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

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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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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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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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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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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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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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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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40 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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68 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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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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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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35 views

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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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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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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166 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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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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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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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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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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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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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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64 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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52 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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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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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 ...
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Computing regular representations

I'm doing a reading of representation theory (not a class) but don't really understand what $GL(\mathbb{F}(G))$ is. We defined $GL(n,\mathbb{C})$ in the usual way (set of matrices with nonzero ...
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Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
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Why center acts by scalars?

Let $\mathfrak{g}$ be a Lie algebra. Let $U(\mathfrak{g})$ be its universal enveloping algebra. Let $Z(U(\mathfrak{g}))$ be the center of $U(\mathfrak{g})$. Let $V$ be an irreducible representation of ...
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$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
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Real or complex representation

How can one know for a given algebra $\frak{g}$ if a specific representation is real or complex? For example if $\frak{g}=so(10)$ how can one know that the representation $\underline{16}$ is complex? ...
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Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
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Structure of $G$-invariant bilinear forms over (finite?) fields

I have a question about the structure of $G$-invariant bilinear forms. Let $G$ be an arbitrary finite group and $\mathbb{F}_q$ a finite field such that $2|G|$ is not divisible by the characteristic of ...
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For a nilpotent Lie subalgebra, $\mathfrak{h}$, is $ad(\mathfrak{h})$ simultaneously diagonalizable if each $ad(H)$ is diagonalizable?

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}\subseteq \mathfrak{g}$ be a nilpotent subalgebra such that for every $H \in \mathfrak{h}$, the adjoint map $ad(H): \mathfrak{g} \rightarrow ...
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Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
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A question on universal Coxeter group

In the set up of Lusztig's Hecke algebra with unequal parameters, let $W$ be a universal Coxeter group with finite many simple reflections, that is, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
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Ordinary irreducible representations of semi-direct products

What is the best source for learning about constructing irreducible representations of semi-direct product $G=N \rtimes_\phi H$ from irreducible representations of $H$ and $N$ over field of complex ...
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Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
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proof of basic fact that torus actions are diagonalizable

Suppose a torus $T=(\mathbb{C}^\ast)^n$ acts on a finite dimensional vector space $W$, and define for $m \in M$ ($M$ is the character lattice of $T$) the eigenspace $W_m$ by $$W_m = \{w \in W \mid ...