Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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Can we drop commutativity assumption?
Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
- 25
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103
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Relation between $\Gamma$-percuspidal parabolic subgroups and split parabolic subgroups of real semisimple Lie groups
Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. Let $\Gamma$ be a lattice in $\mathcal{G}:= G(\mathbb{R})$. I am interested in knowing under what conditions on either of $\mathcal{G}...
- 61
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144
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Unitarizability of group representations
Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...
- 21.8k
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Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099
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270
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Clifford-Mackey theory, references
I am working on a problem related to the local Langlands correspondence and I am interested in certain smooth representations of locally profinite groups (in particular of the Weil group of a local ...
- 303
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56
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Non-trivial summand in End(\rho)
Given a finite group representation $\rho:G\to GL_n(\mathbb C)$ one knows that the trivial representation $\mathbb 1$ is contained in $End(\rho)$.
Let $\rho'$ be the other summand, i.e., $\rho'$ is ...
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148
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Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$
Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) \...
- 6,201
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the complex representations of $B(2, \overline{\mathbb{F}_p})$
as the title, I want to know the complex representations of the $B(2，\overline{\mathbb{F}_p})$, i.e. invertible upper triangle matrix groups over $\mathbb{F}_p$'s algebraic closure $\overline{\mathbb{...
- 11
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331
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Differences between primitive central idempotents and primitive orthogonal idempotents
If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$.
If we ...
- 6,201
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120
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Framed braids and local systems
Let me start by admitting that my question is going to be somewhat vague. But hopefully it is one of these vague questions that can be immediately answered by an expert in the appropriate area.
...
- 1,704
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241
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Reference about a formula of coroot in an affine root system
Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that
$$
(\alpha + p\delta)^{\vee} = \alpha^{\vee} + \frac{2p}{(\alpha,...
- 6,201
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373
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Central extensions of SL2(R) by U(1) ?
Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you
- 69
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140
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Weyl group representation
Let $G$ be a reductive p-adic group.
Let $W$ be a weyl group. if $w$, and $w_o \in W$.
I want to know in which case we have $w w_o w^{-1}= w_o$ ?
in case if $w_o(\theta)=\theta $ where $\theta$ is a ...
- 19
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177
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Connection between Lie algebras and fusion rings
Example: Take the irreps of $SU(2)$: $0,1/2,1,...$ (Spin notation.) The quantum dimensions are $1,q+1/q,q^2+q+1+1/q+1/q^2,...$. At $q=(-1)^{1/5}$ this evaluates to $1,\phi,0,...$ and you get the ...
- 4,829
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163
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Generalized weight space
In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space:
If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
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260
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Is the projector to irreducible tensor modules of SO(N) known?
To project a generic tensor to an irreducible module of SO(N) one has to (anti)symmetrize the indices and then subtract traces, e.g. for symmetric traceless 2-tensors
$$
\frac{1}{2} (\delta_{I_1 J_1} ...
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143
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Soluble group algebras and centralizers
Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every $p'$-...
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266
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PBW proof proposal
One version of the PBW theorem states:
$\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras.
I am curious if this is a possible proof for the PBW theorem, part is taken ...
- 179
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189
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Action of the (special) orthogonal group on differential forms
I was told that the following facts are true. I am looking for a reference to them.
1) The action of $O(n,\mathbb{C})$ on $\wedge^l\mathbb{C}^n$ is irreducible for any $l$.
2) The action of $SO(n,\...
- 21.8k
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146
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"Multiplying" Clebsch-Gordan series
Assume you have a Lie algebra $G$ and a Clebsch-Gordan series $A\bigotimes{B}=C\bigoplus{D}\bigoplus...$
Assume you have a Lie algebra $g$ and a Clebsch-Gordan series $a\bigotimes{b}=c\bigoplus{d}\...
- 4,829
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167
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Are Generalized Verma modules natural w.r.t isometries?
Let $H$ be a subgroup of $G$ with Lie algebras $\mathfrak{h}$ and $\mathfrak{g}$ respectively. If I have 2 representations $V, W$ of $\mathfrak{h}$ equipped with a $\mathfrak{h}$ invariant inner ...
- 257
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177
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Existence of special pants decompositions for non-elementary representations into PSL(2,R)
A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation $\rho\...
- 3,749
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444
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Cartan involutions of su(n)
I have a question regarding Cartan involutions of su(n). Some sources say that there is only one up to equivalence (Wikipedia on Cartan Decomposition). Others say there are Types I, II, III. I looked ...
- 103
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914
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On a technical fact used in the proof of density of smooth vectors in a representation
My question concerns a fact used in Knapp's Representation Theory of Semisimple Groups implicitly in the proof of Lemma 3.13, which is crucial for the proof of density of smooth vectors in a ...
- 486
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109
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solve the singularities of parabolic orbits of schubert cells
Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}.
For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of $X_{w}=\...
- 3,472
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420
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Intertwining number
Hello,
I am wondering about a statement on this page.
Especially about this part:
If $\pi_1$ and $\pi_2$ are irreducible and finite dimensional or unitary, then the intertwining number $c(\pi_1,\...
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165
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Cotorsion theory and its relative homology
Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong \...
- 11
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326
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Endomorphism ring of a direct sum of tilting modules
I have found that a category of modules over a Lie algebra has an infinite number of (partial) tilting modules and that direct sum of these tilting modules is also an object in this category.
