Pieri rule for classical groups, a new perspective

TL;DR

This paper reframes Pieri-type rules for classical groups by connecting them to Kostant's tensor product theorem and to GL branching rules. It extends Kostant's theorem to parabolic settings and proves a uniform Pieri description for classical groups. It then proves an equivalence between Pieri rules for GL and branching rules for GL(n+1) to GL(n) via Howe duality and seesaw duality, suggesting a Levi-subgroup framework underlying these phenomena. The results unify combinatorial, geometric, and representation-theoretic aspects and point toward broader connections with eigencones and Horn-type inequalities.

Abstract

We study a new perspective on a certain Pieri rules for classical groups. Furthermore, we extend a fundamental theorem of Kostant concerning tensor products for classical groups. We show that a certain form of the Pieri rule is equivalent to the converse of this extended version of Kostant's theorem. In addition, we show an equivalence between the Pieri rule and the branching rule for general linear groups.

Theorems & Definitions (30)

• Theorem 2.1: Weyl Character Formula
• Theorem 3.1
• proof
• Theorem 3.2: Pieri Rule
• Remark 3.3
• proof
• Theorem 3.4
• Corollary 3.5
• proof
• Remark 3.6