Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
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$\operatorname{Res}(V+W)=\operatorname{Res}(V)+\operatorname{Res}(W)$?
if there are two $R[G]$ Modules $V,W$ and $R$ some ring, $S$ subgroup of $G$. Is the formula $$\operatorname{Res}_S (V \oplus W) = \operatorname{Res}_S (V) \oplus \operatorname{Res}_S (W) $$
true? I ...
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1answer
16 views
Representations - Tensor Product prove properties of tensor product
I have a problem:
Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
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1answer
30 views
Braid Group of a Weyl Group
I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman.
The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of ...
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3answers
103 views
Are Clifford groups very *non-commutative*?
Clifford groups seem to be very non-commutative by the relation \begin{equation}
\gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}.
\end{equation} But is it really so? Can we put this degree of ...
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47 views
Deciding whether or not a class of modules is “big enough”
For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
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17 views
Computing a restriction of a representation
It is known (Fulton and Harris p.427 among other papers) that the restriction of $\mathrm{GL}_n$ to $\mathrm{O}_n$ yields the following branching rule
$$
...
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32 views
Duality of $Z(G)$ and $[G,G]$ in representation?
This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group.
I was thinking about its manifestation in group ...
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46 views
Frobenius reciprocity
I would like to ask a question on Theorem 8.6 on page 246 in this book.
There is the claim that
the multiplicity of $F$ in $E^G$ is equal to the multiplicity of $E$ in $F_H$.
Why is this just ...
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1answer
31 views
Show that for a finite field $F$ and finite group $G$, $F$ is a splitting field for $FG$
Let $F=\mathbb{F}_q$ be the field of $q$ elements and $G$ be a finite group. I'm trying to show that for an irreducible $FG$-module $V$, we have $\mathrm{End}_{FG}(V)=F \cdot 1$, i.e. that $F$ is a ...
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1answer
23 views
Question about the top of a bound representation of a bound quiver.
I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1.
I have a on page 77. In (d) of Lemma 2.2 on Page 77, it is said that $$ L_a=\sum_{\alpha: a\to b} ...
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1answer
27 views
How to compute the weights of $\Gamma_{3,1}$ the irrep of $\mathfrak{sl}_3\Bbb C$
I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest ...
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1answer
40 views
Weights versus roots
I am not sure of the difference between weights and roots. Am I correct in thinking that the weights are the eigenvalues of the action of the maximal torus on a given representation, and the roots are ...
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0answers
32 views
Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?
Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation}
d_{\alpha}|\#G,
\end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
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2answers
25 views
Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$
Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state
Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...
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1answer
19 views
Questions about maximal submodules.
Let $A$ be a $K$-algebra and $M$ a right $A$-module, where $K$ is a field. Suppose that $M=C\oplus D$, where $C, D$ are right $A$-modules. If $C', D'$ are maximal right $A$-submodules of $C, D$ ...
