Multi-atoms and Monotonicity of Generalized Kostka Polynomials

Abstract

There is a certain family of Poincaré polynomials that arise naturally in geometry. They satisfy a monotonicity property and admit a combinatorial description in terms of a graded poset whose elements are called Littlewood–Richardson tableaux. The purpose of this article is to give a combinatorial explanation of the monotonicity by exhibiting embeddings of these graded posets. In particular, these posets embed into the cyclage poset of Lascoux and Schützenberger, which was introduced to explain the monotonicity of the Kostka–Foulkes polynomials. The image of the embedding into the cyclage poset is characterized, giving another combinatorial description of the Poincaré polynomials in terms of the catabolizable tableaux defined by the author and J. Weyman.