Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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How to define the action of $U(G)$ in this situation?
The usual action of $fg$ on $u⊗v$ , where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by $fg(u⊗v)...
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Morphisms between representations
I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this:
"For any permutation matrix $P$ ...
- 1,327
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Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules
Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial?
...
- 327
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Sum of weights of an irreducible representation of $U(N)$
Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.
Firstly, I would like to know ...
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Quiver representations over any commutative ring
I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image.
Towards the end, he has this ...
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About block $\mathcal{O}_\lambda$ of Category $\mathcal{O}$
The blocks of $\mathcal{O}$ are precisely the subcategories consisting of modules whose composition factors all have highest weights linked by $W_{[\lambda]}$ to an
antidominant weight $\lambda$.
I ...
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About indecomposability and nilpotence
Transferred from MSE where it now received a complete answer.
Maybe the following is easy, but I am not an expert in finite-dimensional Lie algebras and was stuck on the following problem.
Can ...
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Algebra with all modules non-rigid 2
Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective indecomposable module $M$, with the condition that at least one such module exists....
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Specialization (of parameters) of Macdonald polynomials and characters of classical groups
It's well known that the $A_{n-1}$ Macdonald polynomial $P_\lambda(x;q,t)$ becomes the Schur function when $q=t$. Schur functions are characters of the general linear groups. What about other ...
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Representations of smash products with $p$-groups
I am trying to find more generalized counterparts of some well-known results from modular group representations.
My question is the following:
Suppose that $H$ is a finite $p$-group acting as ...
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The projective and injective modules in tensor product of algebra
Let $R,S$ be two K algebras, where K is a fixed field. Then we can get a new algebra $R \otimes S$, i.e. the tensor product of these two algebras. Suppose the following sequence $$0 \rightarrow R\...
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Whitehead's lemma (Lie algebras) for reductive Lie algebras [closed]
Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear ...
- 6,201
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A problem with pointwise stabilizer subgroups of fixed-point subspaces I
Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...
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Absolute irreducibility of a symmetric square?
This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so $G:=\...
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Possible degrees of faithful projective representations of $\mathrm{PSL}(k,q)$ and $\mathrm{Sp}(2k,q)$ over complex numbers
Let $q$ be a prime power and $k$ a positive integer. What are the possible degrees of faithful projective representations of the projective special linear group $\mathrm{PSL}(k,q)$ (over the Galois ...
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$I/N$ is finitely presented module
Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.
...
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Decomposition of $G$-harmonic polynomials
Let $G$ be a finite group and let $H_G$ denote the $G$-harmonic polynomials. What is the structure of $H_G$ as a $G$-module? Is it isomorphic to the regular representation?
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when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
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Orbital integrals of pseudo coefficients of supercuspidal reps
Let $\pi$ be a supercuspidal representation of $G =GL_2(F)$ for a non-archimedean local field $F$, then there exists a maximal subgroup $K$ of $G$, which is compact modulo the center, and a ...
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Cartan decomposition of a unitary group?
For local fields $F$, we consider two case
1) $E$=quadratic extension of $F$ , 2) $E = F \times F$.
Let V be a 2-dim hermition space over E.
In 1) case, by Cartan decompostion $U(2)$ can be ...
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Classification of K-algebras over an algebraically closed field
Hi. In Assem's book et al "Elements of Representation Theory I" it is an exercise to classify all 3 dimensional basic, connected K-algebras where K is an algebraically closed field. Unless I'm wrong I ...
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Reference on a result on representation of moderate growth
Let G be a real reductive group, and P any parabolic subgroup. In the paper 'Canonical extensions of Harish-Chandra modules to representations of $G$' by Casselman, a result says that if we begin with ...
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Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
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Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
- 6,201
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About cluster variables obtained by (sequentially) mutating at exchangeable variables from an initial cluster
Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a ...
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Group algebras and group automorphisms
Say, we have a countable ICC group $G$, a Hilbert space $H$ with a basis indexed by the group elements, the group algebra generated by the left regular representation of $G$ on this Hilbert space, and ...
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When does the Kazhdan-Lusztig polynomial $P_{x,w}(q)$ not vanish at $q=1$?
Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra. For any $\lambda\in \mathfrak{h}^{*}$ let $M(\lambda)$ and $L(\lambda)$ be the Verma module and the simple ...
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About regular infinitesimal character
Let $Z(\mathfrak{g})$ be the centre of $U(\mathfrak{g})$ and let $\chi_\lambda$ be an algebra homomorphism
$Z(\mathfrak{g}) \to \mathbb{C}$ such that
$
z \cdot v = \chi_\lambda(z)v
$
for all $z \in Z(\...
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Dualizing the trivial action on a $C^*$-algebra
Let $G$ be a finite abelian group (cosidered as a discrete topological group), $A$ a unital separable $C^*$-algebra. Let $T\colon G\to \operatorname{Aut}(A)$, $T_g(a)=a$ for all $g\in G$ the trivial ...
