Question 4, chapter III, section 7 in Vinberg “Linear representations of groups. ” [closed]Prove that the linear span of some functions coincide with the space of the following functions on the unit circle.General question about representations of groupsSection 0 in Ernest B. Vinberg, “Linear Representations of groups” Q.7(e)$GL_2(mathbb R)$ acting on $hatmathbb R=mathbb Rcup infty$.Finite-dimensional representations of the integers (2)A difficulty in understanding the solution of #2 section 1 Vinberg.A difficulty in understanding an example in Vinberg.Finding all subspaces invariant under F.A difficulty in understanding the universal property of modules.A difficulty in understanding the definition of “Spaces of Matrix Elements.”prove that any central function of $SU_2$ is uniquely determined by its restriction to the following subgroup.
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Question 4, chapter III, section 7 in Vinberg “Linear representations of groups. ” [closed]
Prove that the linear span of some functions coincide with the space of the following functions on the unit circle.General question about representations of groupsSection 0 in Ernest B. Vinberg, “Linear Representations of groups” Q.7(e)$GL_2(mathbb R)$ acting on $hatmathbb R=mathbb Rcup infty$.Finite-dimensional representations of the integers (2)A difficulty in understanding the solution of #2 section 1 Vinberg.A difficulty in understanding an example in Vinberg.Finding all subspaces invariant under F.A difficulty in understanding the universal property of modules.A difficulty in understanding the definition of “Spaces of Matrix Elements.”prove that any central function of $SU_2$ is uniquely determined by its restriction to the following subgroup.
$begingroup$
The question and its answer is given below:
Where T is the unit circle, and $Phi_n$ is described below:
But I do not understand the solution,could anyone explain it for me please? or give me understandable solution as I am stucked in this problem.
representation-theory lie-groups lie-algebras physics topological-groups
asked Mar 29 at 10:00
hopefullyhopefully
270215
$endgroup$
closed as off-topic by Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer Mar 31 at 1:35
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer
add a comment |
$begingroup$
The question and its answer is given below:
Where T is the unit circle, and $Phi_n$ is described below:
But I do not understand the solution,could anyone explain it for me please? or give me understandable solution as I am stucked in this problem.
representation-theory lie-groups lie-algebras physics topological-groups
asked Mar 29 at 10:00
hopefullyhopefully
270215
$endgroup$
closed as off-topic by Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer Mar 31 at 1:35
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer
add a comment |
$begingroup$
The question and its answer is given below:
Where T is the unit circle, and $Phi_n$ is described below:
But I do not understand the solution,could anyone explain it for me please? or give me understandable solution as I am stucked in this problem.
representation-theory lie-groups lie-algebras physics topological-groups
asked Mar 29 at 10:00
hopefullyhopefully
270215
$endgroup$
The question and its answer is given below:
Where T is the unit circle, and $Phi_n$ is described below:
But I do not understand the solution,could anyone explain it for me please? or give me understandable solution as I am stucked in this problem.
representation-theory lie-groups lie-algebras physics topological-groups
representation-theory lie-groups lie-algebras physics topological-groups
asked Mar 29 at 10:00
hopefullyhopefully
270215
asked Mar 29 at 10:00
hopefullyhopefully
270215
edited Mar 29 at 10:18
hopefully
asked Mar 29 at 10:00
hopefullyhopefully
270215
asked Mar 29 at 10:00
hopefullyhopefully
270215
asked Mar 29 at 10:00
hopefullyhopefully
270215
270215
closed as off-topic by Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer Mar 31 at 1:35
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer
closed as off-topic by Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer Mar 31 at 1:35
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Dietrich Burde, mrtaurho, Leucippus, Cesareo, Eevee Trainer
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A basis of $Phi_n$ is given by the monomials $f_m=u_1^mu_2^n-m$, and for $A(z)=mathrmdiag(z,z^-1)$, we have $ Phi_n(A(z))(f_m)=z^m(z^-1)^n-mf_m$. Therefore $$mathrmtr,Phi_n(A(z))=chi_n(A(z))=z^n+z^n-1z^-1+cdots zz^-n+1+z^-n,$$
which is equal to $(z^n+1-z^-n-1)/(z-z^-1)$.
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
$endgroup$
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A basis of $Phi_n$ is given by the monomials $f_m=u_1^mu_2^n-m$, and for $A(z)=mathrmdiag(z,z^-1)$, we have $ Phi_n(A(z))(f_m)=z^m(z^-1)^n-mf_m$. Therefore $$mathrmtr,Phi_n(A(z))=chi_n(A(z))=z^n+z^n-1z^-1+cdots zz^-n+1+z^-n,$$
which is equal to $(z^n+1-z^-n-1)/(z-z^-1)$.
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
$endgroup$
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
add a comment |
$begingroup$
A basis of $Phi_n$ is given by the monomials $f_m=u_1^mu_2^n-m$, and for $A(z)=mathrmdiag(z,z^-1)$, we have $ Phi_n(A(z))(f_m)=z^m(z^-1)^n-mf_m$. Therefore $$mathrmtr,Phi_n(A(z))=chi_n(A(z))=z^n+z^n-1z^-1+cdots zz^-n+1+z^-n,$$
which is equal to $(z^n+1-z^-n-1)/(z-z^-1)$.
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
$endgroup$
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
add a comment |
$begingroup$
A basis of $Phi_n$ is given by the monomials $f_m=u_1^mu_2^n-m$, and for $A(z)=mathrmdiag(z,z^-1)$, we have $ Phi_n(A(z))(f_m)=z^m(z^-1)^n-mf_m$. Therefore $$mathrmtr,Phi_n(A(z))=chi_n(A(z))=z^n+z^n-1z^-1+cdots zz^-n+1+z^-n,$$
which is equal to $(z^n+1-z^-n-1)/(z-z^-1)$.
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
$endgroup$
A basis of $Phi_n$ is given by the monomials $f_m=u_1^mu_2^n-m$, and for $A(z)=mathrmdiag(z,z^-1)$, we have $ Phi_n(A(z))(f_m)=z^m(z^-1)^n-mf_m$. Therefore $$mathrmtr,Phi_n(A(z))=chi_n(A(z))=z^n+z^n-1z^-1+cdots zz^-n+1+z^-n,$$
which is equal to $(z^n+1-z^-n-1)/(z-z^-1)$.
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
answered Mar 30 at 23:19
Stefan DawydiakStefan Dawydiak
41429
41429
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
add a comment |
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
Could you please look at this question when you have time? math.stackexchange.com/questions/3168577/…
$endgroup$
– hopefully
Mar 30 at 23:30
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
how is the trace of $phi _n$ leads to this expression on the right?
$endgroup$
– Smart
5 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
$begingroup$
The matrix $Phi_n(A(z))$ is diagonalisable with eigenbasis given by the monomials $f_m$. The trace of a matrix is the sum of its eigenvalues.
$endgroup$
– Stefan Dawydiak
2 hours ago
add a comment |
