# $\mathfrak{sl}_3$ webs without faces having a multiple of 4 sides

In settling the main conjecture of Cyclic action on Kreweras walks, see https://arxiv.org/abs/2005.14031, a rather interesting object popped up.

Recall from

Kuperberg, Greg, Spiders for rank 2 Lie algebras, Commun. Math. Phys. 180, No. 1, 109-151 (1996). ZBL0870.17005.

that an $$\mathfrak{sl}_3$$-web $$W$$ is a planar graph, embedded in a disk, with boundary vertices labeled $$1,2,\ldots,m$$ arranged on the rim of the disk in counterclockwise order, and any number of (unlabeled) internal vertices such that

• $$W$$ is trivalent: all the boundary vertices have degree one, while all the internal vertices have degree three;
• $$W$$ is bipartite: the vertices (both boundary and internal) are colored white and black, with edges only between oppositely colored vertices.

An $$\mathfrak{sl}_3$$-web is non-elliptic or irreducible if all of its internal faces have at least $$6$$ sides. Non-elliptic $$\mathfrak{sl}_3$$-webs with all boundary vertices white are in bijection with rectangular standard Young tableaux having $$3$$ rows:

Khovanov, Mikhail; Kuperberg, Greg, Web bases for $$\text{sl}(3)$$ are not dual canonical, Pac. J. Math. 188, No. 1, 129-153 (1999). ZBL0929.17012.

Tymoczko, Julianna, A simple bijection between standard $$3\times n$$ tableaux and irreducible webs for $$\mathfrak{sl}_{3}$$, J. Algebr. Comb. 35, No. 4, 611-632 (2012). ZBL1242.05277.

We call a non-elliptic $$\mathfrak{sl}_3$$-web Kreweras, if it has no faces with a multiple of $$4$$ sides.

One reason why these objects are interesting is the fact that $$\sum_W 2^{\kappa(W)}=\frac{4^n}{(n+1)(2n+1)}\binom{3n}{n},$$ where the sum is over all Kreweras webs $$W$$ with $$3n$$ boundary vertices and $$\kappa(W)$$ connected components. This is the number of Kreweras words, see http://oeis.org/A006335.

The sequence of numbers of connected Kreweras webs, with $$3,6,9,12,15,18,\dots$$ boundary vertices, is $$1,2,12,104,1088,12768,\dots$$, unknown to the OEIS. However, Lagrange inversion shows that this equals $$2^{n+1}\cdot\frac{(4n+1)!}{(3n+2)!(n+1)!}$$. Note that this is $$2^n$$ times http://oeis.org/A000260.

The sequence of numbers of connected Kreweras webs up to rotation, with $$3,6,9,12,15,18,\dots$$ boundary vertices, is $$1,1,2,10,76,714,\dots$$, also unknown to the OEIS.

Question: Can the condition, that a Kreweras web only has faces whose number of sides is not divisible by $$4$$, be given a representation theoretic meaning?

• is there a typo in the question: do you mean "... condition that a $\mathfrak{sl}_3$-web only has faces ... not divisible by 4 [and hence is a Kreweras web]" rather than "... condition that a Kreweras web ..." ? – Noam Zeilberger Jun 3 at 14:54
• (In any case, your preprint is interesting.) – Noam Zeilberger Jun 3 at 14:56
• Well, it's a language problem. But I mean precisely what you are saying. – Martin Rubey Jun 3 at 18:06
• @NoamZeilberger: we updated to include some explicit discussion of OEIS sequences you might be interested in- see what is now Section 6.2 of arxiv.org/abs/2005.14031. – Sam Hopkins Jun 11 at 11:35