Rotation invariant webs for three row flamingo Specht modules

TL;DR

This paper constructs a new rotation-invariant web basis for certain Specht modules, enabling combinatorial understanding of symmetric group actions and deriving cyclic sieving results related to hook length formulas.

Contribution

It introduces a novel web basis for Specht modules, extending previous invariants and connecting to combinatorial skein relations and cyclic sieving phenomena.

Findings

Established a rotation-invariant web basis for specific Specht modules.

Demonstrated combinatorial interpretation of symmetric group actions via skein relations.

Derived a cyclic sieving result for a q-analog of the hook length formula.

Abstract

We introduce a new rotation-invariant web basis for a family of Specht modules $S^{(d^3, 1^{n-3d})}$, indexed by normal plabic graphs satisfying a degree condition and resembling $A_2$ webs. We show that the $\mathfrak{S}_n$ action on our basis can be understood combinatorially via a set of skein relations. From this basis, we obtain a cyclic sieving result for a $q$-analog of the hook length formula for $\lambda$. Our construction extends the jellyfish invariants of Fraser, Patrias, Pechenik, and Striker and is closely related to the weblike subgraphs of Lam.