Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets

In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J. Wildberger, studied certain distributive lattice models for the "Weyl bialternants" (aka "Weyl characters") associated with the rank two root systems/Weyl groups. These distributive lattices were uniformly described as lattices of order ideals taken from certain grid-like posets, although the arguments connecting the lattices to Weyl bialternants were case-by-case depending on the type of the rank two root system. Using this connection with Weyl bialternants, these lattices were shown to be rank symmetric and rank unimodal, and their rank generating functions were shown to have beautiful quotient-of-products expressions. Here, these results are re-derived from scratch using completely uniform and elementary combinatorial reasoning in conjunction with some new combinatorial methodology developed elsewhere by the second listed author.Comment: 15 page

Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner

What Ice Cube’s Song “Endangered Species” Meant for Four Generations of Black Males

Taking inspiration from Public Enemy\u27s lead vocalist Chuck D - who once declared that \u27rap is the CNN of young Black America\u27 - this volume brings together leading legal commentators to make sense of some of the most pressing law and policy issues in the context of hip-hop music and the ongoing struggle for Black equality. Written to \u27say it plain\u27, this collection will be valuable not only to students and scholars of law, African-American studies, and hip-hop, but also to everyone who cares about creating a more just society. Includes Maurer Professor Kevin Brown’s chapter, “What Ice Cube’s Endangered Species Meant for Four Generations of Black Males,” co-authored with Robert Pervine.https://www.repository.law.indiana.edu/facbooks/1300/thumbnail.jp

What Ice Cube’s Song “Endangered Species” Meant for Four Generations of Black Males

Taking inspiration from Public Enemy\u27s lead vocalist Chuck D - who once declared that \u27rap is the CNN of young Black America\u27 - this volume brings together leading legal commentators to make sense of some of the most pressing law and policy issues in the context of hip-hop music and the ongoing struggle for Black equality. Written to \u27say it plain\u27, this collection will be valuable not only to students and scholars of law, African-American studies, and hip-hop, but also to everyone who cares about creating a more just society. Includes Maurer Professor Kevin Brown’s chapter, “What Ice Cube’s Endangered Species Meant for Four Generations of Black Males,” co-authored with Robert Pervine.https://www.repository.law.indiana.edu/facbooks/1300/thumbnail.jp

Solitary and edge-minimal bases for representations of the simple lie algebra G2

AbstractWe consider two families of weight bases for “one-rowed” irreducible representations of the simple Lie algebra G2 constructed in Donnelly et al [Constructions of representations of o(2n+1,C) that imply Molev and Reiner–Stanton lattices are strongly Sperner, Discrete Math. 263 (2003) 61–79] using two corresponding families of distributive lattices (called “supporting graphs”), here denoted LG2LM(k) and LG2RS(k). We exploit the relationship between these bases and their supporting graphs to give combinatorial proofs that the bases enjoy certain uniqueness and extremal properties (the “solitary” and “edge-minimal” properties, respectively). Our application of the combinatorial technique we develop in this paper to obtain these results relies on special total orderings of the elements and edges of the lattices. We also apply this technique to another family of lattice supporting graphs to re-derive results obtained in Donnely et al. using different, more algebraic methods