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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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      Free almost commutative vertex algebras

      Given a commutative $k$-algebra $A$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $CAlg_{k} \rightarrow CVAlg_{k} $, which I'll denote $\...
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      44 views

      About the geometry of the set of weights that is strongly linked to $\lambda$

      Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\...
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      1answer
      98 views

      A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules

      Does anyone have a proof for the following Lemma? Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
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      48 views

      Auslander-Solberg algebras from non-rigid modules

      Let $A$ be a Nakayama algebra and $M$ be the direct sum of all indecomposable $A$-modules $N$ with $Ext_A^1(N,N) \neq 0$. The following is suggested by computer experiments with QPA: Question: Is ...
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      1answer
      87 views

      Shapovalov form on Verma modules

      I will formulate my question first; below is the relevant background information and notation. I have asked this question on stack, but I think the odds are better that I actually get an answere here (...
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      45 views

      Simple trace formula with different spectral footprint?

      A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple ...
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      2answers
      142 views

      Partial ordering on $\mathfrak{h}^*$ and Bruhat ordering

      In section 5.2 (p.95) of Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. Let $\mu\le \lambda$ if $\lambda-\mu\in \Gamma$, where $\Gamma$ is the set of $\mathbb{Z}^{\ge ...
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      votes
      1answer
      111 views

      Combinatorial problem on periodic dyck paths from homological algebra

      edit: I added conjecture 2 that looks much more accessible. Here is the elementary combinatorial translation of the problem (read below for the homological background): Let $n \geq 2$. A Nakayama ...
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      41 views

      Selfinjective algebras with loops

      Given a selfinjective finite dimensional algebra $A$ with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$. Question: Is A derived equivalent to an algebra with a loop in the quiver in ...
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      42 views

      Strong no loop conjecture for uniserial modules

      Let $A$ be a an Artin algebra. The strong no loop conjecture states that a simple $A$-module with $Ext_A^1(S,S) \neq 0$ has infinite projective dimension. This conjecture was recently proved for ...
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      31 views

      On monomial and $\Omega^d$-finite algebras

      Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra. It is well known that monomial algebras ...
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      93 views

      Intersection of Levi subgroups via intersection of their Weyl groups

      Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T \subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
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      2answers
      89 views

      Confusion about $\lambda\in\mathfrak{h}^*$ such that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$

      I am reading this paper: Representation type of the blocks of category $\mathcal{O}_S$ On p. 199, it said that While on p. 183 (Section 9.2) of Representations of Semisimple Lie Algebras in the BGG ...
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      1answer
      160 views

      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 ...
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      0answers
      47 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{\...

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