Contravariant form for reduction algebras and Pieri rule

TL;DR

This paper investigates contravariant forms on reduction algebras and applies these to compute norms of highest weight vectors in tensor products, deriving Pieri rules from their zeros.

Contribution

It introduces methods for constructing contravariant forms on reduction algebras and uses them to explicitly compute norms related to Pieri rules.

Findings

Computed norms of highest weight vectors in tensor products.

Zeros of these norms describe Pieri rules.

Provides new insights into reduction algebra structures.

Abstract

We study properties and constructions of contravariant forms on reduction algebras. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gl(n) with a symmetric or wedge tensor power of its fundamental representation. Their zeroes describe Pieri rules.