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.
Revised:
Accepted:
Published online:
Keywords: Web basis, tensor diagram, Specht module
Fraser, Chris 1; Patrias, Rebecca 2; Pechenik, Oliver 3; Striker, Jessica 4
CC-BY 4.0
@article{ALCO_2025__8_1_235_0,
author = {Fraser, Chris and Patrias, Rebecca and Pechenik, Oliver and Striker, Jessica},
title = {Web invariants for flamingo {Specht} modules},
journal = {Algebraic Combinatorics},
pages = {235--266},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {1},
year = {2025},
doi = {10.5802/alco.407},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.407/}
}
TY - JOUR AU - Fraser, Chris AU - Patrias, Rebecca AU - Pechenik, Oliver AU - Striker, Jessica TI - Web invariants for flamingo Specht modules JO - Algebraic Combinatorics PY - 2025 SP - 235 EP - 266 VL - 8 IS - 1 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.407/ DO - 10.5802/alco.407 LA - en ID - ALCO_2025__8_1_235_0 ER -
%0 Journal Article %A Fraser, Chris %A Patrias, Rebecca %A Pechenik, Oliver %A Striker, Jessica %T Web invariants for flamingo Specht modules %J Algebraic Combinatorics %D 2025 %P 235-266 %V 8 %N 1 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.407/ %R 10.5802/alco.407 %G en %F ALCO_2025__8_1_235_0
Fraser, Chris; Patrias, Rebecca; Pechenik, Oliver; Striker, Jessica. Web invariants for flamingo Specht modules. Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 235-266. doi : 10.5802/alco.407. https://alco.centre-mersenne.org/articles/10.5802/alco.407/
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