## Some Invariant Theory

We continue the previous discussion. Recall that for $\mu = \overline\lambda$ we have a $GL_n\mathbb{C}$-equivariant map

$\displaystyle \bigotimes_j \text{Alt}^{\mu_j} \mathbb{C}^n \to \bigotimes_i \text{Sym}^{\lambda_i} \mathbb{C}^n \subset\mathbb{C}[z_{i,j}]^{(\lambda_i)}, \quad e_T^\circ \mapsto D_T$

which induces an isomorphism between the unique copies of $V(\lambda)$ in both spaces. The kernel Q of this map is spanned by $e_T^\circ - \sum_S e_S^\circ$ for various fillings T with shape $\lambda$ and entries in [n].

E.g. suppose $\lambda = (4, 3, 1)$ and $n=5$; then $\mu = (3, 2, 2,1)$ and the map induces:

$\displaystyle \text{Alt}^3 \mathbb{C}^5 \otimes \text{Alt}^2 \mathbb{C}^5 \otimes \text{Alt}^2 \mathbb{C}^5\otimes \mathbb{C}^5 \longrightarrow \mathbb{C}[z_{1,j}]^{(4)} \otimes \mathbb{C}[z_{2,j}]^{(3)}\otimes \mathbb{C}[z_{3,j}]^{(1)}, \ 1\le j \le 5,$

with kernel Q. For the following filling T, we have the correspondence:

Lemma. If $\mu_j = \mu_{j+1} = \ldots = \mu_{j+m-1}$, then the above map factors through

$\displaystyle \text{Alt}^{\mu_j}\mathbb{C}^n \otimes \ldots \otimes \text{Alt}^{\mu_{j+m-1}}\mathbb{C}^n\longrightarrow \text{Sym}^m \left(\text{Alt}^{\mu_j} \mathbb{C}^n\right).$

E.g. in our example above, the map factors through $\text{Alt}^3 \mathbb{C}^5 \otimes \text{Sym}^2 \left(\text{Alt}^2 \mathbb{C}^5 \right)\otimes \mathbb{C}^5$.

Proof

Indeed if $\mu_j = \mu_{j+1}$, then swapping columns $j$ and $j+1$ of $T$ gives us the same $D_T$. ♦

Now suppose $0\le a\le n$ and $m\ge 0$; pick $\lambda = (m, \ldots, m)$ with length a. Now $\mu$ comprises of m copies of a so by the above lemma, we have a map:

$\displaystyle \text{Sym}^m \left(\text{Alt}^a \mathbb{C}^n\right) \longrightarrow \mathbb{C}[z_{1,j}]^{(m)} \otimes \ldots \otimes \mathbb{C}[z_{a,j}]^{(m)} \cong \mathbb{C}[z_{i,j}]^{(m,\ldots,m)}$

where $1 \le i \le a$ and $1 \le j \le n$. Taking the direct sum over all m we have:

$\displaystyle \text{Sym}^* \left(\text{Alt}^a \mathbb{C}^n\right) := \bigoplus_{m\ge 0} \text{Sym}^m \left(\text{Alt}^a \mathbb{C}^n\right)\longrightarrow \bigoplus_{m\ge 0}\mathbb{C}[z_{i,j}]^{(m,\ldots,m)} \subset \mathbb{C}[z_{i,j}]$

which is a homomorphism of $GL_n\mathbb{C}$-representations. Furthermore, $\text{Sym}^* V$ for any vector space V has an algebra structure via $\text{Sym}^m V \times \text{Sym}^n V \to \text{Sym}^{m+n}V.$ The above map clearly preserves multiplication since multiplying $e_T^\circ e_{T'}^\circ$ and $D_T D_{T'}$ both correspond to concatenation of T and T’. So it is also a homomorphism of $\mathbb{C}$-algebras.

Question. A basis of $\text{Alt}^a\mathbb{C}^n$ is given by

$\displaystyle e_{i_1, \ldots, i_a} := e_{i_1} \wedge \ldots \wedge e_{i_a}$

for $1 \le i_1 < \ldots < i_a \le n.$ Hence $\text{Sym}^*\left(\text{Alt}^a \mathbb{C}^n\right) \cong \mathbb{C}[e_{i_1,\ldots,i_a}]$, the ring of polynomials in $n\choose a$ variables. What is the kernel $P$ of the induced map:

$\mathbb{C}[e_{i_1, \ldots, i_a}] \longrightarrow \mathbb{C}[z_{1,1}, \ldots, z_{a,n}]?$

We have seen that for any $1 \le i_1 < \ldots < i_a \le n$ and $1 \le j_1 < \ldots < j_a \le n$ we have:

$\displaystyle e_{i_1, \ldots, i_a} e_{j_1, \ldots, j_a} = \sum e_{i_1', \ldots, i_a'} e_{j_1', \ldots, j_k', j_{k+1},\ldots, j_a}$

where we swap $j_1, \ldots, j_k$ with various sets of k indices in $i_1, \ldots, i_a$ while preserving the order to give $(i_1', \ldots, i_a')$ and $(j_1', \ldots, j_k')$. Hence, P contains the ideal generated by all such quadratic relations.

On the other hand, any relation $e_T = \sum_S e_S$ is a multiple of such a quadratic equation with a polynomial. This is clear by taking the two columns used in swapping; the remaining columns simply multiply the quadratic relation with a polynomial. Hence P is the ideal generated by these quadratic equations. ♦

Since the quotient of $\mathbb{C}[e_{i_1, \ldots, i_a}]$ by $P$ is a subring of $\mathbb{C}[z_{i,j}]$, we have:

Corollary. The ideal generated by the above quadratic equations is prime.

