Question

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions. This is motivated by some representation theory.

The naive idea is to start with the sequence of symmetric functions $s_{n,n}$ and take the Hankel determinants using the inner product (that is product in the group ring of $S(2n)$) instead of the usual outer product. However this doesn't make sense.

Take the $2 \times 2$ case. Then the naive determinant is $$ \left|\begin{array}{cc} s_{n-1,n-1} & s_{n,n} \\ s_{n,n} & s_{n+1,n+1}\end{array}\right|$$ The inner product of the two diagonal terms is defined but the inner product of the two off-diagonal terms is not.

The idea that I want to test is that this is $\sum_\lambda s_\lambda$ where the sum is over the conjugates of the partitions $4^a2^{n-2a}$.

Any suggestions on how to fix this? If this does get fixed then I would like to know how to calculate the result. The difficulty is that I have not seen an implementation of the inner product in the computer algebra systems I use, Magma and Sage (which I think both use the same source for symmetric functions).

Answer 1

It is unlikely to obtain for such a determinant the sum of all Schur functions indexed by partitions of $2n$ with four parts all even or all odd. Indeed, this sum is already equal to the inner product $s_{n,n}\ast s_{n,n}$ (see arXiv:0809.3469).

About your second question: you can compute the inner product of symmetric functions in SAGE using the "kronecker_product" command. For instance, compute the inner product of Schur functions $s_{6,2}\ast s_{5,3}$ as follows:

s=SymmetricFunctionAlgebra(QQ,basis='schur')

s([6,2]).kronecker_product(s([5,3]))

You may also use Maple with John Stembridge's package SF and the command "itensor".