# Questions tagged [rt.representation-theory]

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

4,536 questions

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**1**answer

198 views

### Applications of the idea of deformation in algebraic geometry and other areas?

The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...

**2**

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**0**answers

40 views

### Minimal Embedding for flags varieties

I would like to understand how to construct a parametization of a flag
variety $F(V,n_1,\ldots,n_r)\subseteq \mathbb{P}^N$ in its minimal embedding.
First, I would like to know if there is a closed ...

**2**

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**0**answers

59 views

### Generic representation of PGL(3)

Let $G$ be the group $PGL(3,F)$, where $F$ is non-archimedean locally compact field, and $(\widetilde{H}_{n})_{n\in\mathbb{N}}$ the decreasing sequence of open and compact subgroups given by (image in ...

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51 views

### Artin’s theorem on induced representations and the kernel

Let $G$ be a finite group, and let $X$ be a family of subgroups of $G$ closed under conjugation and under passage to subgroups. Suppose further that $G$ is the union of the elements of $X,$ and denote ...

**2**

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**0**answers

86 views

### Satake correspondence for groups over finite field

I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...

**-1**

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**0**answers

84 views

### Quick easy question - Fermat's little theorem [on hold]

I'm sure this has an extremely obvious answer, so I'm a little embarrassed to ask it, but... are there any fields $k$, other than $\mathbb{Z}_p$, for which $c^p = c$ for all $c \in k$? In other words,...

**7**

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**0**answers

198 views

### What are the character tables of the finite unitary groups?

I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...

**4**

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**3**answers

154 views

### Real points of reductive groups and connected components

Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...

**6**

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167 views

+200

### On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...

**6**

votes

**1**answer

89 views

### Bijection from $S^2$ to itself interchanging actions of $A_5$

Let $X$ and $Y$ be two copies of $S^2$, and let $A_5$ act on each of them (as a group of rotations). Call these actions $\theta_X$ and $\theta_Y$.
Moreover, let $g \in A_5$ be a fixed element of ...

**4**

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**1**answer

140 views

### On definitions and explicit examples of pure-injective modules

I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the ...

**3**

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**0**answers

82 views

### Recognizing a restriction from $SL_2(\mathbb{C})$ to $SL_2(\mathbb{Z})$

I am aware that classifying all $SL_2(\mathbb{Z})$ representations is more or less completely intractable, but I was wondering what is known about the following simpler question: How do I recognize ...

**2**

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**0**answers

56 views

### Do the values of the global dimension constitute an interval?

Let $Q$ be a fixed finite connected quiver and $k$ a fixed field. Set $Z_Q:= \{ gldim(kQ/I) < \infty | I $ an admissible ideal $\}$.
Question: Is $Z_Q$ an intervall?
This is true for example in ...

**0**

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**0**answers

68 views

### Find representation set of orbits when group acts on a set

Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....

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**0**answers

21 views

### Formal character and unit

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$.
...