer
74 views

Decomposition of representation of Problem In Artin

I am trying to solve the problem: Decompose the standard representation of the cyclic group $C_{n}$ in $\mathbb{R}^{2}$ by rotations into a direct sum of irreducible representations. What I have ...
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Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
3
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1answer
257 views

Nice applications of the Haar measure

The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is ...
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1answer
310 views

Multiplicity of a completely reducible representation in another irreducible representation.

I have got the next question that I am pondering the answer to. Let $\tau$ be a completely reducible representation of finite dimension of a group $G$, and let $\pi$ be another irreducible ...
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1answer
98 views

$S_n$ has only four (irred.) representations with degree $<n$ (for $n>6$)

I'm working on the following exercise: For $n\ge 7$, $S_n$ has no irreducible representations of dimension $m$ with $2\le m\le n-2$. There is a solution here but I'd like to follow the ...
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1answer
66 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
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2answers
306 views

Irreducible Representations

Let $G$ be a group, $F$ a field, and $V$ be an $F[G]$ module (equivalently $F$-representation of $G$). The following definition is well-known. Definition 1. We say that $V$ is irreducible (or simple ...
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1answer
144 views

Non-Isomorphic induced representations (from the same representation of a subgroup)

I believe that it is true that if we have a group $G$, and two copies $H_1$, $H_2$ of some group $H$ as subgroups of $G$, we can fix a representation $V$ of $H$ and have the situation: ...
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1answer
55 views

isomorphism of an irreducible $\mathbb{R}$ represetation

Theorem: Every irreducible $\mathbb{R}$-representation of the real algebra $\mathbb{R}(n)$ is isomorphic to $\mathbb{R}^n$, where the matrix A ∈ $\mathbb{R}(n)$ acts via left matrix multiplication. I ...
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1answer
430 views

Difference between the SU(2) and SO(3) lie groups and their lie algebras

In many places I have seen the SU(2) and SO(3) lie algebras used interchangeably. How are they exactly identical? Moreover, what about their lie groups? Are they identical as well. It would be great ...
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353 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
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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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2answers
61 views

Checking that $\rho$ is a representation

Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
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1answer
97 views

Error in Weyl character formula computation.

I need someone with a keen eye for errors. I am trying to use the Weyl character formula for the symplectic group Sp$(4,\mathbb{C})$ on certain matrices coming from 2x2 quaternion matrices. Summing ...
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2answers
753 views

Finding invariant subspaces

Let $x$ be a variable. Denote by $V$ the vector space consisting of all polynomials $P(x)=ax^2+bx+c$ of degree not more than 2, with complex coefficients. For any real number $t$ determine an operator ...
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94 views

Cartan decomposition of unitary group

For number field $F$, we consider two case 1) $E$=quadratic extension of $F$ , 2) $E = F \times F$. Let V be a 2-dim hermition space over E. In 1) case, by Cartan decompostion $U(2)$ can be ...
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Proofs that the degree of an irrep divides the order of a group

It is a theorem in basic representation theory that the degree of an irreducible representation on $G$ over $\mathbb{C}$ divides the order of $G$. The usual proof of this fact involves algebraic ...
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1answer
134 views

Why is the tensor product of two pseudoreal representations real?

Let G be a group, and $\rho : G \to GL(n, \mathbb{C})$ be a representation of $G$. Then we also get the conjugate representation $\rho^* : G \to GL(n, \mathbb{C})$, where $\rho^*(g) = ...
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1answer
82 views

Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
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1answer
141 views

Cartan or Coxeter matrix of an algebra of infinite global dimension

Let $(Q, I)$ be a bound quiver such that $A=KQ/I$ has infinite global dimension. I want to ask the following questionss: (1) Is the Cartan matrix $C_A$ of $A$ invertible in the matrix ring ...
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1answer
83 views

harish chandra for sl(2,C)

Is it true that each irreducible sl(2,$\mathbb{C}$)-module, $P(\lambda,\mu)$ with $\lambda \in \mathbb{Z}$ appears as the harish chandra module of some $(\pi_{\chi},V_{\chi})$ And given ...
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69 views

Exactness Properties of Schur Functors

The title says it all: What are the exactness properties of Schur Functors? Thanks!
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241 views

how to find all simple modules for the given path algebra

Let $A = KQ$, where $Q$ is the quiver $$\begin{array}{ccc} & \alpha & \\ 1 & \rightleftarrows & 2 \\ & \beta& \end{array}$$ are there simple right $A$-modules with dimension ...
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Trying to get a character table of $S_{4}$ from a character table of $A_{4}$.

I have constructed a character table for $A_{4}$ and need to use induced representations to get a character table for $S_{4}$. I'm not very confident with the concept of induced representations, but ...
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1answer
106 views

Isomorphic representations on exterior powers

Exercise from F+H, Exercise 1.3: Let $\rho : G \rightarrow GL(V)$ be any representation of the finite group $G$ on a $n$-dimensional vector space $V$ and suppose that for any $g \in G$ the ...
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1answer
72 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
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When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
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1answer
76 views

Formula for idempotents in $\mathbb{C}G$

Let $G$ be a finite group with $|G|=n$. Label the irreps $V_1,\ldots , V_t$ of $G$ over $\mathbb{C}$; let $d_i$ denote the degree of $V_i$. By Maschke's theorem we have $\mathbb{C}G\cong ...
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1answer
50 views

Relating Modules and Representations

I am currently studying representation theory and am struggling with the concepts which relate modules and representations. The specific question I am looking at right now is this: but while ...
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1answer
577 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
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1answer
345 views

how to get the injective envelope and projective cover of a given module

Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous ...
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38 views

Representations of $\mathrm{SU}(n)$

I have been given the following representation of $\mathrm{SU}(n)$: Let $V_{k,n} \leq \mathbb{C}[z_1,\dots,z_n]$ be the subspace spanned by the degree-$k$ homogeneous polynomials and define ...
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A natural way of thinking of the definition of an Artin $L$-function?

Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 ...
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2answers
114 views

The character of a representation

Let $\chi$ be the character of a representation of a simple group $G$ and let $g\in G$. If $g$ has order two and $G\neq C_2$ then show that $\chi(g)\equiv \chi (e)$ modulo 4. The hint I get is to ...
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58 views

Different orderings for highest weights of a representation

Recall that given a representation $\pi$ of $\mathfrak{sl}_n$, a weight $\mu$ is said to be of highest weight if its corresponding weight vector is annihilated by all the positive root spaces (1). ...
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356 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
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218 views

the semisimple and local properties of path algebras

Let $Q$ be a finite quiver. Then the following hold: (a) If $KQ$ is semisimple, then $|Q_1| = 0$. If, moreover, $Q $ is connected, show that: (b)$KQ$ is local only if $|Q_0| = 1$ and $|Q_1| = 0$,
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235 views

Examples: Representations over finite rings and Maschke's theorem

Is there a possibility to get the simple $R[G]$-modules, if $R$ is the ring $\mathbb{Z}/n\mathbb{Z}$, $G$ a finite group and $\operatorname{ord}(G)$ and $n$ are relatively prime? For which groups ...
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65 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
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1answer
350 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...