# Tagged Questions

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

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### Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V}$

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V}$$ is an algebraic integer. In the start of this proof we have: ...
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### Isomorphism from $\mathbb{C}[G]$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.

What I want to ask is the proof of the Proposition 10. in "Linear Representations of Finite Groups" by Jean-Pierre Serre. Let $\rho_i : G \rightarrow GL(W_i)$ be the distinct irreducible ...
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### Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
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### Schur and Weyl modules.

Let $m$ be a non-negative integer and $\lambda=(\lambda_1, \cdots, \lambda_s)$ a partition of $m$. If $V$ is a vector space of dimension $n$ (over a field $\mathbb{K}$), we can consider the Schur ...
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### Replacing entries of dice by average of it neighbours

I am interested in Representation Theory. I came across the following answer while reading this question on Mathoverflow. An example from Kirillov's book on representation theory: write numbers ...
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### Inverting the the decomposition of tensor product representation into irreps

Suppose I have two unitary representations $U_V, U_W$ of a group $G$ on finite-dimensional vector spaces $V$ and $W$. I know that the tensor product representation $U_V\otimes U_W$ need not be ...
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### Invariant subspace vs. irreducible subspace (terminology)

In a course in representation theory I was presented the following proposition: Let $(\pi,V)$ be a finite dimensional irreducible representation with a cyclic vector. $V$ has a unique max. proper ...
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### explicit components of regular representation of $S_4$

Consider (left) regular complex representation of $S_4$. It has two 2-dimensional irreducible components. I need exact form of elements in those components (probably, having one element I may get ...
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### Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
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### Reordering indexed expressions (combinatorics)

To me, it appears always as a little 'magic' when people reorder expressions, indexed by highly complex combinations of permutations and I would like to know in deep and formally what really is going ...
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### Why Jacobson, but not the left (right) maximals individually?

When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ...
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### Regge symmetry and outer automorphisms of Dynkin diagrams

Quantum $6j$-symbols are the coefficients of the change of basis matrix in the central extension of Temperley-Lieb algebra(see the book by Kauffman and Lins). It is my understanding that Ocneanu has ...
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### The set of non-conjugate elements

I have $H \leq G$ where $G$ is a group. Now for any $t \notin H$ we have $H \cap tHt^{-1} = e$ Now $N$ is a set of all elements of $G$ which are not conjugate to any element of $H$ I want to ...
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### Is the trivial representation a subrepresentation of a tensor power of any irreducible complex representation of a finite group?

Let $G$ be a finite group, $V$ an irreducible complex representation and $\mathbb{1}$ the trivial representation. Question: $\exists n >0$ such that $\mathbb{1} \le V^{\otimes n}$?
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### Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...