Branching rules of minuscule representations via a new partial order

TL;DR

The paper introduces a new partial order $\le_k$ on the set $\mathcal{A}_k(P)$ of antichains of fixed size in a poset, and analyzes its relation to the Dilworth-induced order $\le_J$. While $\le_k$ is generally coarser than $\le_J$, it yields distributive lattices for all nonempty $\mathcal{A}_k(P)$ when $P$ is a minuscule poset, linking combinatorics of antichains to representation-theoretic branching data. In the minuscule type $A$ case, $\mathcal{A}_k([a]\times[b])$ is shown to be isomorphic to a product of binomial posets $\mathcal{C}(a,k)\times\mathcal{C}(b,k)$ (and to $\mathcal{D}_k([a]\times[b])$), with explicit bijections that illuminate Durfee-diagram interpretations. The paper then extends these results to other minuscule posets and uses the $\le_k$ framework to describe branching rules of minuscule representations up to diagram automorphisms, providing a unified combinatorial method to read off decomposition data from the poset structure.

Abstract

We introduce a new partial order on the set of all antichains of a fixed size in any poset. When applied to minuscule posets, these partial orders give rise to distributive lattices that appear in the branching rules for minuscule representations of complex simple Lie algebras.

Figures (8)

• Figure 1: Minuscule posets.
• Figure 2: An ideal in the poset $[6]\times [5]\subset {\mathbb Z}_+^2$.
• Figure 3: Decomposition of the ideal $I$ from Example \ref{['eg:seq']}.
• Figure 4: Isomorphism between $P=J^3([2]\times[3])$ and $\mathcal{A}_2(P)$, with each element $\{\alpha,\beta\}\in \mathcal{A}_2(P)$ written as $\alpha\beta$.
• Figure 5: Simple Lie algebras admitting minuscule representations.

Theorems & Definitions (30)

• Definition 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Proposition 3.3