## Fundamental Theorems of Invariant Theory

Recall that $g = (g_{i,j}) \in GL_n\mathbb{C}$ takes $z_{i,j} \mapsto \sum_k z_{i,k} g_{k,j}$, which is a left action; here $1\le i\le a$, $1\le j\le n$. We also let $GL_a\mathbb{C}$ act on the right via:

$\displaystyle g=(g_{i,j}) \in GL_a\mathbb{C} :z_{i,j} \mapsto \sum_k g_{i,k}z_{k,j}$

so that $\mathbb{C}[z_{i,j}]$ becomes a $(GL_n\mathbb{C}, GL_a\mathbb{C})$-bimodule. A basic problem in invariant theory is to describe the ring $\mathbb{C}[z_{i,j}]^{SL_a\mathbb{C}}$ comprising of all f such that $g\cdot f = f$ for all $g\in SL_a\mathbb{C}$.

Theorem. The ring $\mathbb{C}[z_{i,j}]^{SL_a\mathbb{C}}$ is the image of:

$\displaystyle \mathbb{C}[D_{i_1, \ldots, i_a}] \cong \mathbb{C}[e_{i_1, \ldots, i_a}]/P \hookrightarrow \mathbb{C}[z_{i,j}]$

where $(i_1, \ldots, i_a)$ runs over all $1 \le i_1 < \ldots < i_a \le n$.

In other words, we have:

• First Fundamental Theorem : the ring of $SL_a\mathbb{C}$-invariants in $\mathbb{C}[z_{i,j}]$ is generated by $\{D_{i_1, \ldots, i_a}: 1 \le i_1 < \ldots < i_a \le n\}.$
• Second Fundamental Theorem : the relations satisfied by these polynomials are generated by the above quadratic relations.

## Proof of Fundamental Theorems

Note that g takes $D_{i_1, \ldots, i_a}$ to:

$\displaystyle \det{\small \begin{pmatrix} z_{1, i_1} & z_{1, i_2} & \ldots & z_{1, i_a} \\ z_{2, i_1} & z_{2, i_2} & \ldots & z_{2, i_a} \\ \vdots & \vdots & \ddots & \vdots \\ z_{a, i_1} & z_{a, i_2} & \ldots & z_{a, i_a}\end{pmatrix}} \mapsto \det {\small \begin{pmatrix}\sum_{j_1} g_{1,j_1} z_{j_1, i_1} & \ldots & \sum_{j_a} g_{1,j_a} z_{j_a, i_a} \\ \sum_{j_1} g_{2,j_1} z_{j_1, i_1} & \ldots & \sum_{j_a} g_{2,j_a} z_{j_a, i_a} \\ \vdots & \ddots & \vdots \\ \sum_{j_1} g_{a,j_1} z_{j_1, i_1} & \ldots & \sum_{j_a} g_{a,j_a} z_{j_a, i_a} \end{pmatrix}} = \det(g) D_{i_1, \ldots, i_a}$

which is $D_{i_1, \ldots, i_a}$ if $g \in SL_a\mathbb{C}$. Hence we have $\mathbb{C}[D_{i_1, \ldots, i_a}] \subseteq \mathbb{C}[z_{i,j}]^{SL_a\mathbb{C}}$. To prove equality, we show that their dimensions in degree d agree. By the previous article, the degree-d component of $\mathbb{C}[e_{i_1, \ldots, i_a}]/P$ has a basis indexed by SSYT of type $\lambda = (\frac d a, \ldots, \frac d a)$ and entries in [n]; if d is not a multiple of a, the component is 0.

Next we check the degree-d component of $\mathbb{C}[z_{i,j}]^{SL_a\mathbb{C} }$. As $GL_a\mathbb{C}$-representations, we have

$\displaystyle \mathbb{C}[z_{i,j}] \cong \text{Sym}^*\left( (\mathbb{C}^a)^{\oplus n}\right)$

where $GL_a\mathbb{C}$ acts on $\mathbb{C}^a$ canonically. Taking the degree-d component, once again this component is 0 if d is not a multiple of a. If $a|d$, it is the direct sum of $\text{Sym}^{d_1} \mathbb{C}^a \otimes \ldots \otimes \text{Sym}^{d_n}\mathbb{C}^a$ over all $d_1 + \ldots + d_n = \frac d a$. The $\psi$ of this submodule is $h_{\lambda'} = \sum_{\mu'\vdash \frac d a} K_{\mu'\lambda'} s_{\mu'}$ where $\lambda'$ is the sequence $(d_1, \ldots, d_n).$ Hence

$\displaystyle \text{Sym}^{d_1} \mathbb{C}^a \otimes \ldots \otimes \text{Sym}^{d_n}\mathbb{C}^a \cong \bigoplus_{\mu'\vdash \frac d a} V(\mu')^{\oplus K_{\mu'\lambda'}}.$

Fix $\mu'$ and sum over all $(d_1, \ldots, d_n)$; we see that the number of copies of $V(\mu')$ is the number of SSYT with shape $\mu'$ and entries in [n]. The key observation is that each $V(\mu')$ is an $SL_a\mathbb{C}$-irrep.

• Indeed, $\mathbb{C}^* \subset GL_a\mathbb{C}$ acts as a constant scalar on the whole of $V(\mu')$ since $\psi_{V(\mu')} = s_{\mu'}$ is homogeneous. Hence any $SL_a\mathbb{C}$-invariant subspace of $V(\mu')$ is also $GL_a\mathbb{C}$-invariant.

Hence $V(\mu')^{SL_a\mathbb{C}}$ is either the whole space or 0. From the proposition here, it is the whole space if and only if $\mu' = (\frac d a, \ldots, \frac d a)$ with a terms (which corresponds to $\det^{d/a}$). Hence, the required dimension is the number of SSYT with shape $(\frac d a, \ldots)$ and entries in [n]. ♦

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