# Questions tagged [representation-theory]

For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.

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### Borel condition vs strong continuity in unitary representations of finite dimensional Lie groups

I am trying to find a completely rigorous treatment of strongly continuous projective unitary representations of (analytic) Lie groups on separable Hilbert spaces in full generality. Up to my ...
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### Serre's Representation Theory exercise 7.3(d)

I am trying to solve Exercise 7.3(d) in Serre's Linear Representation of Finite Groups. I have solved all other parts. The important points of where I am stuck at boils down to the following facts. (I ...
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### Mapping between vectors in irreducible Sp-representation

Let $V$ be the standard Sp-representation with symplectic basis $\{ a_i, b_i \}$. I believe the vector $(b_1 \wedge b_2) \otimes (b_1 \wedge b_2 \wedge a_3 \wedge a_4)$ lies inside the irreducible ...
• 326
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### Does the formal character determine the representation?

Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$. Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
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1 vote
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### "Linear independency" of Lie Brackets

I was watching this eigenchris video. At 21:49, he says: $$[g_i, g_j]=\sum_k {f_{ij}}^{k}g_k$$ for $\mathfrak{so}(3)$. Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What about ...
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### Complexification of complex Lie algebras like $\mathfrak{su}(2)$

I'm reading Brian Hall's book on Lie theory. He defines the complexification $V_{\mathbb{C}}$ of a real vector space $V$ as the linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. Next, he proceeds ...
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### Prime ideals dividing the Artin conductor

Let $L/K$ be a Galois extension of number fields, and let $(\phi,V)$ be a representation of $\operatorname{Gal}(L/K)$. Let $\mathfrak{f}_\phi$ be the Artin conductor of this representation, which is ...
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### Real Commutant Algebra of a Set of Matrices

Suppose I have a collection of $N\times N$ real, symmetric matrices $R_1, R_2, \dots$ and I want to find their orthogonal commutant---that is, the group of real, orthogonal matrices that commute with ...
1 vote
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### Which transitive $G$-sets appear when repeatedly inducing and restricting $G/H$, where $H\subseteq G$ is an inclusion of finite groups?

Let $G$ be a finite group and $H$ be a subgroup. Then $G/H$ is a transitive $G$-set. We can define a sequence of $G$-sets as follows: $$X_0=G/H$$ $$X_{n+1}=Ind_H^GRes^G_HX_n$$ Like every finite $G$-...
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### How to prove that submodules of $L = b(n,F)$ indecomposable

Let $F$ be a field,$L = b(n, F)$ and $V = F^n$. Let $e_1, ..., e_n$ be the standard basis of $F^n$. Define $W_r = Span\{e_1,...,e_r\}$, with $1 \leq r \leq n$. It can be proved that $W_r$ a submodule ...
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### What are the irreducible representations of $A_3$?

I've got some notes saying the Character table of the alternating group $A_3$ is given as in the attached image. I can't seem to figure out what the representations $\rho_1, \rho_2$ are supposed to be....
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### Dualizing a finitely generated bimodule

Let $k$ be a semisimple associative ring and $V$ be a $(k,k)$-bimodule. Suppose that $V$ is both left and right $k$-finite. Let $V^* := \mathrm{Hom}_k(V, k)$ be the dual of $V$ as a left $k$-module. ...
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### Decomposition of primitive central idempotents in group algebras

Let $W$ be an irreducible $\mathbb{C}$-representation of a finite group $G$ with character $\chi_W$. A primitive central idempotents of the group algebra $CG$ is: e=\frac{\dim_{\mathbb{C}}(W)}{|G|}\...
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1 vote
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### Eigenvectors of a matrix that commutes with subgroup of permutation matrices

I have a certain symmetric matrix $M_{ij}$, that is invariant under certain permutations of some of the indices $i=1,\dots,N$. More precisely, there exists a subgroup of permutations $G$, such that if ...
• 6,771
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### Why Is There No Oscillator Representation for Operators in Planar N=4 SYM Theory?

Im studying the planar N=4 Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum systems, ...
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