The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders). In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner. In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups. Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.

HIGHER DEFORMATIONS OF LIE ALGEBRA REPRESENTATIONS II

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

A Generalization of Tychonoff's Product Theorem

A set-theoretical generalization of Tychonoff's theorem on compactness of the product of compact topological spaces is proved.