The root posets and their rich antichains

2013/06/07 by Ringel, Claus Michael
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1306.1593

Abstract

Let Δ be a (connected) Dynkin diagram of rank n≥ 2 and Φ₊ = Φ₊(Δ) the corresponding root poset (it consists of all positive roots with respect to a fixed root basis). The width of Φ₊ is n. We will show that Φ₊ is "conical": it is the disjoint union of n solid chains. The rich antichains in Φ₊ are the antichains of cardinality n-1. It is well known that the number of rich antichains is equal to the cardinality of Φ₊. The set \mathcal R(Δ) of rich antichains in Φ₊ can itself be considered as a poset which is quite similar, but not always isomorphic, to Φ₊. We will show that there always exists a unique rich antichain A such that any rich antichain is contained in the ideal generated by A. For Δ≠ \Bbb E₆ all roots in A have the same length, namely e₂, where e₁ ≤ e₂ ≤ … ≤ e_n are the exponents of Δ. For Δ= \Bbb E₆, the antichain A consists of four roots of length e₂ = 4 and one root of length 5.

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