All Questions
Tagged with rt.representation-theory reference-request
823 questions
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BGG-like resolutions and translations
This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional ...
CommunityBot
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1
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translation functors in parabolic category $\mathcal{O}$
I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.
I am mainly interested in the ...
CommunityBot
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3
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Lie subgroups of SU(4)
Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.
Additionally, which of these subgroups admit four dimensional ...
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0
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239
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Resolution of singularities of this cubic surface?
Let $A = \mathcal O(Y)^{SL_2(\mathbb C)}$ be the ring of invariant functions on $Y := \mathrm{Hom}(\mathbb Z^2, SL_2(\mathbb C))$. We can identify $A$ with the quotient of $\mathbb C[x,y,z]$ by the ...
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$\lambda$-rings and hopf-rings
The direct sum of complex representation rings $R_*\oplus R\Sigma_n$, for $\Sigma_n$ the $n$th symmetric group is also the free $\lambda$-ring on one generator. Here, we take a product obtained from ...
- 31
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What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
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5
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2
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452
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"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 52.9k
6
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1
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Which functions are linear combinations of irreducible characters for a given field $\Bbbk$?
Let $G$ be a finite group. Then it is well known that a function $f\colon G\to \mathbb C$ is a linear combination of irreducible characters iff it is constant on conjugacy classes. What is the ...
- 44.4k
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1
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Decomposition of an induced representation
If there is a finite group $G$ with a cyclic normal subgroup $C_n$, one can describe the indecomposable representations of $G$ through induction. How does $Ind_{C_n}^G$ decompose? For representations ...
- 44.4k
7
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Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
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Good book on representation theory of GL(n)
I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers).
I know only a basic representation theory.
The question I am ...
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Origin of notion of "split Grothendieck group"?
In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$...
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Is there formula name and proof for this theorem ? [closed]
The formula answers: how many tuples $(\sigma_1,\sigma_2,...,\sigma_n)$ of elements of a given group G such that
(1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class.
(2) $\sigma_1\...
- 44.4k
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answer
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Explicit method to compute Macdonald/Koornwinder functions
I'd like to compute explicitly symmetric Macdonald functions associated to arbitrary (possibly non-reduced) root systems, using Computer Algebra System.
Unfortunately Sage seems to only implement ...
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Convex PBW bases
Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, $\...
- 8,866
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Previous work on this generalization of continued fractions?
The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...
CommunityBot
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1
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Finite subgroups of SO(3)
There are several proofs of the famous classification of finite subgroups of $SO(3)$. I heard that there is a "purely algebraic" one attributed to Camille Jordan. Does anybody know of a reference?
...
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
CommunityBot
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1
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Is the following construction of the 0-Hecke monoid (well) known?
Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
- 3,532
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Character theory of $2$-Frobenius groups.
This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
CommunityBot
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2
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What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
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Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`
The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
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Rational automorphisms of semisimple algebraic groups
Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...
- 2,624
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Reference request for character formula between tensor products of Weyl modules.
So it is well known that when you tensor together two induced modules for an algebraic group
$\nabla(\lambda) \otimes \nabla(\mu)$ that the result has a filtration by other induced modules, (I.e. it ...
- 52.9k
11
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Is there a Dedekind-Frobenius group determinant for infinite groups?
If $G$ is a finite group and $\lbrace x_{g} \rbrace_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x_{gh^{-1}}$. The determinant of this matrix is a ...
- 96.4k
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Are irreducible representations subrepresentations of a symmetric power representation?
First of all I am far from being an expert in representation theory, so it is possible (likely) that the following question is trivial (in fact a trivial reference question):
Let $\Gamma$ be a, let's ...
- 1,691
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The proof of the splitting principle in equivariant K-theory via flag manifolds
In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:
Let $j: T\...
- 8,559
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Conjugacy classes in Aut(G)
Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of conjugacy classes of $G$ is described in many places.
Now, I'd like to know the structure/...
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Classifying algebras with two idempotent generators and involution
Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$.
For example,...
- 1,021
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A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
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S-matrix for the BMW category
This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See here for example.)
The minimal idempotents of ...
CommunityBot
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4
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Textbook source for finite group properties deducible from character table?
Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
- 6,357
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searching for text for studying representation theory
I'm a graduate student studying algebraic geometry.
Recently, When I studying Hodge theory, I saw sl2-representation is used in Hodge theory.
So I think that studying representation theory may be ...
- 2,450
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1
answer
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Schur Weyl duality for sl_n representations
Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
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Good books in Modular Representation Theory
Hi every one!
I am reading some paper and it uses Modular Representation Theory. I even dont really know about Representation Theory and I am looking for a good book for beginner. Could you please ...
- 16.2k
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1
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213
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Positive definite functions on G from Hilbert space vectors?
Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$?
This question is rather vague and ...
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Is there a brief name for the symmetric space $SL_{2n} / Sp_{2n}$?
Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$.
(1)...
- 1,780
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Subgroups of GL_2 over a finite field
I've come across the phrase "by the classification of subgroups of $GL_2(F_q)$" in multiple papers, but never with a reference. Here $F_q$ is a finite field of size $q$. Does anyone know a good ...
- 11.2k
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632
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Pseudo coefficients and orbital integrals
I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:
"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...
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Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
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Has there been any application of tensor species?
Joyal's combinatorial species, endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" (...
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Reference for a dual version of the Cauchy decomposition.
By "Cauchy decomposition" I mean the following identity, both sides in which are representations of $GL_n(\mathbb C)\times GL_m(\mathbb C)$:
$$\mathrm{Sym}^p(V\otimes W)=\bigoplus_{\lambda\vdash p} V^\...
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Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator
Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
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Why can I divide an affine variety by the action of the general linear group?
Let $G\subseteq\mathrm{Gl}_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine ...
- 26.3k
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Irreducible mod-p representation of a semidirect product with trivial p-core
Consider the group $G=H\rtimes{}C$, where $H$ has order prime with $p$ and $C$ is cyclic of order $p^k$, and $C\rightarrow{}\mathrm{Aut}(H)$ is faithful (or equivalently $G$ has trivial $p$-core). ...
- 6,357
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Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
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Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
- 52.9k
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Induced character for non-injective homomorphisms
Any group homomorphism $\phi\colon H\to G$ gives rise to an induction/restriction adjunction between $G$-representations and $H$-representations:
$$ \hom_G(\phi_! M, N) \cong \hom_H(M, \phi^* N) $$
...
- 11.2k
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Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$
This question is out of plain curiosity. The first sentence of Deligne's
Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$ (1984) reads (in rough translation) as ...
- 13.3k
6
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1
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What is the name for a finite-group representation that is the sum of all the irreducibles (once)?
I vaguely remember seeing a paper studying the concept of a totally multiplicity-one representation of a finite group, which concept, I recall, had a particular name, which I forget. What is this ...
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