@article{liu_smith_2021, title={Up- and Down-operators on Young's Lattice}, volume={28}, ISSN={["1077-8926"]}, DOI={10.37236/10099}, abstractNote={The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.}, number={3}, journal={ELECTRONIC JOURNAL OF COMBINATORICS}, author={Liu, Ricky Ini and Smith, Christian}, year={2021}, month={Jul} }
 @article{liu_smith_2020, title={The algebra of Schur operators}, volume={87}, ISSN={["1095-9971"]}, DOI={10.1016/j.ejc.2020.103130}, abstractNote={We study a representation of the (local) plactic monoid given by Schur operators ui, which act on partitions by adding a box in column i (if possible). In particular, we give a complete list of the relations that hold in the algebra of Schur operators.}, journal={EUROPEAN JOURNAL OF COMBINATORICS}, author={Liu, Ricky Ini and Smith, Christian}, year={2020}, month={Jun} }