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

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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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18 views

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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1answer
35 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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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 ...
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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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150 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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40 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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33 views

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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36 views

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
23 views

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
45 views

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
43 views

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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38 views

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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92 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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21 views

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
49 views

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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1answer
43 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
17 views

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
58 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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42 views

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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18 views

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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30 views

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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32 views

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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31 views

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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25 views

$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-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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21 views

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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1answer
64 views

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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26 views

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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1answer
27 views

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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20 views

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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1answer
21 views

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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45 views

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 ...
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28 views

Why is U(n) a real form of GL(n)

When $n=1$, we see that $U(1)$ is defined by the equation $z\bar z=1$, hence $a^2+b^2=1$ for $z=a+bi$. Taking complex $a,b$ we see that the solutions are nonzero complex points, hence $U(1)$ is ...
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25 views

Liesubgroups given by representations

Maybe it's a little odd question, but I encountered a classification of homogeneous spaces, and I'm stuck with the following description (c.f. Onishchick, "Topology of transitive transformation ...
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1answer
34 views

Reference request: what is the relation between classical r-matrices and quantum R-matrices?

I learned from a professor that $$ R=Id+(q-1)r+ o(q-1), $$ where $R$ is a quantum $R$-matrix and $r$ is the corresponding classical $r$-matrix. Here $o(q-1)$ denotes a term of the form $A(q-1)^2$, ...
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1answer
48 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
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1answer
28 views

Characters on a torus.

Let $T = (\mathbb{C}^*)^n$. It is said that the characters on $T$ must be of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some $a_1,\ldots, a_n \in \mathbb{Z}$. Is it possible that $a_i ...
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1answer
22 views

Uniqueness of a map.

Let $f: \mathbb{Z}^n \to \mathbb{C}^*$ be a homomorphism. Where $(\mathbb{Z}^n,+)$ is considered as an additive group and $(\mathbb{C}^*$ is considered as an multiplicative group. Fix $b_1,\ldots, b_n ...
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What is the free algebra $A=k\langle X_1,…,X_n \rangle$? And why is it an algebra?

I'm reading a book, where they claim that the free algebra $A=k\langle X_1,...,X_n\rangle$ is an algebra. I've never seen this notation and I've never heard of the free algebra, so I wonder how this ...
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1answer
35 views

Conjugacy class name of the product in ATLAS

Well, I'm trying to read "ATLAS of Finite Groups". To be more precise, I'm interested in character tables of some Weyl groups. Is it possible to determine the conjugacy class name of the product of ...
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1answer
51 views

When does a short exact sequence of representations exist?

The context for this question is that I am trying to determine the Grothendieck group of finite-dimensional complex representations of $T = (\mathbb{C}^*)^n$, where $\mathbb{C}^*$ denotes the ...
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1answer
152 views

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
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34 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
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1answer
62 views

What is known about the representation theory of the symmetric group over $\mathbb{F}_2$

There is a lot of material available about the representation theory of the symmetric group over $\mathbb{C}$ and fields of characteristic $0$. In particular, there is the decomposition of the group ...
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26 views

Scalar Products on the Rational Function Field

Let $\mathbb{R}(t)$ be the rational function fields over $\mathbb{R}$. Are there scalar products $\langle -,- \rangle$ on $\mathbb{R}(t)$ such that multiplication with $t$ is selfadjoint, i.e. ...
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Is $\mathbb{C}[N]$ isomorphic to $U(\mathfrak{n})$?

Let $G$ be an algebraic group and $N$ its maximal unipotent subgroup consisting of all upper triangular unipotent matrices. Let $\mathfrak{n}$ be the Lie algebra of $N$. It is said that ...