## Some Invariant Theory

We continue the previous discussion. Recall that for we have a -equivariant map

which induces an isomorphism between the unique copies of in both spaces. The kernel *Q* of this map is spanned by for various fillings *T* with shape and entries in [*n*].

E.g. suppose and ; then and the map induces:

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

Lemma. If , then the above map factors through

E.g. in our example above, the map factors through .

**Proof**

Indeed if , then swapping columns and of gives us the same . ♦

Now suppose and ; pick with length *a*. Now comprises of *m* copies of *a* so by the above lemma, we have a map:

where and . Taking the direct sum over all *m* we have:

which is a homomorphism of -representations. Furthermore, for any vector space *V* has an algebra structure via The above map clearly preserves multiplication since multiplying and both correspond to concatenation of *T* and *T’*. So it is also a homomorphism of -algebras.

Question. A basis of is given byfor Hence , the ring of polynomials in variables. What is the kernel of the induced map:

**Answer**

We have seen that for any and we have:

where we swap with various sets of *k* indices in while preserving the order to give and . Hence, *P* contains the ideal generated by all such quadratic relations.

On the other hand, any relation 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 by is a subring of , we have:

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

## Fundamental Theorems of Invariant Theory

Recall that takes , which is a left action; here , . We also let act on the right via:

so that becomes a -bimodule. A basic problem in *invariant theory* is to describe the ring comprising of all *f* such that for all .

Theorem. The ring is the image of:where runs over all .

In other words, we have:

**First Fundamental Theorem**: the ring of -invariants in is generated by

**Second Fundamental Theorem**: the relations satisfied by these polynomials are generated by the above quadratic relations.

## Proof of Fundamental Theorems

Note that *g* takes to:

which is if . Hence we have . To prove equality, we show that their dimensions in degree *d* agree. By the previous article, the degree-*d* component of has a basis indexed by SSYT of type and entries in [*n*]; if *d* is not a multiple of *a*, the component is 0.

Next we check the degree-*d* component of . As -representations, we have

where acts on canonically. Taking the degree-*d* component, once again this component is 0 if *d* is not a multiple of *a*. If , it is the direct sum of over all . The of this submodule is where is the sequence Hence

Fix and sum over all ; we see that the number of copies of is the number of SSYT with shape and entries in [*n*]. The key observation is that each is an -irrep.

- Indeed, acts as a constant scalar on the whole of since is homogeneous. Hence any -invariant subspace of is also -invariant.

Hence is either the whole space or 0. From the proposition here, it is the whole space if and only if with *a* terms (which corresponds to ). Hence, the required dimension is the number of SSYT with shape and entries in [*n*]. ♦