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

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Repeated Irreducible Representations in a representation

I'm reading through Serre's - Linear representations of finite groups. He has the following theorem (theorem 4 of chapter 2): Let $V$ be a linear representation of $G$, with character $\phi$ and ...
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33 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
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Presentation for Kernel of Induction map from $\oplus R(H)$ to $R(G)$

This is an exercise in Serre's representation theory book: Suppose that $X$ is a family of subgroups of $G$ stable under conjugation and taking subgroups Let $N$ be the kernel of map $\mathbb{Q} ...
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14 views

Signed column sum for Young tableau $t$

I am currently working on the book The Symmetric Group by Bruce Sagan. The following passage comes before introducing Specht Modules: Suppose that the tableau $t$ has rows $R_1, R_2, ..., R_l$ ...
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30 views

Irreducible Representations of $<X,Y>/\{[X,Y]=Y\}$

I was doing exercises from Etingof's Introduction to Representation Theory and came across this problem. $2.16.2$ Find all irreducible representations of the Lie algebra $L$ with generators $X$ and ...
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30 views

Basic Manipulation of Adams operations in R(G)

This is part of an exercise in Serre's representation theory book I am self-studying, but mostly it is about manipulation of symmetric polynomials. Let $\rho$ be a representation of a finite group ...
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15 views

Dimension of a weight space which is of weight $0$.

Let $V$ be a module of a Lie algebra $\mathfrak{g}$ and $V_{0}$ be the weight space of $V$ of weight $0$. $$ V_0 = \{ v\in V: h.v = 0, h \in \mathfrak{h} \}, $$ $\mathfrak{h}$ is a Cartan subalgebra ...
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11 views

Matrix representation of Boolean algebra?

Is there such a thing as matrix representations of Boolean algebra? Give a boolean algebra with finite elements {a,b,c...} and operations $\cap, \neg$, we can regard $\cap$ as matrix multiplication ...
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18 views

How to compute $\lambda(h_i)$?

Let $\lambda$ be a weight and $h_i = h_{\alpha_i} \in \mathfrak{h}$, $\alpha_i$ is a simple root. $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $\mathfrak{g}$. How to compute $\lambda(h_i)$? ...
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35 views

Representations of group algebra and its centre

Are the irreducible representations of the algebra $Z(\mathbb{C}G)$ for a finite group G all irreducible representations of the algebra $\mathbb{C}G$, i.e. are the representations of the group algebra ...
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39 views

Classification of separable algebras up to Morita equivalence

Is there a simple classification of separable algebras up to Morita equivalence, working over a particular field $k$? For example, over $\mathbb{C}$, every separable algebra is Morita equivalent to ...
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24 views

Measure in dual group - Kirillov theory

Let $G$ be a nilpotent connected, simply connected lie group. With the orbit method Kirillov describes the classes of equivalence of all irreducible unitary representations. Hence one identifies the ...
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35 views

Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
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81 views

Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
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76 views

Decomposing a matrix representation

I am currently working on the following problem: Assume that $X$ is a reducible matrix representation of the form \begin{equation} X(g)=\left( \begin{array}{c|c} A(g) & B(g)\\ \hline ...
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95 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
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precise definition of “irreducible representation” (of associative algebras with unit)

Let $K$ be a field and $A$ an associative $K$-Algebra with unit. By a representation of $A$ I mean a homomorphism of $K$-Algebras with unit $f\colon V\rightarrow{End}_K(V)$ where $V$ is a finite ...
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68 views

Is the tensor product of two representations a representation?

I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie ...
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35 views

Simultaneous diagonalisable matrices

I am well aware that there are already several questions and posts regarding the following topic. However, I could not find any answer to the following problem in Bruce Sagan's book The Symmetric ...
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28 views

Unitary matrix for matrix representation

In the book The Symmetric Group the author says: Let $\chi$ and $\psi$ be characters of the $G$-module $V$. By picking an orthonormal basis for $V$, we obtain a matrix representation $Y$ for ...
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5answers
109 views

Getting an intuitive feel for induced representations

I'm reading about induced representations for research. Particularly, I'm trying to get a firm grasp on the finite group case before venturing on to the locally compact case. I've been looking at ...
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68 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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24 views

Multiplicity of G-module

I am currently working on Bruce Sagan's The Symmetric Group. The following proposition is given without proof: Let $V$ and $W$ be $G$-modules with $V$ irreducible. Then dim Hom($V$,$W$) is the ...
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Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
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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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Invariants of the symmetric group

Let $V_\lambda$ be an irreducible representation of the symmetric group $S_n$ as usual labeled by parition $\lambda$ of $n.$ Question. Is there any general information about the algebra of ...
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25 views

