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      Questions tagged [rt.representation-theory]

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

      4
      votes
      1answer
      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
      votes
      0answers
      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
      votes
      0answers
      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 ...
      1
      vote
      0answers
      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
      votes
      0answers
      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 ...
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      votes
      0answers
      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
      votes
      0answers
      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
      votes
      3answers
      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. $\mathbf G = \operatorname{GL}_n$. ...
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      votes
      0answers
      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
      1answer
      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
      votes
      1answer
      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
      votes
      0answers
      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
      votes
      0answers
      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
      votes
      0answers
      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....
      1
      vote
      0answers
      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 $\mathfrak{g}$. ...

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