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Rings of Invariants

###Rings of Invariants### ConsiderConsider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free polynomial ring in $\dim(V)$-many variables. Then the ring of invariants of $V$ is the subring $$\mathbb{C}[V]^G:=(f\in\mathbb{C}[V] s.t. \forall g\in G, f(gv)=f(v))$$

The Conjugation Representation

###The Conjugation Representation### LetLet $M_2(\mathbb{C})$ denote the space of $2\times 2$ complex matrices. The conjugation representation $V_C$ of $G$ on $M_2(\mathbb{C})$ is defined by $g\cdot m:= gmg^{-1}$. Thus, $V_C$ is a 4-dimensional $G$-representation.

The ring of invariants $\mathbb{C}[V_C]^G$ is well-known. It is freely generated by the functions $tr$ and $det$; this is a special case of the general fact that the only conjugation-invariant functions on $M_n(\mathbb{C})$ are symmetric functions of eigenvalues.

The Tautological Module

###The Tautological Module### LetLet $V_T$ denote the $2$-dimensional $G$-representation where $G$ acts in the 'obvious' way, by the inclusion $G=SL_2(\mathbb{C})\subset GL_2(\mathbb{C})$. This is called the tautological $G$-representation. Since $SL_2(\mathbb{C})$ acts on $V_T$ with a dense orbit, the ring of invariants $\mathbb{C}[V_C]^G$ is boring; it is just $\mathbb{C}$.

Now, let $V_T^n$ denote the $n$-fold direct sum of $V_T$. This is a natural $2n$-dimensional $G$-representation. We can choose a $G$-invariant skew-symmetric bilinear form $\omega$ on $V_T$; this is big words for the usual scalar cross product $v_1\times v_2$. This defines a natural $G$-invariant function on the space of pairs $(v_1,v_2)\in V_T^2$.

Then the ring of invariants $\mathbb{C}[V_T^n]^G$ is generated by $\omega(v_i,v_j)$ for $1\leq i j\leq n$. However, these do not (in general) freely-generate the ring of invariants, there are relations between them. The relations are all of the form $$ \omega(v_i,v_k)\omega(v_j,v_l)=\omega(v_i,v_j)\omega(v_k,v_l)+\omega(v_l,v_i)\omega(v_j,v_k)$$ for $1\leq ijkl\leq n$.

Both of these facts can be deduced by observing that $\mathbb{C}[V_T^n]^G$ can be identified with the homogeneous coordinate ring of the Grassmanian $Gr(2,n)$. Then the generators above are the Plucker coordinates, and the relations are the 3-term Plucker relations (there are no higher Plucker relations here).

The Question

###The Question### BothBoth of these examples have clever solutions and pretty answers. I am curious about the combination of both cases.

Let $m$ and $n$ be positive integers, and consider the direct sum $G$-representation $V^m_C\oplus V_T^n$. What is the ring of invariants $\mathbb{C}[V_C^m\oplus V_T^n]^G$?

I am aware of general procedures for producing these rings, for arbitrary finite-dimensional $G$-reps; see e.h. Sturmfel's Algorithms in Invariant Theory. However, I suspect that there is a clever solution to this particular problem. Not only because it is a combination of two problems with a clever solution, but because I have a guess as to what the answer is, and all the relations seem to be similar to the 3-term Plucker relations. I also suspect the answer is ancient (like much invariant theory), which is why I am trying to find an answer rather than try to prove my guess is correct by brute force.

###Rings of Invariants### Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free polynomial ring in $\dim(V)$-many variables. Then the ring of invariants of $V$ is the subring $$\mathbb{C}[V]^G:=(f\in\mathbb{C}[V] s.t. \forall g\in G, f(gv)=f(v))$$

###The Conjugation Representation### Let $M_2(\mathbb{C})$ denote the space of $2\times 2$ complex matrices. The conjugation representation $V_C$ of $G$ on $M_2(\mathbb{C})$ is defined by $g\cdot m:= gmg^{-1}$. Thus, $V_C$ is a 4-dimensional $G$-representation.

The ring of invariants $\mathbb{C}[V_C]^G$ is well-known. It is freely generated by the functions $tr$ and $det$; this is a special case of the general fact that the only conjugation-invariant functions on $M_n(\mathbb{C})$ are symmetric functions of eigenvalues.

