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Results tagged with rt.representation-theory
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user 5301
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
5
votes
Recovering representation from its character
I assume you are talking about complex finite dimensional representations of finite groups. Philosophically speaking, one should be able to recover a representation from a character but, in practice, …
- 12.1k
3
votes
When is there a $g$-module isomorphism between a semi-simple Lie algebra $g$ and an exterior...
No (2nd question). Take $k=1$ and $I=g$ as a $g$-module and the extension is trivial.
- 12.1k
1
vote
Are there infinite groups which have only a finite number of irreducible representations?
No way, doc! Take $PSL_2$ over a field of cardinality alef-2012. It has exactly one complex irreducible rep up to an equivalence...
- 12.1k
1
vote
Is decomposition of a representation of a group G over a vector space V into irreducible sub...
Take the trivial representation of your favourite $G$ on ${\mathbb C}^{2012}$. What kind of uniqueness are you talking about?
- 12.1k
2
votes
Recovering representation from its character
expat, as you don't care about nothing, here is a similar (to David's) way of doing it. It will be more efficient. Your group is $G$ and your character is $\chi$. You can work over rationals only if y …
- 12.1k
1
vote
Getting the Weyl dimension formula geometrically
Publish it if you can find one :-)) You have no convenient basis for cohomology with easily described multiplication. You have a clean formula for Chern character and not so clean formula for Todd cla …
- 12.1k
4
votes
dominant weights
What do you mean by "why"? Do you want a complete proof - it is rather straightforward and an easy exercise.
If you want a moral reason then this is because it is rather small or miniscule, to be pr …
- 12.1k
1
vote
Constructing inequivalent irreps of finite groups
Why representations? A better question is to find all irreducible characters or to fill the character table. You can do it efficiently for any given group.
After you have characters, you may want to …
- 12.1k
3
votes
Classification of finite complex reflection groups
Answering the first question, if the field has characteristic zero then the classification will be reduced to Shephard-Todd. By this I mean that every finite reflection group will be on Shephard-Todd …
- 12.1k
3
votes
tensor product of projective irreducible modules
It is, if the sum of dimensions is less then $p+2$, by Serre's Theorem (Sur la semi-simplicité des produits tensoriels de représentations de groupes).
- 12.1k
3
votes
Accepted
Sub-representations of the affine group
As Victor explained consider the functions $X^m$ where $X^m(\alpha)=\alpha^m$. As $m$ runs between $0$ and $p^k-1$, these functions form a basis of your space of functions. This is a nice wavy basis, …
- 12.1k
0
votes
Who colored in my Dynkin diagrams?
It is the bypartite graph structure on the corresponding Coxeter diagram.
- 12.1k
4
votes
0
answers
150
views
Nilpotent orbits and subspaces
Let ${\mathbb g}$ be a simple complex finite dimensional Lie algebra, $X\subseteq{\mathbb g}$ a nilpotent orbit. Did anyone study maximal vector subspaces of the closure $\overline{X}$?
In particular …
- 12.1k
1
vote
Brauer's permutation lemma — extending to some other finite groups?
You may be able to salvage something by looking at the Bursnide ring of your group. The isomorphism class of the permutation representation of a set $X$ is governed by products $[X]e$ with some of the …
- 12.1k
2
votes
Purely algebraic proof for unitarizability of representations of a compact real semisimple L...
Why not, doc? Take a unitary representation $V$ of $G$. Its tensor power $T^nV$ is unitary as well via the obvious form
$$
<a\otimes b\otimes \ldots , a^\prime \otimes b^\prime \ldots> =
<a,a^\prime> …
- 12.1k
4
votes
A reference book for Schur's lemma
Doc, if you want to be anal with your references, you should quote
Amitsur, A. S. Algebras over infinite fields. Proc. Amer. Math. Soc. 7 (1956), 35–48.
Otherwise, this is a well-known fact and you c …
- 12.1k
1
vote
Irreducible representations of W-algebra in case $\mathfrak sl_3$
All (not only finite dimensional!!) irreducible reps of $W$-algebra of $sl_2$ at its subregular nilpotent (read $U(sl_2)$) were classified by Block. A similar classification for the minimal $W$-algebr …
- 12.1k
8
votes
2
answers
687
views
Irreducible representations of simple complex groups
Let $G$ be a simple complex algebraic group. What are its complex irreducible finite-dimensional representations?
Before you start voting to close the question, I never said "rational". I am asking a …
- 12.1k
4
votes
0
answers
221
views
Second symmetric square of the adjoint representation
I have just come across the following experimental fact. Let ${\mathfrak g}$ be a simple complex Lie algebra.
