A web basis of invariant polynomials from noncrossing partitions

Rebecca Patrias, Oliver Pechenik, Jessica Striker

Published 2021-12-10, updated 2022-07-18Version 3

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules $S^\lambda$. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases -- essentially, only the cases $\lambda=(d,d)$ and $\lambda=(d,d,d)$. Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the $2$-parameter family of Specht modules with $\lambda$ of the form $(d,d,1^\ell)$. The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.