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# Rotation invariant webs for three row flamingo Specht modules

Jesse Kim  Department of Mathematics
University of California, San Diego
La Jolla, CA, 92093-0112, USA<EMAIL_ADDRESS>
###### Abstract.

Fraser, Patrias, Pechenik, and Striker introduced jellyfish invariants living
within the flamingo Specht module $S^{(d^{r},1^{n-rd})}$ indexed by ordered
set partitions of $n$ with $d$ blocks of size at least $r$. They show that
jellyfish invariants corresponding to noncrossing partitions form a
rotationally invariant linearly independent set. We show that a weakening of
the noncrossing conditions extends their construction to a basis, though it
comes at the cost of rotation invariance. In the case $r=3$, we recover
rotation invariance by extending their construction to a certain set of normal
plabic graphs we call augmented webs, due to their resemblance to Kuperberg’s
$A_{2}$ webs.

##  1\. Introduction

This paper introduces a new web basis for a family of irreducible
$\mathfrak{S}_{n}$ modules indexed by partitions $(d,d,d,1^{n-3d})$. A web
basis is a basis with a number of useful properties, the systematic study of
which began with work of G. Kuperberg [7]. What exactly constitutes a web
basis differs somewhat between authors, we will use a list of properties laid
out by C. Fraser, R. Patrias, O. Pechenik, and J. Striker in [4]. The
properties they give are:

  1. (1)

Each basis element is indexed by a planar diagram with $n$ boundary vertices,
embedded in a disk.

  2. (2)

There is a topological criterion allowing identification of basis diagrams.

  3. (3)

The action of the long cycle $c=(12\dots n)$ on the basis is by rotation of
diagrams.

  4. (4)

The action of the long element $w_{0}\in n(n-1)\dots 1$ on the basis is by
reflection of diagrams.

  5. (5)

There is a finite list of ‘skein relations’ describing the action of a simple
transposition $s_{i}$ on a basis diagram.

The oldest web basis, which predates the term, is the Temperley-Lieb basis for
two-row rectangle shapes $(d,d)$, indexed by noncrossing perfect matchings of
$2d$ vertices and studied by a variety of authors [20, 22, 7, 14]. Kuperberg
introduced a basis for three-row rectangle shapes $(d,d,d)$ indexed by
bipartite trivalent planar graphs with no faces of degree less than 6, called
nonelliptic $SL_{3}$ webs as they describe the $SL_{3}$-invariant space of a
certain tensor space [7]. Rhoades gave a basis for two-row pennant shapes
$(d,d,1^{n-2d})$ indexed by noncrossing set partitions and described in terms
of skein relations [15]. The relationship between our basis and Kuperberg’s
$SL_{3}$ webs is in a sense analogous to the relationship between Rhoades’
basis and the Temperley-Lieb basis.

In [9], R. Patrias, O. Pechenik, and J. Striker introduced jellyfish
invariants, giving an alternate construction of Rhoades’ basis within the
homogeneous coordinate ring of a 2-step partial flag variety. Along with C.
Fraser, they further develop these jellyfish invariants, reinterpreting them
within the homogeneous coordinate ring of a Grassmanian [4]. They also extend
their definitions to give jellyfish invariants living within a Specht modules
indexed by a partition of shape $(d^{r},1^{n-rd})$. They dub these flamingo
Specht modules as the partitions indexing appear to stand on one leg. In the
case $r>2$, they do not give a basis for this module. Instead, they give a
linearly independent set indexed by noncrossing set partitions and a spanning
set indexed by all set partitions.

The first goal of this paper is to extend their linearly independent set to a
basis of $S^{(d^{r},1^{n-rd})}$ in the case $r>2$. We do so by replacing the
noncrossing condition with a weaker one based on ideas introduced by P.
Pylyavsky in [12], which we call $r$-weakly noncrossing. This basis fails to
have the rotation and reflection invariance desired of a web basis, however.
The second main goal of this paper is to remedy this in the case $r=3$ by
introducing a second basis indexed by a certain rotationally invariant set
$AW(n,d)$ of normal plabic graphs we call augmented webs, as they closely
resemble Kuperberg’s $SL_{3}$ webs with extra edges. This resolves a
conjecture made in [6]. Plabic graphs were first introduced by A. Postnikov
[11] in order to study the totally nonnegative Grassmanian; we use the
combinatorial machinery developed for them to show that our indexing set has
the correct enumeration. Our basis will be related to objects called weblike
subgraphs introduced by T. Lam to define $SL_{3}$ web immanants and later used
by C. Fraser, T. Lam, and I. Le to introduce a higher rank version of
Postnikov’s boundary measurement map [8, 3].