###The Tautological Module### Let $V_T$ denote the $2$-dimensional $G$-representation where $G$ acts in the 'obvious' way, by the inclusion $G=SL_2(\mathbb{C})\subset GL_2(\mathbb{C})$. This is called the tautological $G$-representation. Since $SL_2(\mathbb{C})$ acts on $V_T$ with a dense orbit, the ring of invariants $\mathbb{C}[V_C]^G$ is boring; it is just $\mathbb{C}$.

Now, let $V_T^n$ denote the $n$-fold direct sum of $V_T$. This is a natural $2n$-dimensional $G$-representation. We can choose a $G$-invariant skew-symmetric bilinear form $\omega$ on $V_T$; this is big words for the usual scalar cross product $v_1\times v_2$. This defines a natural $G$-invariant function on the space of pairs $(v_1,v_2)\in V_T^2$.

Then the ring of invariants $\mathbb{C}[V_T^n]^G$ is generated by $\omega(v_i,v_j)$ for $1\leq i j\leq n$. However, these do not (in general) freely-generate the ring of invariants, there are relations between them. The relations are all of the form $$ \omega(v_i,v_k)\omega(v_j,v_l)=\omega(v_i,v_j)\omega(v_k,v_l)+\omega(v_l,v_i)\omega(v_j,v_k)$$ for $1\leq ijkl\leq n$.

Both of these facts can be deduced by observing that $\mathbb{C}[V_T^n]^G$ can be identified with the homogeneous coordinate ring of the Grassmanian $Gr(2,n)$. Then the generators above are the Plucker coordinates, and the relations are the 3-term Plucker relations (there are no higher Plucker relations here).

###The Question### Both of these examples have clever solutions and pretty answers. I am curious about the combination of both cases.

Let $m$ and $n$ be positive integers, and consider the direct sum $G$-representation $V^m_C\oplus V_T^n$. What is the ring of invariants $\mathbb{C}[V_C^m\oplus V_T^n]^G$?

I am aware of general procedures for producing these rings, for arbitrary finite-dimensional $G$-reps; see e.h. Sturmfel's Algorithms in Invariant Theory. However, I suspect that there is a clever solution to this particular problem. Not only because it is a combination of two problems with a clever solution, but because I have a guess as to what the answer is, and all the relations seem to be similar to the 3-term Plucker relations. I also suspect the answer is ancient (like much invariant theory), which is why I am trying to find an answer rather than try to prove my guess is correct by brute force.

Rings of Invariants

Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free polynomial ring in $\dim(V)$-many variables. Then the ring of invariants of $V$ is the subring $$\mathbb{C}[V]^G:=(f\in\mathbb{C}[V] s.t. \forall g\in G, f(gv)=f(v))$$

The Conjugation Representation

Let $M_2(\mathbb{C})$ denote the space of $2\times 2$ complex matrices. The conjugation representation $V_C$ of $G$ on $M_2(\mathbb{C})$ is defined by $g\cdot m:= gmg^{-1}$. Thus, $V_C$ is a 4-dimensional $G$-representation.

The ring of invariants $\mathbb{C}[V_C]^G$ is well-known. It is freely generated by the functions $tr$ and $det$; this is a special case of the general fact that the only conjugation-invariant functions on $M_n(\mathbb{C})$ are symmetric functions of eigenvalues.

The Tautological Module

Let $V_T$ denote the $2$-dimensional $G$-representation where $G$ acts in the 'obvious' way, by the inclusion $G=SL_2(\mathbb{C})\subset GL_2(\mathbb{C})$. This is called the tautological $G$-representation. Since $SL_2(\mathbb{C})$ acts on $V_T$ with a dense orbit, the ring of invariants $\mathbb{C}[V_C]^G$ is boring; it is just $\mathbb{C}$.

Now, let $V_T^n$ denote the $n$-fold direct sum of $V_T$. This is a natural $2n$-dimensional $G$-representation. We can choose a $G$-invariant skew-symmetric bilinear form $\omega$ on $V_T$; this is big words for the usual scalar cross product $v_1\times v_2$. This defines a natural $G$-invariant function on the space of pairs $(v_1,v_2)\in V_T^2$.

Then the ring of invariants $\mathbb{C}[V_T^n]^G$ is generated by $\omega(v_i,v_j)$ for $1\leq i j\leq n$. However, these do not (in general) freely-generate the ring of invariants, there are relations between them. The relations are all of the form $$ \omega(v_i,v_k)\omega(v_j,v_l)=\omega(v_i,v_j)\omega(v_k,v_l)+\omega(v_l,v_i)\omega(v_j,v_k)$$ for $1\leq ijkl\leq n$.

