Overview
This research investigates the Schur positivity of the product of chains, denoted $\mathbf m\times\mathbf n$. The study establishes specific conditions under which these products are not Schur positive, identifying a particular negative Schur coefficient and explicitly evaluating its magnitude.
Research Context
The work directly addresses a conjecture proposed by Li, Qiu, Yang, and Zhang. This conjecture posits a two-case interpretation regarding the Schur positivity of the product of chains $\mathbf m\times\mathbf n$. Previous findings are referenced as part of the broader context for this interpretation.
Approach
The research methodology involved proving conditions for the absence of Schur positivity. This included identifying a specific negative Schur coefficient. For calculation purposes, the parameter $m$ was expressed as $m=n+k$. The identified negative Schur coefficient was indexed by the partition $(2n+k-1,2n+k-3,\ldots,k+5,k,k,4)$.
Findings
- The product of chains $\mathbf m\times\mathbf n$ is proven not to be Schur positive under the conditions $n\ge4$ and $m\ge3n-1$.
- A specific negative Schur coefficient was identified and explicitly evaluated. This coefficient, indexed by $(2n+k-1,2n+k-3,\ldots,k+5,k,k,4)$ (where $m=n+k$), is equal to $(n-2)!$ multiplied by a polynomial of degree three in $k$.
- These results, when considered alongside earlier findings, support the two-case interpretation of the conjecture by Li, Qiu, Yang, and Zhang. According to this interpretation, these products are not Schur positive under two scenarios: (1) when $n=2$ and $m\ge8$, or (2) when $n\ge3$ and $m\ge n+5$.
Why This Matters
The findings contribute to the understanding of Schur positivity within combinatorial lattice theory, specifically by providing concrete proofs and evaluations for a long-standing conjecture. This work advances the formal mathematical framework concerning the properties of product lattices, refining the conditions under which certain algebraic properties, such as Schur positivity, are absent.