- 1,188
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Quadratic Casimir of symplectic group
Does anyone know the formula for the value of quadratic Casimir of the symplectic group $Sp(2N)$ in the fundamental representation? In this definition, $Sp(2)=SU(2)$. Thanks a lot.
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Strange modules part II
Let $A$ be a finite dimensional symmetric algebra over a field (we can also assume that it is connected).
Call a non-projective indecomposable module $M$ strange in case $Ext^i(M,M)=0$ for all but ...
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Are simple Poisson $A$-modules finitely generated as $A$-modules?
Let $A$ be a Poisson $\mathbb{C}$-algebra: $A$ has a the structure of a complex commutative algebra and at the same time it carries the structure of a Lie algebra, with Lie bracket $\{\cdot, \cdot\}$. ...
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Questions about Lowey length
Let $\Lambda$ be an artin algebra.
If $M$ is a finitely generated $\Lambda$-module with Loewy length 2 and finite projective dimension. How to get the exact sequence $$0 \rightarrow A \rightarrow P/...
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Different computation methods to determine the conjugacy classes of a finite extension group N.G
I am looking for methods to compute the conjugacy classes of any finite extension group N.G from the classes of G.
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The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
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References request: vector representations of Lie superalgebras
Are there some references of fundamental representations of Lie superalgebras (in particular for the Lie superalgebra $sl(m|n)$? Thank you very much.
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Canonical basis of cluster algebras
Let $x_{k+1} = \frac{x_k^{d_k}+1}{x_{k-1}}$, $k \in \mathbb{Z}$, where $d_{k+2} = d_k \in \mathbb{Z_{>0}}$. Let $b=d_1$ and $c=d_2$. Define the cluster algebra $A = A(\left( \begin{matrix} 0 & ...
- 6,201
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Evaluation modules of $U_q(L(sl_2))$
Let $a \in \mathbb{C}^{\times}$, $r \in N$. Let $W = V_q(r)$ be the $r$-dimensional irreducible type 1 representation of $U_q(gl_2(\mathbb{C}))$. In the usual basis $\{v_0, \ldots, v_r\}$, the action ...
- 6,201
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Maximal possible dimension of abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$? [closed]
Fix $n \in \mathbb{N}$, and let $\mathfrak{h}_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}_n$ is the Lie algebra with basis $x_1, \dots, ...
user74564
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Highest weight formulas for quadratic Casimir and dimension for the simply laced Lie algebras
Intro (tldr-ish):
In the meantime, in the literature I dug up the formulae not only for the dimension D of a $G_2$ module, but also its quadratic Casimir C2 (eigenvalue). After some playing, I ...
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Non Lie-group ribbon categories
I learnt here that a) Reshitikhine-Turaev works with any ribbon category but
b) those not coming from Lie groups are rare.
Can someone give an actual example (and preferrable with purely graphic ...
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Decomposition of semi simple local systems
I found A question similar to this, but the answer wasn't clear to me and I'm not supposed to ask for further clarification in the answer section.
Let $L$ a semi simple local system defined over an ...
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Reductive subgroup and its derived subgroup with an irreducible represenation
Could you please answer the following question: Let $V$ be a faithful irreducible representation of a connected reductive group $H$ defined over $\mathbb{R}$ Is it true that the derived group of $H$, ...
- 601
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Nagakami behavoir
Is the sum of square Nagakami random variables Erlang distributed?
What is the distribution of euclidean norm of complex Nagakami?
Cheers!
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Maximal subgroups of indefinite special orthogonal group SO(p,q)
Can someone answer the following question:
Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions:
...
- 601
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Proof of Cartan's solvability criterion
The following is a key lemma in popular proofs of Cartan's solvability criterion for Lie algebras over a field of characteristic zero.
Given $a\subseteq b\subseteq \mathrm{End}(V)$, let $m=\{x\in\...
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Does the nonvanishing of a Littlewood-Richardson coefficient implies comparability of highest weights?
Let $\mathfrak{g}$ be a semi-simple finite dimensional Lie algebra.
Denote by $L(\lambda)$ an irreducible finite-dimensional $\mathfrak{g}$-module of highest weight $\lambda$. (I.e. $\lambda$ is ...
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Continuation of homomorphisms of representations...
Hi all.
If $G$ is a finite group and $\varrho : G \to \text{GL}(V), \eta : G \to \text{GL}(W)$
are finite dimensional representations, $V_0$ is a $G$-invariant subspace of $V$
and $f : V_0 \to W$ is ...
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Commutativity of nilpotents in minuscule case
Consider a minuscule representation (in a semisimple case). This gives a decomposition:
g=p\oplus n. Is n commutative?
(Sorry for a stupid question - I am not an expert and it's Sunday and I have no ...
user27328
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Projective modules over Lie (super) algebras
Hi:
Given a finite dimensional Lie superalgebra $G$ over the field of complex numbers with decomposition $G_{-1}+G_{0}+G_{1}$, where $G_{0}$ is the even part and $G_{-1}+G_{1}$ is the odd part. ...
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