One application of our rotationally invariant basis in the case $r=3$ is that
it gives us a cyclic sieving result on the indexing set. Let $X_{n,d}(q)$
denote the $q$-analog of the hook length formula for
$\lambda=(d^{3},1^{n-3d})$,

$X_{n,d}(q):=q^{3(d-1)+\binom{n-3(d-1)}{2}}\frac{[n]!_{q}}{\prod_{(i,j)\in\lambda}[h_{ij}]_{q}}$

where $h_{ij}$ denotes the hook length of a box $(i,j)$ in the Young diagram
for $\lambda$. We show that the triple $(AW(n,d),C,X_{n,d}(q))$ exhibits the
cyclic sieving phenomenon when $n$ is odd, and a signed version of cyclic
sieving holds when $n$ is even. Specializing to the case $n=3d$ recovers a
cyclic sieving result on $SL_{3}$ webs studied by T.K. Petersen, Pylyasky, and
Rhoades in [10].

The rest of the paper is organized as follows. In Section 2, we review
necessary background information. In Section 3 we give a definition of
$r$-weakly noncrossing set partitions and give a basis of
$S^{(d^{r},1^{n-rd})}$ which extends the jellyfish invariant basis. In Section
4, we define augmented webs as a certain subset of normal plabic graphs and
give a combinatorial bijection between them and $3$-weakly noncrossing set
partitions. This bijection will draw on ideas developed by J. Tymoczko and H.
Russel to give a bijection between $SL_{3}$ webs and objects called
$m$-diagrams, a special case of our 3-weakly noncrossing set partitions [16,
21]. In Section 5, we define an $SL_{3}$-invariant polynomial attached to each
normal plabic graph. We show that this definition extends jellyfish invariants
and that the set of invariants attached to augmented webs satisfy properties
(3) and (4) of a web basis. In Section 6, we show that skein relations hold
for our plabic graph invariants. We use these skein relations to show that
augmented web invariants are indeed a basis for $S^{(d^{3},1^{n-3d})}$. In
Section 7, we show that our augmented web invariants can be interpreted in
terms of weblike subgraphs. In Section 8, we discuss the cyclic sieving result
for augmented webs which arises from our rotationally invariant basis. In
Section 9, we discuss some possible future directions for this work.

##  2\. Background

###  2.1. Specht modules

The irreducible representations of the symmetric group $\mathfrak{S}_{n}$ are
indexed by integer partitions of $n$. The irreducible indexed by a partition
$\lambda\vdash n$ is denoted $S^{\lambda}$ and is called the Specht module of
$\lambda$. We give a construction of $S^{\lambda}$ following the approach of
[4]. See [17] for a more detailed treatment of Specht modules.

Let $\lambda\vdash n$, and let $\lambda^{\prime}$ be its transpose. Consider a
matrix of $n\lambda_{1}^{\prime}$ variables,

$M=\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\\
x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\\ \vdots&&\ddots&\vdots\\\
x_{\lambda_{1}^{\prime},1}&x_{\lambda_{1}^{\prime},2}&\cdots&x_{\lambda_{1}^{\prime},n}\end{bmatrix}$

The symmetric group $\mathfrak{S}_{n}$ acts on this matrix, and thus on
$\mathbb{C}[M]$, by permuting columns of $M$. Let
$\pi=\\{\pi_{1},\pi_{2},\dots,\pi_{\lambda_{1}}$ denote a set partition of $n$
with shape $\lambda^{\prime}$. For each such set partition $\pi$, let
$p_{\pi}$ be the polynomial

$p_{\pi}=\prod_{i=1}^{\lambda_{1}}M_{[\lambda_{i}^{\prime}]}^{\pi_{i}}$

where $M_{[\lambda_{i}^{\prime}]}^{\pi_{i}}$ denotes the matrix minor of $M$
whose rows are indexed by $[\lambda_{i}^{\prime}]$ and columns are indexed by
$\pi_{i}$.

