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Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$.
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Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
beginequation
D(g) D(h) = e^i omega(g,h) D(gh)
endequation
These can be classified by the equivalence relation $omega(g,h) sim omega(g,h)+theta(g)+ theta(h) - theta(gh)$ subject to the condition $omega(g,h)+omega(gh,l)-omega(h,l)-omega(g,hl) = 0$.
The distinct equivalent classes of projective representations are labeled by elements of $H^2(G,U(1))$.
I know that for every such finite group, there is atleast one finite covering group C with the property that every projective representation of G can be lifted to an ordinary representation of C [1].
My question is about the relationship between the irreducible representations (irreps) of C and the group $H^2(G,U(1))$. Specifically, is the following statement true?:
Every irrep $Gamma_i$ of C can be associated an element of $nu in H^2(G,U(1))$ like $Gamma^nu_i$. The group property of $H^2(G,U(1))$ is reflected in the Clebsch-Gordan decomposition of tensor product of irreps of C:
beginequation
Gamma^nu_i otimes Gamma^mu_j cong bigoplus_k Gamma^nu+mu_k
endequation
I have noticed that this is true for all cases I have seen when $ H^2(G,U(1)) cong mathbbZ_2$ like $G = mathbbZ_2 times mathbbZ_2$, $C = D_8$ but I am unsure if this is true in general.
[1] https://en.wikipedia.org/wiki/Schur_multiplier#Relation_to_projective_representations
group-theory finite-groups representation-theory mathematical-physics group-cohomology
asked Jul 20 at 16:33
sawd
233
add a comment |Â
up vote
2
down vote
favorite
Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
beginequation
D(g) D(h) = e^i omega(g,h) D(gh)
endequation
These can be classified by the equivalence relation $omega(g,h) sim omega(g,h)+theta(g)+ theta(h) - theta(gh)$ subject to the condition $omega(g,h)+omega(gh,l)-omega(h,l)-omega(g,hl) = 0$.
The distinct equivalent classes of projective representations are labeled by elements of $H^2(G,U(1))$.
I know that for every such finite group, there is atleast one finite covering group C with the property that every projective representation of G can be lifted to an ordinary representation of C [1].
My question is about the relationship between the irreducible representations (irreps) of C and the group $H^2(G,U(1))$. Specifically, is the following statement true?:
Every irrep $Gamma_i$ of C can be associated an element of $nu in H^2(G,U(1))$ like $Gamma^nu_i$. The group property of $H^2(G,U(1))$ is reflected in the Clebsch-Gordan decomposition of tensor product of irreps of C:
beginequation
Gamma^nu_i otimes Gamma^mu_j cong bigoplus_k Gamma^nu+mu_k
endequation
I have noticed that this is true for all cases I have seen when $ H^2(G,U(1)) cong mathbbZ_2$ like $G = mathbbZ_2 times mathbbZ_2$, $C = D_8$ but I am unsure if this is true in general.
[1] https://en.wikipedia.org/wiki/Schur_multiplier#Relation_to_projective_representations
group-theory finite-groups representation-theory mathematical-physics group-cohomology
asked Jul 20 at 16:33
sawd
233
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
beginequation
D(g) D(h) = e^i omega(g,h) D(gh)
endequation
These can be classified by the equivalence relation $omega(g,h) sim omega(g,h)+theta(g)+ theta(h) - theta(gh)$ subject to the condition $omega(g,h)+omega(gh,l)-omega(h,l)-omega(g,hl) = 0$.
The distinct equivalent classes of projective representations are labeled by elements of $H^2(G,U(1))$.
I know that for every such finite group, there is atleast one finite covering group C with the property that every projective representation of G can be lifted to an ordinary representation of C [1].
My question is about the relationship between the irreducible representations (irreps) of C and the group $H^2(G,U(1))$. Specifically, is the following statement true?:
Every irrep $Gamma_i$ of C can be associated an element of $nu in H^2(G,U(1))$ like $Gamma^nu_i$. The group property of $H^2(G,U(1))$ is reflected in the Clebsch-Gordan decomposition of tensor product of irreps of C:
beginequation
Gamma^nu_i otimes Gamma^mu_j cong bigoplus_k Gamma^nu+mu_k
endequation
I have noticed that this is true for all cases I have seen when $ H^2(G,U(1)) cong mathbbZ_2$ like $G = mathbbZ_2 times mathbbZ_2$, $C = D_8$ but I am unsure if this is true in general.
[1] https://en.wikipedia.org/wiki/Schur_multiplier#Relation_to_projective_representations
group-theory finite-groups representation-theory mathematical-physics group-cohomology
asked Jul 20 at 16:33
sawd
233
Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
beginequation
D(g) D(h) = e^i omega(g,h) D(gh)
endequation
These can be classified by the equivalence relation $omega(g,h) sim omega(g,h)+theta(g)+ theta(h) - theta(gh)$ subject to the condition $omega(g,h)+omega(gh,l)-omega(h,l)-omega(g,hl) = 0$.
The distinct equivalent classes of projective representations are labeled by elements of $H^2(G,U(1))$.
I know that for every such finite group, there is atleast one finite covering group C with the property that every projective representation of G can be lifted to an ordinary representation of C [1].
My question is about the relationship between the irreducible representations (irreps) of C and the group $H^2(G,U(1))$. Specifically, is the following statement true?:
Every irrep $Gamma_i$ of C can be associated an element of $nu in H^2(G,U(1))$ like $Gamma^nu_i$. The group property of $H^2(G,U(1))$ is reflected in the Clebsch-Gordan decomposition of tensor product of irreps of C:
beginequation
Gamma^nu_i otimes Gamma^mu_j cong bigoplus_k Gamma^nu+mu_k
endequation
I have noticed that this is true for all cases I have seen when $ H^2(G,U(1)) cong mathbbZ_2$ like $G = mathbbZ_2 times mathbbZ_2$, $C = D_8$ but I am unsure if this is true in general.
[1] https://en.wikipedia.org/wiki/Schur_multiplier#Relation_to_projective_representations
group-theory finite-groups representation-theory mathematical-physics group-cohomology
asked Jul 20 at 16:33
sawd
233
asked Jul 20 at 16:33
sawd
233
asked Jul 20 at 16:33
sawd
233
asked Jul 20 at 16:33
sawd
233
233
add a comment |Â
add a comment |Â
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