Fact: ${\mathfrak g}$ is a constituent of $S^2{\mathfrak g}$ if and only if ${\mathfrak …
- 12.1k
3
votes
Accepted
Irreducibility of the $\mathfrak{g}$-module $\mathfrak{o}(k)/ad(\mathfrak{g})$
Note that ${\mathfrak o}(k)\cong \wedge^2 {\mathfrak g}$. It has ${\mathfrak g}$ as a summand, coming from the Lie bracket $\wedge^2 {\mathfrak g} \rightarrow {\mathfrak g}$ . Calculation of the rest …
- 12.1k
9
votes
1
answer
525
views
Unitary Representations of $GL_2({\mathbb Q}_p)$
I was reading about the classification of unitary representations of $G=GL_2({\mathbb Q}_p)$ in Automorphic representations and ... by Goldfeld-Hundley yesterday and could not understand a very basic …
- 12.1k
3
votes
Real representations of SO(n) and U(n)
Read about Frobenius-Schur anywhere. In a nutshell a complex irreducible $V$ of complex dimension $n$ can give 1 or 2 real irreducibles, whose real dimensions are $n$ or $2n$. This can be easily deter …
- 12.1k
9
votes
3
answers
1k
views
Why is symmetric group not matrix?
I have just learned from mighty Wikipedia that the symmetric group of an infinite set is not a matrix group. Why?
I can see rep-theory reasons: sizes of minimal non-trivial, non-sign representations …
- 12.1k
3
votes
Accepted
Is a smallness condition necessary in the Tannaka reconstruction theorem?
Yes, of course. You still have a natural homomorphism $A\rightarrow END(U_A)$. Since ${}_AA$ is a free $A$-module, an endomorphism $x\in END(U_A)$ is determined by its value $x_A$ on ${}_AA$. This pro …
- 12.1k
5
votes
Accepted
adjoint action of a Levi subalgebra
Not sure that there would be a closed formula as Jim has pointed out. You can easily calculate it in examples because you can determine highest weight vectors in $g$ for $m$. Indeed, if $\alpha_1, \ld …
- 12.1k
1
vote
Invariant subspaces and isotypic decomposition (reference request)
For compact groups you can quote IV.2.7 in Naimark-Stern. There the $T$-isotypical component is described as the image of an operator $E^T$.
For general semisimple categories it may be better to giv …
- 12.1k
1
vote
Tensor-indecomposable modules
I think Q1 is clear as soon as $A\neq K$. BTW, $A=K$ is a silly counterexample.
Your quiver has a sink $a$ and a source $b$. There are no arrows from $a$ to $b$. Then $A$ has no element $x\neq 0$ suc …
- 12.1k
2
votes
Accepted
Universal property of induced representation
You are writing a right adjoint to restriction so you have a natural $H$-module map
$$
iE\rightarrow E, \ (f(x):G\rightarrow E)) \mapsto f(1) .
$$
To cook up a map in the opposite direction, you need …
- 12.1k
1
vote
Algebraicity of holomorphic representations of a semisimple complex linear algebraic group
I can think of a funny shortcut but you will be there on your own in the mathematical jungle (it is not written anywhere as far as I know) and you may lose your students to tigers and anakondas:-))
W …
- 12.1k
13
votes
Finite-dimensional faithful representations of compact groups
No, it is false! Take a product of alef-2011 copies of $C_2$. It is a bit too big to fit into $GL_n (C)$...
- 12.1k
-1
votes
Bernstein's presentation for the Hecke algebra
MathSciNet search for Hecke and Bernstein Presentation gives
this ...
- 12.1k
12
votes
Accepted
Can we bound degrees of complex irreps in terms of the average conjugacy class size?
Yes, take 2-extraspecial group $2^{2n+1}$, plus or minus should not matter. It has $2^{2n}$ irreducible representations of degree 1 and one of degree $2^n$. So your $K_G$ is about 2 while $d(G)=2^n$.
- 12.1k
1
vote
Accepted
Representations of reductive Lie group
You need to be over a field of zero characteristic and your representation needs to be rational, i.e. matrix entries need to be algebraic functions on $G$. Then it is completely reducible, see any boo …
- 12.1k
1
vote
Is there a category of representations of a simple Lie algebra on which its Weyl group natur...
