One Garnir to rule them all: on Specht modules and the CataLAnKe theorem

simplify a dual straightening algorithm which gives a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. show that this presentation can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our result generalizes a recent result about staircase partitions to all partitions. We then apply this result to the representation of the symmetric group on the multi-linear component of the free LAnKe with $2n-1$ generators, denoted $rho_{n,3}$. Friedmann, Hanlon, Stanley and Wachs recently proved that $rho_{n,3}$ is isomorphic to the Specht module $S^{2^{n-1}1}$ and hence has dimension given by the Catalan numbers. provide a new proof of this result which has the advantage of introducing an explicit isomorphism between the two spaces. Doing so allows us to find a basis for $rho_{n,3}$ corresponding to standard Young tableaux of shape $2^{n-1}1.$