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Questions tagged [rt.representation-theory]

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

2
votes
2answers
63 views

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 ...
1
vote
1answer
80 views

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 $G$. ...
8
votes
1answer
235 views

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....
1
vote
1answer
97 views

Duality isomorphism of representations of the maximal torus with respect to Steinberg's basis—is it an involution?

I am trying to apply Steinberg's basis of his paper "On a theorem of Pittie" (MSN) for the case $G$ of type $A_2$ and the maximal torus $T$ itself as a maximal rank subgroup. Denote by $\alpha_1, \...
1
vote
0answers
73 views

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 ...
2
votes
1answer
53 views

Translation functor on parabolic Verma module

I want to prove that following proposition by using Theorems/propositions in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. Define $ \Lambda := \{\nu \in \mathfrak{h}^* ...
4
votes
1answer
154 views

Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
5
votes
2answers
278 views

For a fixed dominant weight $\lambda$, are almost all dominant weights in the same coset above it?

First some notation as in e.g. the book by Humphreys on Lie Algebras. Let $E$ be an Euclidean space with inner product $(-,-)$, and denote $\langle v,w \rangle = \frac{2(v,w)}{(w,w)}$. Let $\Phi$ be ...
32
votes
1answer
1k views

Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$

Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
1
vote
1answer
62 views

About locally finite condition in category $\mathcal{O}^\mathfrak{p}$

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and ...
-1
votes
0answers
72 views

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 ...
3
votes
0answers
34 views

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 ...
4
votes
0answers
35 views

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 ...
0
votes
1answer
259 views

About irreducible morphisms

I have asked the following question in Mathematics stack: https://math.stackexchange.com/questions/2202032/about-irreducible-morphisms. But there is no response, so I repost it here. A morphism $f: X\...
5
votes
1answer
139 views

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. ...
2
votes
1answer
134 views

How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?

As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a ...
3
votes
2answers
169 views

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 ...
1
vote
1answer
59 views

About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$

As is shown in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each ...
1
vote
1answer
112 views

Action is determined by a braiding

Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by \begin{align} & \Psi: V \otimes W \to W \otimes V, \\ &...
5
votes
2answers
370 views

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 ...
4
votes
1answer
71 views

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 ...
15
votes
3answers
3k views

What are the current breakthroughs of Geometric Complexity Theory?

I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods. That program seems ...
4
votes
4answers
291 views

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 ...
11
votes
3answers
439 views

Nonnegativity of an integral over the unitary group

For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let $$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$ Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
2
votes
1answer
67 views

Exhaustion of restrictions of holomorphic / antiholomorphic representations

Let $G$ be a simple Lie group of Hermitian type, and $G'$ be a reductive subgroup of $G$. Suppose that $G'$ is also of Hermitian type and contains the center of the maximal compact subgroup of $G$. ...
4
votes
0answers
70 views

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 ...
-2
votes
0answers
84 views

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 $M=0$?...
6
votes
0answers
262 views

What is the technical difference between a deformation and a perturbation?

What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?
1
vote
1answer
69 views

Well-definedness of translation functor

Fix $\eta, \nu \in \mathfrak{h}^*$ and write $pr_\eta$ and $pr_\nu$ for the natural projections of the category $\mathcal{O}$ onto $\mathcal{O}_{\chi_\eta}$ and $\mathcal{O}_{\chi_\nu}$. If $L$ is ...
1
vote
0answers
45 views

Question on the proof that the Jacquet module preserves admissibility

Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\...
3
votes
0answers
39 views

Bounds for the finitistic dimension

The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. For finite dimensional algebras $A$ with radical cube ...
6
votes
2answers
220 views

Homological questions on monoid algebras

Given a finite monoid G and its group algebra A over a field $K$. I have never really studied such algebras, but I have the following questions (which are probably basic questions on any large class ...
13
votes
1answer
288 views

For which finite groups $G$ is every character a virtual permutation character?

Let $G$ be a finite group. A (complex) character $\chi$ of $G$ is said to be a virtual permutation character if it can be expressed as a $\mathbb{Z}$-linear combination of characters induced from the ...
5
votes
0answers
75 views

Generalisation of the Witt–Berman induction theorem

$\DeclareMathOperator{\Aut}{Aut}\DeclareMathOperator{\Ind}{Ind}$I believe I can prove the following induction theorem (modulo carefully checking a few details), and I would like to know whether this ...
1
vote
0answers
108 views

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 ...
0
votes
1answer
56 views

Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$

Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s_1\dots s_n$ ...
3
votes
0answers
80 views

Adams operation on the rational homology

The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an ...
5
votes
3answers
363 views

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 $\mathfrak h$, ...
1
vote
0answers
46 views

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 ...
3
votes
2answers
119 views

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 ...
2
votes
0answers
71 views

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 ...
5
votes
1answer
228 views

Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $G=SU(N)$. We know there is an exact sequence: $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$ For $G=SU(2)...
2
votes
1answer
87 views

Schur index of a representation and its divisors

We fix following objects: (1) $G$ is a finite group. (2) $\chi$ is complex irreducible character of $G$. (3) $m$ is the Schur index of $\chi$ w.r.t. the rational field $\mathbb{Q}$. (4) All the ...
3
votes
1answer
64 views

Convex Hulls of Demazure Modules

Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V_{\...
18
votes
3answers
1k views

When is a coadjoint orbit an integrable system (in a weak sense explained below)?

Let $X$ be an affine holomorphic symplectic variety of dimension $2n$, with the associated Poisson bracket { , }. Let's say it's an integrable system when there are $n$ algebraically independent ...
3
votes
2answers
249 views

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 ...
5
votes
2answers
231 views

Transitive embedding of the projective space $\Bbb R P^2$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the 2-dimensional projective space $\Bbb R P^2$ into the $4$-sphere, that is ...
11
votes
0answers
623 views

Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
5
votes
1answer
128 views

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 ...
5
votes
2answers
147 views

Indecomposable, non-simple, modules of quantum groups at roots of unity

Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...