What ...
- 829
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214
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Program for minimal projective resolution
Hi,
I have the following problem.
I am searching for a program that can compute a minimal projective resolution of an arbitrary finitely
generated module of a quotient of a path algebra ...
- 1,755
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133
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Building a representation out of a generalized Verma module
I am trying to figure out representations of loop groups in order to understand conformal blocks. I am currently trying to figure out are the weight spaces of a representation with a given lowest ...
- 257
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272
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minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,472
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153
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(g,k) modules examples
I can't find any online references on these harish chandra modules and I have a hard time starting this question. Does anyone have any good references or some examples I can see.
Let the group be $...
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572
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why a projective module is a projective cover for its largest semisimple quotient?
Why a projective module is a projective cover for its largest semisimple quotient? That is - why the projection on the quotient is an essential morphism in this case?
- 63
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155
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complex reductive Lie groups which are not defined over the real numbers
Hello
Someone knows something about the complex reductive Lie groups (the complexification of a compact Lie group) which are not defined over the real numbers. I would like to know, if is it possible,...
- 1
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110
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Quaternion algebras over $k[[u,v]]$
Let $R=k[[u,v]]$ be a power series ring over algebraically closed
field of characteristic zero. The quaternionic $R$-algebra is
$A=R\langle x,y\rangle/I$, where $I=(x^2-a, y^2-b, xy+yx-2c)$ and $a,b,c\...
- 1
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948
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How do I find the irreducible representations of a group of matrices?
I have a set of permutation matrices (n x n) of a graph which form a group (the automorphisms). They are obviously a subgroup of the symmetric group S_n. Is there a way to find the irreducible ...
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167
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representations of the Lorentz group in 4 dimensions
Hi,
First of all I should say I am quite uneducated in group theory, so my question can be very naive. Sorry about that.
I'm reading Srednicki's "Quantum Field Theory" and I have a bit of trouble ...
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158
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Get $H^1(S,sl(2,R)_{Ad\phi}$) dimension directly from differential forms
For a genus g surface S with fundamental group $\pi$, consider Teichmuller space $Hom(\pi,SL(2,R))/SL(2,R)$, we identify tangent space at point $\phi\in Hom(\pi,SL(2,R))/SL(2,R)$ as $H^1(S,sl(2,R)_{...
- 11
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78
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What is the image of L-parameter on inertia subgroup?
Let $F$ be a local field. Is there any reference talking about the the image of L-parameter on inertia subgroup $I_F$ of Weil group $W_F$? Many Thanks!
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289
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Modular representations of the symplectic group
Let G=Sp(2m,2) be a finite symplectic group acting on $F_2^{2m}$. This group G acts 2-transitively on $\Omega_{+}$ and on $\Omega_{-}$. Let $F$ be an algebraic closure of $F_2$.
I am interested to ...
- 2,179
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429
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[]-infinity algebra and Projective representation
This is a very vague question.
We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...
- 1,271
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217
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References needed for representation theory of certain unipotent algebraic groups in characteristic zero
Due to my preoccupation with characteristic p > 0 representation theory of algebraic groups, I am quite ignorant of some basic facts pertaining to their characteristic zero theory, mostly concerning ...
- 511
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115
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Short Clebsch-Gordan expansions
For any rep R of a Lie group G you have $R\otimes{R}=R_1\oplus{R_2}\oplus...\oplus{R_n}$
(eh, is this correct even if G is non-compact, non-simple, non-reductive or
non-whatever?). Can you, for any n, ...
- 4,829
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185
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Characterization of Complex Group Algebras
Is there a way to characterize which complex algebras arise as the group algebra of some locally compact group? To make this more concrete, say $A$ is a sub-algebra of $\text{Mat}(n,\mathbb{C})$, $n\...
- 445
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241
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Does regularity of a D-module for an unusual filtration imply regularity for the usual one?
One definition of regular D-modules on affine space is that a D-module is regular if it has a filtration compatible with the order filtration on differential operators whose associated graded is ...
- 44.7k
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240
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Orbits of Infinite Grassmannian
"Any $L^+G$-orbit in Gr is known to be an orbit of the form Oλ for some λ ∈ $X_*(T)$, and Oλ = Oμ iff λ and μ are conjugate by W and the orbits form a finite stratification of each of the $Gr_i$."
...
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176
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Group theoretical properties and symmetry based representation of common functions
Dear All,
I'm searching for some references, whether someone already studied the group theoretical properties of functions. There are some very basic symmetries, like parity, but is there a set of ...
- 1
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168
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The derived representation and $H_\pi^\infty$
There is some (probably stupid) thing that I did not get in Serge Lang's $SL_2(\mathbf{R})$: On page 93 he considers a representation $\pi:G\to GL(H)$ of a group $G$ in a Banach space. Then he ...
- 12.7k
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685
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Unitary and injective representation of a free group
Is there an example of an injective homomorphism
$\pi: F_2\to U(n)$ of the 2 generator free group $F_2$
in some unitary group of matrices $U(n)$?
- 155
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491
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Linear Representations of the Groups
Does anyone know a good book on Linear Representations of the finite Groups which does not assumes a lot of background. Book which will be good to study for computer science and will cover all( at ...
- 81