# Tag Info

## New answers tagged representation-theory

1 vote

### Is the regular representation of a given finite group unique?

The course syllabus defined it as a representation that contains all irreducible representations of a group. This is the wrong definition. The regular representation of a (finite, for simplicity) ...
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### Lattice in noncompact simple group is Ad-irreducible

The desired fact follows from a Borel density theorem type result. Thm 3 of https://www3.nd.edu/~andyp/notes/BorelDensity.pdf states "Let G be a connected semisimple $\mathbb{R}$-algebraic group ...
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### complex vs modular representation theory

No, there cannot be such an equivalence, because the category of finite-dimensional representations of $G$ over a field $K$ is sensitive to $K$: the endomorphism ring of every irreducible ...
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### Row orthogonality vs Column orthogonality

$\newcommand{\Size}{\left\lvert #1 \right\rvert}\newcommand{\Irr}{\mathrm Irr}\newcommand{\norm}{\trianglelefteq}$Let me rewrite the column orthogonality relations as \begin{equation*} \sum_{...
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### $\chi$ is an irreducible faithful character of $G$ then $Z(G)=\{g \in G: |\chi(g)|=\chi(1)\}$

$g\in Z(G)$ iff $|C_G(g)|=|G|$. We have that $|G|=\displaystyle\sum_{\chi\in Irr(G)}\chi(1)^2$, and so $|C_G(g)|=\displaystyle\sum_{\chi\in Irr(G)}|\chi(g)|^2$, by a direct consequence of the Second ...
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### Let $X$ be compact metric space and $\mu_n$ be sequence of complex measures on it. Prove the following two representations of $C(X)$ is equivalent.

Your $U$ is onto because your $\mu_n$'s are mutually singular: let $\{X_n\}$ be a partition of $X$ by Borel sets such that $\mu_n=0$ outside $X_n$. Then, for any $g=\{g_n\}\in H$, wlog (up to equality ...
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### The character of the alternating sum of the exterior powers of the standard representation

Let $S^1$ be the unit circle in $\mathbb C$, let $T=(S^1)^n$ be the $n$-torus, and let $$U=\{(z_1,\dots,z_n)\in T:z_1\cdots z_n=1\}.$$ This is a subspace of $T$ homeorphic to an $(n-1)$-torus. The ...
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### The character of the alternating sum of the exterior powers of the standard representation

If $V$ is a representation of a group $G$ of degree $d$, $g\in G$ and $p\geq0$, then the character of $\Lambda^pV$ evaluated at $g$ is $$\chi_{\Lambda^pV}(g)=\sigma_p(\lambda_1,\dots,\lambda_m),$$ ...
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### Understanding the 4 dimensional irrep of $SU_2$

In general, the $(n+1)$-dimensional irrep of $SU(2)$ is the $n^{th}$ symmetric power $S^n(\mathbb{C}^2)$, which can be described explicitly as the space of homogeneous polynomials of degree $n$ in two ...
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### Schur functors are pairwise non isomorphic

The character can be defined for finite-dimensional representations of an arbitrary group $G$ (or more generally for finite-dimensional modules over an arbitrary algebra), and it's an invariant of ...
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