Both of these facts can be deduced by observing that $\mathbb{C}[V_T^n]^G$ can be identified with the homogeneous coordinate ring of the Grassmanian $Gr(2,n)$. Then the generators above are the Plucker coordinates, and the relations are the 3-term Plucker relations (there are no higher Plucker relations here).

The Question

Both of these examples have clever solutions and pretty answers. I am curious about the combination of both cases.

Let $m$ and $n$ be positive integers, and consider the direct sum $G$-representation $V^m_C\oplus V_T^n$. What is the ring of invariants $\mathbb{C}[V_C^m\oplus V_T^n]^G$?

I am aware of general procedures for producing these rings, for arbitrary finite-dimensional $G$-reps; see e.h. Sturmfel's Algorithms in Invariant Theory. However, I suspect that there is a clever solution to this particular problem. Not only because it is a combination of two problems with a clever solution, but because I have a guess as to what the answer is, and all the relations seem to be similar to the 3-term Plucker relations. I also suspect the answer is ancient (like much invariant theory), which is why I am trying to find an answer rather than try to prove my guess is correct by brute force.

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Greg Muller
Greg Muller
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The ring of SL_2 invariants in sums of conjugation and tautological modules

###Rings of Invariants### Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free polynomial ring in $\dim(V)$-many variables. Then the ring of invariants of $V$ is the subring $$\mathbb{C}[V]^G:=(f\in\mathbb{C}[V] s.t. \forall g\in G, f(gv)=f(v))$$

###The Conjugation Representation### Let $M_2(\mathbb{C})$ denote the space of $2\times 2$ complex matrices. The conjugation representation $V_C$ of $G$ on $M_2(\mathbb{C})$ is defined by $g\cdot m:= gmg^{-1}$. Thus, $V_C$ is a 4-dimensional $G$-representation.

The ring of invariants $\mathbb{C}[V_C]^G$ is well-known. It is freely generated by the functions $tr$ and $det$; this is a special case of the general fact that the only conjugation-invariant functions on $M_n(\mathbb{C})$ are symmetric functions of eigenvalues.

###The Tautological Module### Let $V_T$ denote the $2$-dimensional $G$-representation where $G$ acts in the 'obvious' way, by the inclusion $G=SL_2(\mathbb{C})\subset GL_2(\mathbb{C})$. This is called the tautological $G$-representation. Since $SL_2(\mathbb{C})$ acts on $V_T$ with a dense orbit, the ring of invariants $\mathbb{C}[V_C]^G$ is boring; it is just $\mathbb{C}$.

Now, let $V_T^n$ denote the $n$-fold direct sum of $V_T$. This is a natural $2n$-dimensional $G$-representation. We can choose a $G$-invariant skew-symmetric bilinear form $\omega$ on $V_T$; this is big words for the usual scalar cross product $v_1\times v_2$. This defines a natural $G$-invariant function on the space of pairs $(v_1,v_2)\in V_T^2$.

Then the ring of invariants $\mathbb{C}[V_T^n]^G$ is generated by $\omega(v_i,v_j)$ for $1\leq i j\leq n$. However, these do not (in general) freely-generate the ring of invariants, there are relations between them. The relations are all of the form $$ \omega(v_i,v_k)\omega(v_j,v_l)=\omega(v_i,v_j)\omega(v_k,v_l)+\omega(v_l,v_i)\omega(v_j,v_k)$$ for $1\leq ijkl\leq n$.

Both of these facts can be deduced by observing that $\mathbb{C}[V_T^n]^G$ can be identified with the homogeneous coordinate ring of the Grassmanian $Gr(2,n)$. Then the generators above are the Plucker coordinates, and the relations are the 3-term Plucker relations (there are no higher Plucker relations here).

###The Question### Both of these examples have clever solutions and pretty answers. I am curious about the combination of both cases.

Let $m$ and $n$ be positive integers, and consider the direct sum $G$-representation $V^m_C\oplus V_T^n$. What is the ring of invariants $\mathbb{C}[V_C^m\oplus V_T^n]^G$?

I am aware of general procedures for producing these rings, for arbitrary finite-dimensional $G$-reps; see e.h. Sturmfel's Algorithms in Invariant Theory. However, I suspect that there is a clever solution to this particular problem. Not only because it is a combination of two problems with a clever solution, but because I have a guess as to what the answer is, and all the relations seem to be similar to the 3-term Plucker relations. I also suspect the answer is ancient (like much invariant theory), which is why I am trying to find an answer rather than try to prove my guess is correct by brute force.