Then the Specht module $S^{\lambda}$ is the span of these polynomials as $\pi$
ranges over all set partitions of shape $\lambda$.

###  2.2. Jellyfish invariants

An ordered set partition of $n$ is a set partition with a total order on its
blocks. Two blocks $A,B$ of an ordered set partition cross if there exist
$a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B$ such that $a_{1}<b_{1}<a_{2}<b_{2}$
or $b_{1}<a_{1}<b_{2}<a_{2}$. An ordered set partition is noncrossing if no
two of its blocks cross. Let $\mathcal{OP}(n,d,r)$ denote the set of all
ordered set partitions with exactly $d$ blocks and blocks of size at least
$r$, and let $\mathcal{NCOP}(n,d,r)$ denote the set of all such partitions
which are also noncrossing.

In [4], Fraser, Pechenik, Patrias, and Striker define a polynomial living in
$S^{(d^{r},1^{n-rd})}$ attached to any ordered set partition
$\pi\in\mathcal{OP}(n,d,r)$. We include their basic definitions here and refer
the reader to [4] for examples and further exposition.

######  Definition 2.1.

Let $\pi=\\{\pi_{1},\dots,\pi_{d}\\}\in\mathcal{OP}(n,d,r)$ be an ordered set
partition. Define the set of $r$-jellyfish tableaux, $\mathcal{J}_{r}(\pi)$ to
be the set of generalized tableau $T$ with $d$ columns and $n-(d-1)r$ rows
with the following constriants:

  1. (1)

$T_{ij}\in[n]$ or $T_{ij}$ is nonempty

  2. (2)

If $i\in[r],T_{ij}$ is nonempty.

  3. (3)

If $i>r$, there exists exactly one $j$ such that $T_{ij}$ is nonempty

  4. (4)

The nonempty entries in column $j$ are exactly the elements of $\pi_{j}$ in
increasing order.

For each $T\in\mathcal{J}_{r}(\pi)$, define a polynomial

$J(T)=\prod_{i=1}^{d}M_{R_{i}(T)}^{\pi_{i}}$

where $R_{i}(T)$ is the set of rows containing an entry in $\pi_{i}$.

For each $\pi\in\mathcal{OP}(n,d,r)$, the $r$-jellyfish invariant, denoted
$[\pi]_{r}$ is

$[\pi]_{r}=\sum_{T\in\mathcal{J}_{r}(\pi)}\textrm{sign}(T)J(T)$

where $\textrm{sign}(T)$ denotes the sign of the reading word of $T$.

Fraser, Patrias, Pechenik, and Striker prove the following about $r$-jellyfish
invariants:

######  Theorem 2.2 ([4, Theorem 4.24]).

For each ordered set partition $\pi\in\mathcal{OP}(n,d,r)$, the invariant
$[\pi]_{r}$ lies in the flamingo Specht module $S^{(d^{r},1^{n-rd})}$.

######  Theorem 2.3 ([4, Proposition 5.11]).

For any ordered set partition $\pi\in\mathcal{OP}(n,d,r)$ and any permutation
$w\in\mathfrak{S}_{n}$, we have

$w\cdot[\pi]_{r}=\textrm{sign}(w)[w\cdot\pi]_{r}$

Note that this implies the span of jellyfish invariants is closed under the
action of $\mathfrak{S}_{n}$, and must therefore be equal to
$S^{(d^{r},1^{n-rd})}$.

######  Theorem 2.4 ([4, Theorem 5.13]).

For each noncrossing set partition $\gamma\in\mathcal{NC}(n,d,r)$, order the
blocks in any way to create a corresponding ordered set partition
$\pi_{\gamma}$. Then the set
$\\{[\pi_{\gamma}]_{r}:\gamma\in\mathcal{NC}(n,d,r)\\}$ is linearly
independent.

