Immanants and representations of the infinite symmetric group

Wildon's Weblog 2020-10-12

This is a reminder to the author of a wild question. (I won’t even call it an idea.) Given a $n \times n$ matrix $X$ and a symmetric group character $\chi$ , the latex immanant $d_\chi(X)$ is defined by

$\displaystyle \sum_{\sigma \in \mathrm{Sym}_n} \chi(\sigma) \prod_{i=1}^n X_{(i,\sigma(i))}.$

Thus when $\chi$ is the sign character the immanant is the familiar determinant, and when $\chi$ is the trivial character, it is the permanent, of interest partly because of the Permanent dominance conjecture and its starring role in Valiant’s programme to prove the algebraic analogue of the P $\not=$ NP conjecture. Very roughly stated, a more general conjecture is that all immanants, except for the determinant (or multiples of it), are hard to compute.

My wild question is: can one generalize immanants to representations of infinite symmetric groups, and does this give any extra freedom to in some sense `interpolate’ from the trivial to the sign character?