# Questions tagged [rt.representation-theory]

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

4,718
questions

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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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### 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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### 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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### 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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**1**answer

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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**1**answer

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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### 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{\...