Explicit Pieri Inclusions

TL;DR

This paper introduces a new explicit, non-recursive formula for Pieri inclusions in representation theory, enabling efficient polynomial-time computation for certain partitions, improving upon previous exponential algorithms.

Contribution

It provides a novel closed-form, non-recursive description of Pieri inclusions, enhancing computational efficiency for partitions with bounded distinct parts.

Findings

New closed-form formula for Pieri inclusions

Polynomial-time algorithm for partitions with bounded parts

Improved computational efficiency over previous methods

Abstract

By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fl{\o}stad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups. In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.