Mixing
Decomposing $k$th exterior powers $Lambda^kV(omega_1)$
$begingroup$
Let $Phi$ be a $G_2$ root system, $omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V
= V(omega_1)$ of highest weight $omega_1$.
I am asked to decompose in terms of highest weights the exterior powers $Lambda^kV$ for all $k$.
Note $Lambda^kV = V^{otimes k}/N_n$, where:
$N_n = {v_1 otimes dots otimes v_k mid v_i in V ; forall i text{ and } v_i = v_j text{ for some }i text{ and } j}$.
Denote the coset of $u_1 otimes dots otimes u_k$ by $u_1 wedge dots wedge u_k$. It is known that if $V $ has basis $v_1, dots, v_n$, then $Lambda^kV$ has basis ${v_{i_1} wedge dots wedge v_{i_k} mid i_1 < dots < i_k in {1,dots, n} }$.
In our case, $V$ has dimension $n = 7$ and so, I think that it would be correct to say that $lambda^kV = 0, ; forall k geq 8$ since if we can't have eight of the seven basis vectors wedged together without a repeat, meaning it would land within $N_7$.
Further, I think this means that $Lambda^7V$ has dimension $1$ and is exactly $span{v_1wedge dots wedge v_7}$
Now, regarding the specifics of the question, I think this means that for $kgeq 8$ the $k$th exterior powers of $V$ have highest weight $0$ since it is the $0$-space?
For $k=7$ however, I am less certain of how to proceed. I suppose that we could without loss of generality suppose that $v_1$ is the highest weight vector of weight $omega_1$ in $V(omega_1)$, but then we have:
For $min mathbb C$ and $t in mathfrak{t}$ the Cartan Subalgebra,
$t cdot m(v_1wedge dots wedge v_7) = m((t cdot v_1)wedge dots wedge v_7 + dots + v_1 wedge dots wedge (tcdot v_7))$
It is not clear to me that this is a highest weight vector of any weight, because I am not sure how $t$ acts on any of the other basis vectors. Thus I am not sure how to proceed with this case.
Additionally, I am even less sure how to proceed with the cases of $k leq 6$. In these cases, the representation has dimension greater than 1, but aside from that I am struggling to make any helpful or worthwhile observations.
How might I be able to proceed and decompose these exterior powers?
representation-theory lie-algebras exterior-algebra root-systems
asked Jan 9 at 21:48
user366818user366818
949410
$endgroup$
add a comment |
$begingroup$
Let $Phi$ be a $G_2$ root system, $omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V
= V(omega_1)$ of highest weight $omega_1$.
I am asked to decompose in terms of highest weights the exterior powers $Lambda^kV$ for all $k$.
Note $Lambda^kV = V^{otimes k}/N_n$, where:
$N_n = {v_1 otimes dots otimes v_k mid v_i in V ; forall i text{ and } v_i = v_j text{ for some }i text{ and } j}$.
Denote the coset of $u_1 otimes dots otimes u_k$ by $u_1 wedge dots wedge u_k$. It is known that if $V $ has basis $v_1, dots, v_n$, then $Lambda^kV$ has basis ${v_{i_1} wedge dots wedge v_{i_k} mid i_1 < dots < i_k in {1,dots, n} }$.
In our case, $V$ has dimension $n = 7$ and so, I think that it would be correct to say that $lambda^kV = 0, ; forall k geq 8$ since if we can't have eight of the seven basis vectors wedged together without a repeat, meaning it would land within $N_7$.
Further, I think this means that $Lambda^7V$ has dimension $1$ and is exactly $span{v_1wedge dots wedge v_7}$
Now, regarding the specifics of the question, I think this means that for $kgeq 8$ the $k$th exterior powers of $V$ have highest weight $0$ since it is the $0$-space?
For $k=7$ however, I am less certain of how to proceed. I suppose that we could without loss of generality suppose that $v_1$ is the highest weight vector of weight $omega_1$ in $V(omega_1)$, but then we have:
For $min mathbb C$ and $t in mathfrak{t}$ the Cartan Subalgebra,
$t cdot m(v_1wedge dots wedge v_7) = m((t cdot v_1)wedge dots wedge v_7 + dots + v_1 wedge dots wedge (tcdot v_7))$
It is not clear to me that this is a highest weight vector of any weight, because I am not sure how $t$ acts on any of the other basis vectors. Thus I am not sure how to proceed with this case.
Additionally, I am even less sure how to proceed with the cases of $k leq 6$. In these cases, the representation has dimension greater than 1, but aside from that I am struggling to make any helpful or worthwhile observations.
How might I be able to proceed and decompose these exterior powers?
representation-theory lie-algebras exterior-algebra root-systems
asked Jan 9 at 21:48
user366818user366818
949410
$endgroup$
add a comment |
$begingroup$
Let $Phi$ be a $G_2$ root system, $omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V
= V(omega_1)$ of highest weight $omega_1$.
