Return to Question

Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of $V$ --- can be constructed out of $Q$ (and its inverse), $C$ and a nonzero "volume element" $\nu \in \Lambda^{26}V$.

A quick calculation in LiE, however, shows that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

 > alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0]
      1

which I would have a hard time constructing out of $Q$, $C$ and $\nu$.

One possible way to understand $\Phi$ is to think in terms of the $\mathfrak{so}(9)$ subalgebra of $\mathfrak{g}$. Under $\mathfrak{so}(9)$, $V$ breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$ V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta $$

There are precisely two $\mathfrak{so}(9)$-invariants in $\Lambda^9 V$: one is the volume form on $\Lambda^1$ and the other is the "volume" form on $\Lambda^0$ wedged with the $\mathfrak{so}(9)$-invariant 8-form on $\Delta$. Notice that $(\mathfrak{so}(9),\Delta)$ is the holonomy representation for the Cayley plane $F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual $8$-form. Then $\Phi$ is some linear combination of these two $\mathfrak{so}(9)$-invariants, which I have yet to work out.

Questions

I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature:

1. Do $Q,C,\Phi,\nu$ form a complete generating set for the $F_4$-invariants in $\bigotimes V$?

2. Is there a more convenient (for calculations) description of $\Phi$? In particular, I would like to know about the relation of the form $\Phi \otimes \Phi = \cdots$.

Thank you in advance.

This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.

Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$. Since $V$ is irreducible, $Q$ is non-degenerate and we may use it to identify $V$ with $V^*$ as $\mathfrak{g}$ modules.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of in $\bigoplus_{n\geq 0} V^{\otimes n}$ --- can be constructed out of $Q$ (and its inverse), $C$ and a nonzero "volume element" $\nu \in \Lambda^{26}V$ via products in the tensor algebra and contractions.

For example, we can construct six invariant tensors out of $Q$ and $C$ in degree $4$ $$ Q_{ab}Q_{cd} \qquad Q_{ac}Q_{bd} \qquad Q_{ad}Q_{bc} \qquad C_{abe}C_{cdf} Q^{ef} \qquad C_{ace}C_{bdf} Q^{ef} \qquad C_{ade}C_{bcf} Q^{ef} $$ which satisfy a linear relation, since there is only a 5-dimensional space of such tensors.

Now, a quick calculation in LiE reveals that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

 > alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0]
      1

which cannot be constructed out of $Q$, $C$ and $\nu$ in the aforementioned way.

One possible way to understand $\Phi$ is to think in terms of the $\mathfrak{so}(9)$ subalgebra of $\mathfrak{g}$. Under $\mathfrak{so}(9)$, $V$ breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$ V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta $$

There are precisely two $\mathfrak{so}(9)$-invariants in $\Lambda^9 V$: one is the volume form on $\Lambda^1$ and the other is the "volume" form on $\Lambda^0$ wedged with the $\mathfrak{so}(9)$-invariant 8-form on $\Delta$. Notice that $(\mathfrak{so}(9),\Delta)$ is the holonomy representation for the Cayley plane $F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual $8$-form. Then $\Phi$ is some linear combination of these two $\mathfrak{so}(9)$-invariants, which I have yet to work out.

Questions

I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature:

1. Can every invariant tensor be constructed out of $Q$ (and its inverse) $C$, $\nu$ and $\Phi$ by products and contractions?

2. Is there a more convenient (for calculations) description of $\Phi$? In particular, I would like to know about the relation of the form $\Phi \otimes \Phi = \cdots$.

Thank you in advance.

Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of $V$ --- can be constructed out of $Q$, $C$ and a nonzero "volume element" $\nu \in \Lambda^{26}V$.

A quick calculation in LiE, however, shows that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

 > alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0]
      1
 >

which I would have a hard time constructing out of $Q$, $C$ and $\nu$.

One possible way to understand $\Phi$ is to think in terms of the $\mathfrak{so}(9)$ subalgebra of $\mathfrak{g}$. Under $\mathfrak{so}(9)$, $V$ breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$ V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta $$

There are precisely two $\mathfrak{so}(9)$-invariants in $\Lambda^9 V$: one is the volume form on $\Lambda^1$ and the other is the "volume" form on $\Lambda^0$ wedged with the $\mathfrak{so}(9)$-invariant 8-form on $\Delta$. Notice that $(\mathfrak{so}(9),\Delta)$ is the holonomy representation for the Cayley plane $F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual $8$-form. Then $\Phi$ is some linear combination of these two $\mathfrak{so}(9)$-invariants, which I have yet to work out.

Questions

I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature:

1. Do $Q,C,\Phi,\nu$ form a complete generating set for the $F_4$-invariants in $\bigotimes V$?

2. Is there a more convenient (for calculations) description of $\Phi$? In particular, I would like to know about the relation of the form $\Phi \otimes \Phi = \cdots$.

Thank you in advance.

Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of $V$ --- can be constructed out of $Q$ (and its inverse), $C$ and a nonzero "volume element" $\nu \in \Lambda^{26}V$.

A quick calculation in LiE, however, shows that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

 > alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0]
      1

which I would have a hard time constructing out of $Q$, $C$ and $\nu$.

One possible way to understand $\Phi$ is to think in terms of the $\mathfrak{so}(9)$ subalgebra of $\mathfrak{g}$. Under $\mathfrak{so}(9)$, $V$ breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$ V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta $$

There are precisely two $\mathfrak{so}(9)$-invariants in $\Lambda^9 V$: one is the volume form on $\Lambda^1$ and the other is the "volume" form on $\Lambda^0$ wedged with the $\mathfrak{so}(9)$-invariant 8-form on $\Delta$. Notice that $(\mathfrak{so}(9),\Delta)$ is the holonomy representation for the Cayley plane $F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual $8$-form. Then $\Phi$ is some linear combination of these two $\mathfrak{so}(9)$-invariants, which I have yet to work out.

Questions

I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature:

1. Do $Q,C,\Phi,\nu$ form a complete generating set for the $F_4$-invariants in $\bigotimes V$?

2. Is there a more convenient (for calculations) description of $\Phi$? In particular, I would like to know about the relation of the form $\Phi \otimes \Phi = \cdots$.

Thank you in advance.

Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of $V$ --- can be constructed out of $Q$, $C$ and a nonzero "volume element" $\nu \in \Lambda^{26}V$.

A quick calculation in LiE, however, shows that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

 > alt_tensor(9,[0,0,0,1],F4)
 1X[0,0,0,0] +1X[0,0,0,1] +2X[0,0,0,2] +2X[0,0,0,3] +2X[0,0,0,4] +
 1X[0,0,0,5] +1X[0,0,1,0] +2X[0,0,1,1] +2X[0,0,1,2] +1X[0,0,1,3] +
 3X[0,0,2,0] +1X[0,0,2,1] +1X[0,1,0,0] +2X[0,1,0,1] +1X[0,1,0,2] +
 2X[0,1,1,0] +1X[0,2,0,0] +1X[1,0,0,1] +1X[1,0,0,2] +1X[1,0,0,3] +
 3X[1,0,1,0] +3X[1,0,1,1] +1X[1,0,1,2] +1X[1,1,0,0] +1X[1,1,0,1] +
 1X[2,0,0,0] +1X[2,0,0,1] +1X[2,0,0,2]
 >

which I would have a hard time constructing out of $Q$, $C$ and $\nu$.

One possible way to understand $\Phi$ is to think in terms of the $\mathfrak{so}(9)$ subalgebra of $\mathfrak{g}$. Under $\mathfrak{so}(9)$, $V$ breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$ V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta $$

There are precisely two $\mathfrak{so}(9)$-invariants in $\Lambda^9 V$: one is the volume form on $\Lambda^1$ and the other is the "volume" form on $\Lambda^0$ wedged with the $\mathfrak{so}(9)$-invariant 8-form on $\Delta$. Notice that $(\mathfrak{so}(9),\Delta)$ is the holonomy representation for the Cayley plane $F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual $8$-form. Then $\Phi$ is some linear combination of these two $\mathfrak{so}(9)$-invariants, which I have yet to work out.

Questions

I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature:

1. Do $Q,C,\Phi,\nu$ form a complete generating set for the $F_4$-invariants in $\bigotimes V$?

2. Is there a more convenient (for calculations) description of $\Phi$? In particular, I would like to know about the relation of the form $\Phi \otimes \Phi = \cdots$.

Thank you in advance.

Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of $V$ --- can be constructed out of $Q$, $C$ and a nonzero "volume element" $\nu \in \Lambda^{26}V$.

A quick calculation in LiE, however, shows that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

 > alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0]
      1
 >

which I would have a hard time constructing out of $Q$, $C$ and $\nu$.

One possible way to understand $\Phi$ is to think in terms of the $\mathfrak{so}(9)$ subalgebra of $\mathfrak{g}$. Under $\mathfrak{so}(9)$, $V$ breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$ V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta $$

There are precisely two $\mathfrak{so}(9)$-invariants in $\Lambda^9 V$: one is the volume form on $\Lambda^1$ and the other is the "volume" form on $\Lambda^0$ wedged with the $\mathfrak{so}(9)$-invariant 8-form on $\Delta$. Notice that $(\mathfrak{so}(9),\Delta)$ is the holonomy representation for the Cayley plane $F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual $8$-form. Then $\Phi$ is some linear combination of these two $\mathfrak{so}(9)$-invariants, which I have yet to work out.

Questions

I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature:

1. Do $Q,C,\Phi,\nu$ form a complete generating set for the $F_4$-invariants in $\bigotimes V$?

2. Is there a more convenient (for calculations) description of $\Phi$? In particular, I would like to know about the relation of the form $\Phi \otimes \Phi = \cdots$.

Thank you in advance.