  We give a recursive construction of the categorified Young symmetrizer
introduced by Abel-Hogancamp in arXiv:1510.05330 corresponding to the
single-column partition. As a consequence, we obtain new expressions for the
uncolored $y$-ified HOMFLYPT homology of positive torus links and the $y$-ified
column-colored HOMFLYPT homology of positive torus knots. In the latter case,
we compare with the row-colored homology of positive torus knots computed by
Hogancamp-Mellit in arXiv:1909.00418, verifying the mirror symmetry conjectures
of arXiv:1112.0030 and arXiv:1304.3481 in this case.