I am asked to decompose in terms of highest weights the exterior powers $Lambda^kV$ for all $k$.
Note $Lambda^kV = V^{otimes k}/N_n$, where:
$N_n = {v_1 otimes dots otimes v_k mid v_i in V ; forall i text{ and } v_i = v_j text{ for some }i text{ and } j}$.
Denote the coset of $u_1 otimes dots otimes u_k$ by $u_1 wedge dots wedge u_k$. It is known that if $V $ has basis $v_1, dots, v_n$, then $Lambda^kV$ has basis ${v_{i_1} wedge dots wedge v_{i_k} mid i_1 < dots < i_k in {1,dots, n} }$.
In our case, $V$ has dimension $n = 7$ and so, I think that it would be correct to say that $lambda^kV = 0, ; forall k geq 8$ since if we can't have eight of the seven basis vectors wedged together without a repeat, meaning it would land within $N_7$.
Further, I think this means that $Lambda^7V$ has dimension $1$ and is exactly $span{v_1wedge dots wedge v_7}$
Now, regarding the specifics of the question, I think this means that for $kgeq 8$ the $k$th exterior powers of $V$ have highest weight $0$ since it is the $0$-space?
For $k=7$ however, I am less certain of how to proceed. I suppose that we could without loss of generality suppose that $v_1$ is the highest weight vector of weight $omega_1$ in $V(omega_1)$, but then we have:
For $min mathbb C$ and $t in mathfrak{t}$ the Cartan Subalgebra,
$t cdot m(v_1wedge dots wedge v_7) = m((t cdot v_1)wedge dots wedge v_7 + dots + v_1 wedge dots wedge (tcdot v_7))$
It is not clear to me that this is a highest weight vector of any weight, because I am not sure how $t$ acts on any of the other basis vectors. Thus I am not sure how to proceed with this case.
Additionally, I am even less sure how to proceed with the cases of $k leq 6$. In these cases, the representation has dimension greater than 1, but aside from that I am struggling to make any helpful or worthwhile observations.
How might I be able to proceed and decompose these exterior powers?
representation-theory lie-algebras exterior-algebra root-systems
asked Jan 9 at 21:48
user366818user366818
949410
$endgroup$
Let $Phi$ be a $G_2$ root system, $omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V
= V(omega_1)$ of highest weight $omega_1$.
I am asked to decompose in terms of highest weights the exterior powers $Lambda^kV$ for all $k$.
Note $Lambda^kV = V^{otimes k}/N_n$, where:
$N_n = {v_1 otimes dots otimes v_k mid v_i in V ; forall i text{ and } v_i = v_j text{ for some }i text{ and } j}$.
Denote the coset of $u_1 otimes dots otimes u_k$ by $u_1 wedge dots wedge u_k$. It is known that if $V $ has basis $v_1, dots, v_n$, then $Lambda^kV$ has basis ${v_{i_1} wedge dots wedge v_{i_k} mid i_1 < dots < i_k in {1,dots, n} }$.
In our case, $V$ has dimension $n = 7$ and so, I think that it would be correct to say that $lambda^kV = 0, ; forall k geq 8$ since if we can't have eight of the seven basis vectors wedged together without a repeat, meaning it would land within $N_7$.
Further, I think this means that $Lambda^7V$ has dimension $1$ and is exactly $span{v_1wedge dots wedge v_7}$
Now, regarding the specifics of the question, I think this means that for $kgeq 8$ the $k$th exterior powers of $V$ have highest weight $0$ since it is the $0$-space?
For $k=7$ however, I am less certain of how to proceed. I suppose that we could without loss of generality suppose that $v_1$ is the highest weight vector of weight $omega_1$ in $V(omega_1)$, but then we have:
For $min mathbb C$ and $t in mathfrak{t}$ the Cartan Subalgebra,
$t cdot m(v_1wedge dots wedge v_7) = m((t cdot v_1)wedge dots wedge v_7 + dots + v_1 wedge dots wedge (tcdot v_7))$
It is not clear to me that this is a highest weight vector of any weight, because I am not sure how $t$ acts on any of the other basis vectors. Thus I am not sure how to proceed with this case.
Additionally, I am even less sure how to proceed with the cases of $k leq 6$. In these cases, the representation has dimension greater than 1, but aside from that I am struggling to make any helpful or worthwhile observations.
How might I be able to proceed and decompose these exterior powers?
representation-theory lie-algebras exterior-algebra root-systems
representation-theory lie-algebras exterior-algebra root-systems
asked Jan 9 at 21:48
user366818user366818
949410
asked Jan 9 at 21:48
user366818user366818
949410
edited Jan 10 at 0:14
user366818
asked Jan 9 at 21:48
user366818user366818
949410
asked Jan 9 at 21:48
user366818user366818
949410
asked Jan 9 at 21:48
user366818user366818
949410
949410
add a comment |
add a comment |
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