Web invariants for flamingo Specht modules

TL;DR

This work advances the diagrammatic realization of Specht modules by embedding jellyfish invariants into the Grassmannian framework, replacing the earlier 2-step flag variety setting with homogeneous coordinates of the Grassmannian $Gr(n,2n)$. It proves that the $r$-jellyfish invariants $[\pi]_r$, defined from ordered set partitions, are tensor diagram invariants and lie inside flamingo Specht modules $S^{(d^r,1^{n-rd})}$, with a concrete recurrence governing their relations. A key technical achievement is the $(2^r+1)$-term recurrence that mirrors Plücker relations, enabling diagrammatic skein relations, sign-control, and, in the hook case $r=1$, a diagrammatic basis for $S^{(d,1^{n-d})}$ based on noncrossing partitions. These results provide new tools for constructing web bases in broader flamingo shapes and illuminate the connections between combinatorial set partitions, Grassmannians, and representation theory of the symmetric group, with potential applications to cluster-like structures and diagrammatic categorifications.

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.

Figures (8)

• Figure 1: The Young diagram (left) of the flamingo partition $(6^3,1^5)$ and a flamingo (right) blending in.
• Figure 2: A visual depiction of the ordered set partition $(2~5~6\mid 3 \mid 1~4)$ considered in Section \ref{['sec:set_partitions']}. In this picture, the ordering of the blocks is not recorded. In later examples, we will occasionally use color-coding to illustrate the intended ordering, as needed.
• Figure 3: A jellyfish tableau and a cute happy jellyfish.
• Figure 4: A schematic tensor diagram $W_{\pi,r}$ illustrating the algorithm in Definition \ref{['def:tensor_diagram_from_pi']}. Vertices $1,\dots,r$ are not used. Vertices in $S=\{r+1,\ldots,\nu\}$ and $E=\{\nu+1,\ldots,n\}$ are bookkeeping vertices such that each $u_i$ connects to all vertices in $S$ and each $w_i$ connects to all vertices in $E$. The vertices $\pi_i +n$ are schematic, and they encode the set partition $\pi$ in the sense that $w_i$ is connected to all of the $|\pi_i|$ boundary vertices in $\pi_i+n$. Note these vertices will not usually be cyclically ordered as we have drawn them here. See Figure \ref{['fig:running_tensor']} for an explicit example.
• Figure 5: The tensor diagram obtained by Definition \ref{['def:tensor_diagram_from_pi']} from ordered set partition $\pi=(2\ 3\ 6\ 10 \mid 5 \ 7\ 8\ 9 \mid 1\ 4)$ from Examples \ref{['ex:RStableaux']} and \ref{['ex:cap']}. Note for example that $w_1$ connects to boundary vertices $12$, $13$, 16, and 20, which is $\pi_1+n$ in this example.

Theorems & Definitions (56)

• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Example 3.4
• Definition 3.5
• Example 3.6
• Definition 3.7
• Remark 3.8
• Example 3.9