I can cook up such category on my Foreman grill. Let $\tilde{U}(g)=S(h)\otimes_{S(h)^W}U(g)$ be the extended enveloping algebra of $g$. $W$ acts on $\tilde{U}(g)$, hence, it acts on the category of it …
- 12.1k
5
votes
Proving that some principal series representations of SL(2,F) are irreducible
I dont think all of them are irreducible: when you restrict from GL(2) to SL(2) a representation can split into two. I think all the necessary information is in the following paper:
M. Tadic, Notes o …
- 12.1k
3
votes
Introduction to W-Algebras/Why W-algebras?
The motivation did not change. The theory of finite W-algebras advanced quite a bit, imho, mostly thanks to the works of Losev.
There is a good introductory text by Arakawa now.
- 12.1k
2
votes
Accepted
Image of Fourier transform for finite non-abelian groups
The transform, which is a ring isomorphism,
$$
P \mapsto \widehat{P} = \sum_{g\in G} P(g)g
$$
goes from your $L(G)$ to the group algebra ${\mathbb C}G$. The convolution corresponds to the multiplicati …
- 12.1k
14
votes
Two equivalent irreducible representations given by integer matrices
No, it is not even true for matrices, e.g., $\left(\begin{array}{rr}0 & 1\\1&0\end{array}\right)$ and $\left(\begin{array}{rr}1 & 0\\0&-1\end{array}\right)$.
- 12.1k
20
votes
A ring for which the category of left and right modules are distinct
$\begin{pmatrix} {\mathbb Z} & {\mathbb Q}\\ 0 & {\mathbb Q} \end{pmatrix}$
is a canonical example for such things. Let us list all simple right and left modules:
$$R_p=\begin{pmatrix} {\mathbb Z}/(p) …
- 12.1k
3
votes
Accepted
Left-right non-bimodule examples
Suppose $A$ is a non-commutative Hopf algebra. Then you can use $M=A$ with the left adjoint and right regular actions:
$$
a\cdot m = \sum_{(a)}a_{(1)}mS(a_{(2)}), \ m\cdot b = mb .
$$
In particular, y …
- 12.1k
2
votes
Accepted
Category of representations of a tensor product algebra
Yes, it will be exactly Deligne's tensor product of abelian categories. See https://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories
- 12.1k
1
vote
Morphisms of two fully reducible representations of a group
Doc, your language is old-fashioned. Such questions become clearer with higher levels of abstraction. You want to go from representations to modules and then further to categories. It leaves unnecessa …
- 12.1k
4
votes
Representations are determined by characters : Groups and Lie algebras
Doc, I am not sure what your question is, but the answer is yes. Whatever definition of character you are using, any two extensions of $M$ by $N$ will have the same character. Thus, a non-trivial exte …
- 12.1k
3
votes
Dixmier's lemma as a generalisation of Schur's first lemma
If $M-c$ is not invertible, then its kernel is non-zero or its image is smaller than $A$. But $A$ is irreducible. This means any proper submodule is zero. This forces $M-c=0$.
BTW, for finite-dimensi …
- 12.1k
1
vote
How do Jordan algebras help one understand representations of exceptional Lie algebras?
No, you cannot use "Jordan theory" to get working models for the representations of the exceptional Lie algebras. There is a chance of doing it with $F_4$ but not with any $E_n$-s.
Historically, peo …
- 12.1k
2
votes
Building Lie-like algebras from modules over semisimple Lie algebras
The general idea smells like coloured Lie superalgebras. In a nutshell, take the category of $\Gamma$-graded vector spaces and skew braiding by a bicharacter of $\Gamma$. Now consider Lie algebras in …
- 12.1k
8
votes
Accepted
In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure ch...
Think of reps of $D(G)$ as $G$-equivariant vector bundles on $G$, where the group acts by conjugation. In this language, the tensor product is push-forward under multiplication $\mu : G\times G \right …
- 12.1k
4
votes
0
answers
267
views
First Fundamental Theorem for Alternating Group
I know it fails but is there an answer?
More precisely, let $V$ be the standard complex $n$-dimensional representation of the alternating group $A_n$, $kV$ the direct sum of its $k$ copies, $S(kV)$ i …
- 12.1k
3
votes
Which is the correct universal enveloping algebra in positive characteristic?
You forgot the third one: restricted enveloping algebra. Hence, in characteristic p we have 3 enveloping algebras with homomorphisms
U->U_0->U_{dp}
All 3 are Hopf algebras and can be used for differen …
- 12.1k
5
votes
Accepted
Killing form vs its counterpart in a given represenation
They are proportional if $g$ is simple. The form $K_\phi$ defines a homomorphism from the adjoint to the coadjoint representation. If the adjoint representation is irreducible, i.e. $g$ is simple, you …
- 12.1k