Search for: All records

Webs give a diagrammatic calculus for spaces of $$U_{q}(\mathfrak{sl}_{r})$$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $$U_{q}(\mathfrak{sl}_{4})$$-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary $$U_{q}(\mathfrak{sl}_{r})$$. Here, we show that Fraser’s basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant $$U_{q}(\mathfrak{sl}_{r})$$-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general $$U_{q}(\mathfrak{sl}_{r})$$-web bases. 

Thrall’s problem asks for the Schur decomposition of the higher Lie modules $$L_\lambda$$, which are defined using the free Lie algebra and decompose the tensor algebra as a general linear group module. Although special cases have been solved, Thrall’s problem remains open in general. We generalize Thrall’s problem to the free Lie superalgebra, and prove extensions of three known results in this setting: Brandt’s formula, Klyachko’s identification of the Schur–Weyl dual of $$L_n$$, and Kráskiewicz–Weyman’s formula for the Schur decomposition of $$L_n$$. The latter involves a new version of the major index on super tableaux, which we show corresponds to a $q,t$-hook formula of Macdonald.