###  2.3. Noncrossing Tableaux

Noncrossing tableaux were introduced by P. Pylyavsky in [12] to give a non-
crossing counterpart to standard young tableaux. Formally, noncrossing
tableaux are set partitions; Pylyavsky chose the name noncrossing tableaux to
distinguish them from the more standard definition of noncrossing set
partitions given in the previous subsection. As we will be using noncrossing
tableaux in the context of set partitions, we will instead refer to these as
weakly noncrossing set partitions.

######  Definition 2.5.

Let $A=\\{a_{1}<a_{2}<\cdots<a_{|A|}\\}$ and
$B=\\{b_{1}<b_{2}<\cdots<b_{|B|}\\}$ be two disjoint subsets of $[n]$ with
$|A|\leq|B|$. We say $A$ and $B$ are weakly noncrossing if for all $1\leq
i\leq|A|-1$ we do not have

$a_{i}<b_{i}<a_{i+1}<b_{i+1}$

or

$b_{i}<a_{i}<b_{i+1}<a_{i+1}$

and additionally, if $|A|<|B|$, we do not have

$b_{|A|}<a_{|A|}<b_{|B|}$

A set partition $\pi$ is weakly noncrossing if blocks of $\pi$ are pairwise
weakly noncrossing.

Pylyavsky showed that weakly noncrossing set partitions of shape
$\lambda\vdash n$ are in bijection with standard Young tableaux of shape
$\lambda$.

###  2.4. Plabic graphs

Plabic graphs were introduced by Postnikov in order to study the totally
nonnegative Grassmanian. A textbook treatment can be found in [2]. We will
only need combinatorial results about plabic graphs, which we list here.

A plabic graph $G$ is a planar graph embedded in a disk, possibly with loops
and multiple edges between vertices, with interior vertices colored black and
white and boundary vertices labelled clockwise 1 through $n$. A normal plabic
graph is a plabic graph for which white vertices are degree three, boundary
vertices only connect to black vertices, and same colored vertices do not
share an edge. For this paper, we consider only normal plabic graphs and state
results only as they apply to normal plabic graphs, rather than including the
full generality.

Given a normal plabic graph $G$ , the trip at $i$ is the walk in $G$ starting
at boundary vertex $i$ which turns right at every black vertex and left at
every white vertex until it reaches the boundary at avertex we denote
$\textrm{trip}(i)$. The permutation $i\mapsto\textrm{trip}(i)$ is called the
trip permutation of $G$. The exceedances of $G$ are the exceedances of this
permutation, i.e. those trips for which $\textrm{trip}(i)>i$.

Two normal plabic graphs are normal move equivalent if one can be obtained
from the other via a seuquence of normal urban renewal moves and normal flip
moves, which we now define. The normal urban renewal move is the move shown
below, where filled in arcs represent any number of edges leading elsewhere in
the graph

$\leftrightarrow$

The normal flip move is

$\leftrightarrow$

A normal plabic graph is reduced if it is not normal move equivalent to any
plabic graph which contains a forbidden configuration, i.e. a face of degree
two or a leaf vertex not adjacent to the boundary.

A bad feature of a normal plabic graph $G$ is one of the following:

  * •

A roundtrip: A cycle in $G$ which turns left at every whtie vertex and right
at every black vertex.

  * •

An essential self-intersection: A trip in $G$ which passes through the same
edge twice.

  * •

A bad double-crossing: Two trips in $G$ which both pass through edge $e_{1}$
then edge $e_{2}$ in that order.

######  Theorem 2.6 ([2, Theorem 7.8.6]).

A normal plabic graph is reduced if and only if it does not contain any bad
features.

The more common use of this theorem is to test whether a plabic graph is
reduced or not. The plabic graphs we are interested in in, however, will be
clearly reduced as their normal move equivalence class will have size 1. We
will instead apply it to understand the structure of the trips of our plabic
graphs.

