Positroidal aspects of non-nesting rook placements

Abstract: Rook matroids were recently introduced by the author and Alexandersson as matroids whose bases arise from certain restricted rook placements on a skew-shaped board. They were shown to be a subclass of transversal matroids and positroids. We further investigate the structural properties of rook matroids with an emphasis on the positroidal point of view. In particular, we characterize rook matroids in terms of Grassmann necklaces of positroids, answering a question of Lam (2024). Along the way, we give a new proof of the positroidal structure of rook matroids and determine an important subclass of their cyclic flats.