A classification of nonzero skew immaculate functions

Mason, Sarah; Xie, Tianhao

doi:10.2140/involve.2026.19.87

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A classification of nonzero skew immaculate functions

Sarah Mason and Tianhao Xie

Vol. 19 (2026), No. 1, 87–105

DOI: 10.2140/involve.2026.19.87

Abstract

We present conditions under which the skewed version of immaculate noncommutative symmetric functions are nonzero. We are motivated by the quest to determine when the matrix definition of a skew immaculate function aligns with the Hopf algebraic definition. We describe a necessary condition for a skew immaculate function to include a nonzero term, as well as a sufficient condition for there to be at least one nonzero term that survives any cancellation. We bring in several classical theorems such as the pigeonhole principle from combinatorics and Hall’s matching theorem from graph theory to prove our theorems.

Keywords

symmetric functions, NSym, immaculate functions, Jacobi–Trudi formula, Hall's matching theorem

Mathematical Subject Classification

Primary: 05E05

Secondary: 05A05

Milestones

Received: 8 March 2024

Revised: 6 August 2024

Accepted: 8 August 2024

Published: 25 January 2026

Communicated by Joshua Cooper

Authors

Sarah Mason
Department of Mathematics Wake Forest University Winston-Salem, NC United States

Tianhao Xie
Department Of Mathematics Wake Forest University Winston-Salem, NC United States

Vol. 19, No. 1, 2026