Web invariants for flamingo Specht modules

TL;DR

This paper extends the web basis construction for certain Specht modules, simplifying their geometric realization and broadening the class of modules for which these bases can be constructed, including new combinatorial and diagrammatic interpretations.

Contribution

It shows that web polynomials can be realized in the Grassmannian coordinate ring and extends web basis constructions from pennant to flamingo Specht modules, including new bases and spanning sets.

Findings

Web polynomials can be realized as tensor invariants in Grassmannians.

Extended web basis constructions to flamingo Specht modules.

Provided bases and spanning sets for specific Specht modules.

Abstract

Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module $S^{(d,d,1^{n-2d})}$. These polynomials were interpreted as global sections of a line bundle on a 2-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a 2-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module $S^{(d,d,1^{n-2d})}$ to more general flamingo Specht modules $S^{(d^r,1^{n-rd})}$. In the hook case $r=1$, we obtain a spanning set that can be restricted to a basis in various ways. In the case $r>2$, we obtain a basis of a well-behaved subspace of $S^{(d^r,1^{n-rd})}$, but not of the entire module.