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 .
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 by
for Hence , the ring of polynomials in variables. What is the kernel of the induced map:
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]. ♦