# Questions tagged [rt.representation-theory]

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

4,611 questions
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### Symmetry of Casimirs of Lie algebras

The dimensions of the invariant tensors (Casimirs) of the simple Lie algebras are known, but I nowhere could find whether they are completely symmetric or antisymmetric with respect to an variable ...
0answers
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### Hilbert modular form as a representation of Hecke algebra

I am reading some notes by Snowden and I don't understand a sentence. Clearly, if we have an appropriate $R = T$ theorem then we get a modularity lifting theorem, as $\rho$ defines a homomorphism ...
1answer
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### On Ado´s theorem [on hold]

Suppose that $\mathfrak{h}$ is a Lie sub algebra of a finite-dimensional Lie algebra $\mathfrak{g}$. If $\mathfrak{h}$ admits a faithful linear representation $\rho$, can I find a faithful linear ...
0answers
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### Intersection of components in Springer fibre of type A

From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...
0answers
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### Metric structures making the cohomology into a module over a Lie algebra

The cohomology of a closed Kaehler manifold is an $\mathfrak{sl}_2$-module. I think Verbitsky has shown that the cohomology of a closed hyperkaehler manifold is an $\mathfrak{so}_5$-module. For what ...
1answer
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### Matrix of cosecants appearing in equivariant index computations

In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
1answer
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### Long time existence for heat flow in Corlette-Donaldson Theorem

I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14) https://arxiv.org/pdf/1402.4203.pdf For completeness, the statement is as follows. ...
2answers
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### Plucker relations in orthogonal Grassmannian

Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $\varpi_3$. Since $\varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known ...
2answers
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### Compactness of the automorphic quotient and genericity

Let $G$ be a reductive group defined over a field $F$. Let $\mathbf{A}$ denote the ring of adeles of $F$. My question is: Assuming the automorphic quotient $[G]=G(F) \backslash G(\mathbf{A})$ is ...
1answer
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### Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...
0answers
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### Panyushev's conjectured duality for root poset antichains

In his 2004 paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380), Panyushev conjectured (Conjecture 6.1) the ...
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### Trivial modules of group rings [closed]

Let $R=\mathbb{F}_p[D]$ where $D$ is a finite group of order prime to $p$. Let $M$ be any finitely generated (left) $R$-module. If one knows that $\textrm{Hom}_R(\mathbb{F}_p,M)=0$, can one show $M=0$?...
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### Existence of a weight of a representation in the fundamental Weyl chamber

Let $\mathfrak g$ be a complex simple Lie algebra. Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system. Pick a partial order on $\mathfrak h$, ...
0answers
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### Definition of Symplectic Motive

I just started reading a paper "On the Langlands correspondence for symplectic motives" by Benedict H. Gross, which talks about Symplectic Motives. The paper starts with the following Let $M$ be a ...
0answers
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### Uniqueness of Equivariant Harmonic Map for Surface Group Representation

In section 1.2 of https://arxiv.org/pdf/1311.2919.pdf the following result is stated. $\textbf{Theorem}$ (Labourie). Let $S$ be a closed Riemann surface of negative Euler characteristic, $Γ$ its ...
4answers
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### Homology of solvable (nilpotent) Lie algebras

Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...
0answers
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### Peter–Weyl theory for vector fields

Let $G$ be a compact Lie group. The classical Peter-Weyl theorem shows that $L^2(G)$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$. This is a ...
1answer
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2answers
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### Problem based representation theory book

I am trying to find books similar in the spirit of Ram Murty's Problems in Analytic Number theory or Murty Esmonde's Problems in Algebraic number theory in the field of Representation Theory (of ...
0answers
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### Koszul and quadratic algebras with Gorenstein dimension 2

In proposition 2.19. of http://inmabb.criba.edu.ar/revuma/pdf/v48n2/v48n2a05.pdf it was mentioned that a finite dimensional algebra of global dimension 2 is quadratic if and only if it is Koszul. ...
2answers
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### On Auslander algebras

Given a connected quiver algebra $A$ that is representation finite, the Auslander algebra $B_A$ of $A$ is defined as the endomorphism ring of the direct sum of each indecomposable $A$-module. It is ...
1answer
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### Classification of commutative Frobenius algebras

Are there attempts to classify commutative finite dimensional Frobenius algebras? They appear often in mathematics, such as in algebraic geometry and the famous category equivalence between ...
1answer
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### Ideals of commutative Frobenius algebras

Given a finite dimensional commutative (connected=local) Frobenius algebra $A$ over a field $K$. Question 1: Does $A$ have only finitely many ideals? (the answer should be no in the non-commutative ...
0answers
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1answer
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### Relative position and change of torus

Let $G$ be a connected split reductive group over a field $k$ of characteristic $0$. Let $T$ and $T'$ be two split maximal tori of $G$ and $B \supset T, B' \supset T'$ be two Borel subgroups of $G$. ...
3answers
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### Tannaka duality for semisimple groups

Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect_k$ for a field $k$ (of characteristic $0$) ...
0answers
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2answers
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### Subalgebra of a group algebra

Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra. Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$. Question: Is there any ...
0answers
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### When is the category of complexes of finite type?

For a ring $R$ define the category of complexes of length $n \geq 2$ as the category $C_n(R)$ with objects the complexes of the form $0 \rightarrow X_n \rightarrow \cdots X_1 \rightarrow 0$ with the ...