Web invariants for flamingo 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.

31 pages, 8 figures. Proof of Theorem 5.5 simplified; other minor edits

References in corpus (10)

• Webs and quantum skew Howe duality
• Cyclic Sieving of Increasing Tableaux and small Schröder Paths
• Light ladders and clasp conjectures
• A web basis of invariant polynomials from noncrossing partitions
• Web bases for the general linear groups
• Tableau evacuation and webs
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• An embedding of the skein action on set partitions into the skein action on matchings
• Rotation invariant webs for three row flamingo Specht modules