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

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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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1answer
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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 ...
3
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1answer
101 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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1answer
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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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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 ...
3
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1answer
38 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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1answer
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
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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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70 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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2answers
27 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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25 views

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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1answer
29 views

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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0answers
18 views

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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1answer
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$X$ ...
2
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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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20 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) = ...
3
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2answers
93 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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1answer
67 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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3answers
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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
34 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 ...
3
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1answer
65 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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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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1answer
73 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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31 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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30 views

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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1answer
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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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1answer
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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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19 views

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 ...
3
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1answer
58 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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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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1answer
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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 ...
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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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1answer
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What does it mean an isomorphism of a Dynkin diagram induced by some $w \in W$.

I read some papers encounter the concept " an isomorphism of a Dynkin diagram induced by some $w \in W$ ". Let's consider the Dynkin diagram $$ 1 \to 2 \to 3 \to 4. $$ I found that $\phi(1)=4, ...
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1answer
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Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
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1answer
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Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of ...
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Dimension of the space of tensors obtained by making partial symmetrizations and skew-symmetrizations.

Let $A=(a_{i_1\dots i_k})_{i_1,\dots,i_k=1}^n$ be a higher order cubic tensor or hypermatrix. The following two facts are well-known and are easy to prove: ${(\bf 1) }$ The dimension of the ...
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1answer
91 views

On the indecomposable decomposition of the reduction of an integral representation

Here is a problem I have been grappling with all day. I started out thinking it might be true but am now inclined to believe it is false, and would like to see a counterexample. Suppose $G$ is a ...
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Clarification on the wording of a Representation Theory problem in Dummit and Foote, 3rd Edition

I'm a bit confused by problem 21 in section 18.1 in Dummit and Foote, which says the following: "Let $G$ be a noncyclic abelian group acting by conjugation on an elementary abelian $p$-group $V$, ...
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1answer
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Decomposing direct product of irreps

I know characters of two 2-dimentional irreps (U and V) of a group with 6 conjugate classes. The characters are: $\begin{pmatrix} 2&-1&-1&2&0&0\end{pmatrix}$ and $\begin{pmatrix} ...
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1answer
24 views

Show a Representation is Indecomposable

I am working through the open course notes from MIT for Representation Theory. One problem given is Let $A = k[x_1,...,x_n]$ and $I \subset A$ be any ideal in $A$ containing all homogenous ...
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42 views

Decomposition of tensor product into direct sum of fields

If I have tensor product of two fields $V_1\otimes V_2$, what is the general approach to decompose this product into a direct sum of fields? In particular, I have $\bullet\;\Bbb Q(\sqrt 2) ...
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1answer
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Representation of $\mathfrak{sl}_2(\mathbb{C})$ corresponding to Lie algebra representation

We have a representation $R$ of a Lie group $\mathrm{SL}_2(\mathbb{C})$ in the space of polynomials $\mathbb{C}[x,y]$ such that $R\begin{pmatrix} a & b \\ c & ...
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1answer
56 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
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global section of some sheaves

Let $\mathrm{Grass}(r,V)$ be the Grassmannian over a field $k$. What is $H^0(\Sigma^{\alpha}(S))$, where $S$ is the tautological sheaf and $\Sigma$ the Schur functor. In characteristic zero this is ...
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Questions about an action of $U(\mathfrak{g})$.

Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ its universal envoloping algebra. Let $G$ be the Lie group of $\mathfrak{g}$ and $U$, $B^{-}$ the upper unipotent subgroup and lower Borel ...