Aryaman Jal: Rook Schur polynomials, matroid polytopes, and ladder determinantal rings

Abstract: We introduce the rook Schur polynomial of a skew shape and positive integer k as the "rook weight'' enumerator of k-tuples of noncrossing lattice paths inside the skew shape. We will present a smorgasbord of results concerning the combinatorial and algebraic properties of these polynomials. One leitmotif is an intriguing parallel with classical variants of Schur polynomials exemplified by a) an identity relating rook Schur polynomials to ordinary Schur polynomials in the case of rectangular shapes, b) a polytopal description of their supports in analogy with results of Postnikov and Stanley (2010), and c) determinantal identities mirroring those of Merzon and Smirnov (2016) for flagged Schur polynomials. A second theme connects rook Schur polynomials to ladder determinantal rings by providing a new description of their h-vectors. The circle of ideas involved will include rook matroids, the RSK correspondence, and weighted generalizations of the LGV lemma. This is work in progress with Saskia Solotko.