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# Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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### Integral of Schur functions over $SU(N)$ instead of $U(N)$

Schur functions are irreducible characters of the unitary group $U(N)$. This implies $$\int_{{U}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\delta_{\lambda\mu},$$ where the overline means complex ...
32 views

### Determining the groups compatible with given fusion rules

Say I have an unknown group $G$ with a simple, real, faithful representation $\mathbf n$ such that  {\mathbf n}\otimes{\mathbf n} \approx {\mathbf 1}_s\oplus{\mathbf a}_s\oplus{\mathbf b}_s\oplus{\...
113 views

### Complete reducibility and field extension

Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...
60 views

### Root system for semisimple Lie algebras [on hold]

Let $L$ be a semisimple Lie algebra. We can have a decomposition $L=\oplus_{\alpha=1}^tL_\alpha$ where each $L_\alpha$ is a simple ideal. It is well known that if we fix a maximal toral algebra $H$ of ...
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### A characterization of reflections on Euclidean space [migrated]

In the book "Introduction to Lie algebras and Representation theory" by J. Humphreys page 42, the author proves the following lemma: Let $E$ be a finite dimensional Euclidean space over $\mathbb R$ ...
108 views

### Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
186 views

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+... 0answers 49 views ### Representation theoretic characterisation of symmetric spaces Let$G$be a simple compact Lie group and$H$a closed subgroup. Let$\mathfrak{h}\subset \mathfrak{g}$denote the corresponding Lie algebras. Let$\mathfrak{m}$be an orthogonal complement to$\...
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### What subalgebras of $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ are orthogonal to the centre of $\mathfrak{gl}(n)$?

Consider the Lie algebra inclusions $\mathfrak{gl}(n) \subset \mathfrak{so}(2n)$ and $\mathfrak{gl}(n) \subset \mathfrak{sp}(2n)$. Let $\mathfrak{c} \subset \mathfrak{gl}(n)$ denote the centre. ...
73 views

### Branching rules for E6 into SU(3)^3

I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
142 views

### Irreducible factors of primitive permutation group representation

Consider a primitive permutation group $\Gamma\subseteq\mathrm{Sym}(N)$ on the $n$-element set $N=\{1,...,n\}$, that is, $\Gamma$ does not preserve any non-trivial partition of $N$. Consider the ...
191 views

Let $n \geq 0$ be an integer, let $A = (a_{ij})$ be the $(2n+1) \times (2n+1)$ matrix defined by $a_{ij} = 0$ unless $i + j = 2n+2$, in which case $a_{ij} = 1$. Let $G$ be the group scheme over $\... 0answers 56 views ###$End_A(M)$for representation-finite$A$and indecomposable$M$Given a representation-finite finite dimensional algebra$A$and an indecomposable module$M$. Question: What possible algebras can occur as$End_A(M)$? Is it always tame? Note than when$End_A(M)\$...

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