###  2.5. $SL_{3}$ Webs

$SL_{3}$ webs, or $A_{2}$ webs, were introduced by Kuperberg to study $SL_{3}$
invariant tensors and the representation theory of the quantum group
$U_{q}(\mathfrak{sl}_{3})$ [7]. The following is based on both Kuperberg’s
work as well as a nice discussion of the topic by S. Fomin and P. Pylyavsky in
[1]. A sign string of length $n$ is a string containing $n$ letters, all each
$+$ or $-$, e.g. $(++--++-)$. Given a sign string $s=s_{1}s_{2}\cdots s_{n}$,
an $SL_{3}$ web of type $s$ is a bipartite plabic graph with $n$ boundary
vertices in which every interior vertex has degree 3 and boundary vertex $i$
is adjacent to a black vertex if $s_{i}=+$ and a white vertex if $s_{i}=-$.
This is a slightly anachronistic version of the definition, as plabic graphs
were defined after $SL_{3}$ webs, but the comparison will be useful for us
later.

$SL_{3}$ webs have representation theoretic meaning. Let $V$ be the three-
dimensional defining representation of $SL_{3}$, with basis
$\\{e_{1},e_{2},e_{3}\\}$. An $SL_{3}$ web with sign string $(+++--++)$ e.g.
represents an element of the space of $SL_{3}$ invariant tensors of $V\otimes
V\otimes V\otimes V^{*}\otimes V^{*}\otimes V\otimes V$, where $V$ is the
three-dimensional defining representation of $SL_{3}$, $+$ correspond to
copies of $V$ and $-$ correspond to cpoies of $V^{*}$.

The unique $SL_{3}$ web of sign string $(+++)$

$1$$2$$3$

represents the tensor
$\sum_{\sigma\in\mathfrak{S}_{3}}\textrm{sign}(\sigma)e_{\sigma(1)}\otimes
e_{\sigma(2)}\otimes e_{\sigma(3)}$ and the unique $SL_{3}$ web of sign string
$(---)$

$1$$2$$3$

represents the tensor
$\sum_{\sigma\in\mathfrak{S}_{3}}\textrm{sign}(\sigma)e_{\sigma(1)}\otimes
e_{\sigma(2)}\otimes e_{\sigma(3)}$. Concatenation of webs represents tensor
product, and an edge between vertices represents tensor contraction.

We can also give a purely combinatorial desciption of the invariant each web
represents in terms of proper edge colorings. A proper edge coloring $\ell$ of
an $SL_{3}$ web $W$ is a labelling of the edges by the numbers $1,2,3$ such
that no label appears more than once around each vertex. For each labelling,
we get a simple basis tensor $T_{\ell}$ by taking the basis vector or dual
basis vector $e_{j}$ or $e_{j}^{*}$ (depending on th sign string) at boundary
vertex $i$ whose incident edge has label $j$, and a sign $\textrm{sign}(\ell)$
given by $(-1)^{cc(\ell)}$, where $cc(\ell)$ denotes the number of interior
vertices for which $1,2,3$ appear in counterclockwise order in the labelling
$\ell$ . The $SL_{3}$ invariant associated to $W$, which we denote
$[W]_{SL_{3}}$ is

$[W]_{SL_{3}}=\sum_{\textrm{proper labellings
}\ell}\textrm{sign}(\ell)T_{\ell}$

A web is called nonelliptic if it contains no faces of degree 4 or less. The
invariants for the set of all noneeliptic webs form a basis for the space of
$SL_{3}$ invariant tensors.

###  2.6. Cyclic sieving

The cyclic sieving phenomenon was introduced by V. Reiner, D. Stanton, and D.
White in order to unify a number of enumerative results in combinatorics [13].
See their paper and a survey by B. Sagan for further reference [18].

######  Definition 2.7.

Let $X$ be a finite set equipped with an action of the finite cyclic group
$C\cong\mathbb{Z}/n\mathbb{Z}$ with generator $c$, let $X(q)$ be a polynomial,
and let $\zeta$ be an $n^{th}$ root of unity. The triple $(X,C,X(q)$ is said
to exhibit the cyclic sieving phenomenon if $|X^{c^{d}}|=X(\zeta^{d})$ for any
integer $d>0$, where $X^{c^{d}}$ denotes the set of all elements of $X$ fixed
by $c^{d}$.