Commutant Algebra of Matrix Representation

I am currently working on Bruce Sagan's The Symmetric Group. In the following example they show that for a representation that contains 2 different subrepresentations the commutative algebra Com$X$ ...
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1answer
26 views

Irreducible representation - Eigenvalues of Matrix

I am currently working at Bruce Sagan's "The Symmetric Group". The following example is an illustration to show that Maschke's Theorem is not true for infinite groups. The following paragraphs are ...
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18 views

Trace functionals as invariant elements of $R[\mathfrak{g}]$ under $G$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ and let $G$ be its inner automorphism group. Then $G$ acts on $R[\mathfrak{g}]\cong S(\mathfrak{g}^*)$ via $(\sigma\cdot f)(x) = ...
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91 views

Inner product in Maschke's Theorem

I am working through Maschke's Theorem on page 16 in Bruce Sagan's The Symmetric Group: In order to prove the theorem the author constructs an inner product $\langle v, w \rangle' = \sum_{g \in G} ...
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Order of center of character

I am working on a course in representation theory and I've got completely stuck on some exercises regarding $|G:Z|$ where $G$ is a finite group with an irreducible representation $\theta : ...
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65 views

How we apply representation theory to physics.

I want to have a concrete idea of what people do with representation theory in physics. Here is what I think: Corresponding to a specific "physics", there is particularly a Lie group (called G) of ...
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Representation theory in physics

0 down vote favorite I'm sorry if this is somewhat a dumb question. First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements ...
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1answer
25 views

Question in Fulton and Harris regarding induced representation.

I'm confused by the following paragraph: I don't see why $g\cdot W$ depends only on the left coset $gH$. What does he mean precisely by that? Why is it true that $gh\cdot W = g\cdot(h\cdot W) = ...
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Every unitary representation of a compact group is a direct sum of irreducible representations.

I've read nice proofs of a few different variants of the Peter-Weyl theorem and its corollaries. For instance I know that for $G$ a compact group, $L^2(G)$ is a Hilbert space direct sum of the matrix ...
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1answer
31 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
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61 views

Associated idempotents

Let $e$ and $f$ be elements of an associative algebra $A$. We say $e$ and $f$ are associated if there exist elements $x, y \in A$ such that: $$e = xy, f = yx.$$ My teacher said it is an easy exercise ...
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Quadratic Casimir of fundamental irreps of simply-laced Lie algebras [migrated]

I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It ...
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16 views

Constructing all inequivalent faithful irreducible projective representations of finite abelian groups

Let $G$ be a finite abelian group, and $\alpha \in H^2 (G,\mathbb{C}^*)$ a 2-cohomology class. It is known (in Karpilovsky's multi-volume tome or elsewhere) that a finite abelian group admits a ...
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69 views

Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
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28 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
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Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...
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Finding a basis and weight space for $L = so_6(\mathbb{C})= \{x \in End(\mathbb{C}^6)|^txS + Sx = 0 \}$

The question: Let $S = \left(\begin{array}{cc} 0 & I_3 \\ I_3 & 0 \end{array}\right)$ and let $$L = so_6(\mathbb{C})= \{x \in End(\mathbb{C}^6)|^txS + Sx = 0 \}$$ 1) Find a basis for $L$ ...
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Confusion regarding PBW theorem

I was reading up Humphrey's Introduction to Lie Algebras and Representation Theory and have a confusion regarding a consequence of PBW. First some notations: Let $L$ be a Lie algebra over ...
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32 views

In a semi-simple module, any submodule is a direct factor?

I need help understanding the following : In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple) (i) How is this equivalent to the ...
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Decomposition of a specific $SO(n)$-representation into irreducible ones

I need to decompose a specific $SO(n)$-representation into irreducible ones, but my background on representation theory is rather weak, so I post the problem here. Let $V$ be a $n$-dimensional real ...
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54 views

A representation of $G$ over $V$ gives $V$ the structure of a $G$-module?

In Fulton and Harris's book Representation Theory: A First Course, they define a representation of a finite group on $V$ in Lecture 1. Then they say that the representation gives $V$ the structure of ...
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140 views

Why is the Plancherel measure interesting?

One can average a class function $f:G\to\Bbb C$ for a finite group $G$ by interpreting $f$ as a complex-valued function on the space ${\rm cl}(G)$ of conjugacy classes and computing the expectation ...
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35 views

Character Theory Exercise

I am having trouble with the following exercise in character theory: If $\chi, \psi, \zeta$ are irreducible characters of a finite group then $\langle \chi\psi, \zeta \rangle \leq \zeta(1)$. I can ...
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A Question on integration formula on $KAK$ decomposition

The following proposition appears in page 141 in Knapp's book, representation theory of semisimple groups. Let $G$ be linear connected reductive, and fix a positive system $\Sigma^+$ of restricted ...