A web basis of invariant polynomials from noncrossing partitions

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