Hankel determinants of symmetric functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:19:10Z http://mathoverflow.net/feeds/question/28362 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28362/hankel-determinants-of-symmetric-functions Hankel determinants of symmetric functions Bruce Westbury 2010-06-16T08:52:41Z 2010-06-16T11:49:30Z <p>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.</p> <p>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.</p> <p>Take the $2 \times 2$ case. Then the naive determinant is $$\left|\begin{array}{cc} s_{n-1,n-1} &amp; s_{n,n} \\ s_{n,n} &amp; s_{n+1,n+1}\end{array}\right|$$ The inner product of the two diagonal terms is defined but the inner product of the two off-diagonal terms is not.</p> <p>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 $4^a2^{n-2a}$.</p> <p>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).</p> http://mathoverflow.net/questions/28362/hankel-determinants-of-symmetric-functions/28368#28368 Answer by Emmanuel Briand for Hankel determinants of symmetric functions Emmanuel Briand 2010-06-16T09:47:50Z 2010-06-16T11:49:30Z <p>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 <a href="http://front.math.ucdavis.edu/0809.3469" rel="nofollow">arXiv:0809.3469</a>).</p> <p>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:</p> <p>s=SymmetricFunctionAlgebra(QQ,basis='schur')</p> <p>s([6,2]).kronecker_product(s([5,3]))</p> <p>You may also use Maple with John Stembridge's package SF and the command "itensor".</p>