One way of obtaining cyclic sieving results is via the following, which can be
found in Sagan’s survey [18] and follows from a result of Springer [19].

######  Theorem 2.8 ([18, Theorem 8.2], [19]).

Let $W$ be a finite complex reflection group and let $C\leq W$ be cyclically
generated by a regular element $g$. Let $V$ be a $W$-module with a basis $B$
such that $gB=B$. Then the triple

$(B,C,F^{V}(q))$

exhibits the cyclic sieving phenomenon, where $F^{V}(q)$ denotes the fake
degree polynomial for $V$.

See [18] for a complete definition of the fake degree polynomial, we will only
need the following.

######  Proposition 2.9.

Let $\lambda$ be a partition of $n$ and let $S^{\lambda}$ be the corresponding
Specht module. The fake degree polynomial $F^{S^{\lambda}}(q)$ is given by

$F^{S^{\lambda}}(q)=q^{b(\lambda)}\frac{[n]!_{q}}{\prod_{(i,j)\in\lambda}[h_{ij}]_{q}}$

where $b(\lambda)=0\lambda_{1}+\lambda_{2}+2\lambda_{3}+\cdots$ and $h_{ij}$
denotes the hook length of box $(i,j)$ in the Young diagram of $\lambda$.

##  3\. Weakly-noncrossing jellyfish invariants are a basis

In this section, we extend the linearly independent set given in [4] to a
basis by introducing a weaker version of the noncrossing condition for ordered
set partitions. The weaker version is similar to the noncrossing tableau
defined by P. Pylyavsky in [12]. Our version will differ in that it will
depend on $r$.

######  Definition 3.1.

Let $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$ and
$B=\\{b_{1},b_{2},\dots,b_{|B|}\\}$ be two subsets of $[n]$ each of size $\geq
r$. We say that $A$ and $B$ are $r$-weakly noncrossing if the following holds:

  1. (1)

For each $i=1\dots,r-2$, The arc $(a_{i},a_{i+1})$ does not cross the arc
$(b_{i},b_{i+1})$.

  2. (2)

For any $j_{1},j_{2}\geq r$, the arc $(a_{r-1},a_{j_{1}})$ does not cross the
arc $(b_{r-1},b_{j_{2}})$.

An (ordered) set partition is $r$-weakly noncrossing if its blocks are
pairwise $r$-weakly noncrossing.

One can think of this definition as being noncrossing in the sense of
Pylyavsky in the first $r-2$ entries, and noncrossing in the strong sense in
the remaining entries.

We first show that the set of $r$-weakly noncrossing set partitions is the
correct size:

######  Proposition 3.2.

There is a bijection between standard young tableaux of shape
$(d^{r},1^{n-rd})$ and $r$-weakly noncrossing set partitions in $NT(n,d,r)$.

###### Proof.

We show that both sets are in bijection with a set of rectangular tableaux
filled with a subset of $[n]$:

######  Definition 3.3.

Let $T(n,d,r)$ denote the set of all tableaux of shape $\lambda=(d^{r})$
filled with with integers in $[n]$ such that

  1. (1)

Entries increase along rows and down columns.

  2. (2)

No element of $[n]$ appears more than once.

  3. (3)

For any $i$ which does not appear in the tableaux, the number of entries $j<i$
appearing in row $r-1$ strictly exceeds the number of entries $j<i$ appearing
in row $r$.

######  Example 3.4.

Consider the two tableaux below.

1&367

24811

9131416

1&367

*(lightgray)2*(lightgray)41113

*(lightgray)5*(lightgray)81416

The tableau on the left is an element of $T(16,4,3)$. The tableau on the right
is not in $T(16,4,3)$ because 9 does not appear as a filling and there are the
same number of fillings less than 9 in the second and third rows, highlighted
in gray.

The bijection between $SYT(d^{r},1^{n-rd})$ and $T(n,d,r)$ is as follows.
Given a standard young tableaux $t\in SYT(k^{r},1^{n-rd})$, if removing the
tail of blocks produces a tableaux in $T(n,d,r)$, do so. Otherwise, let $i$ be
the maximal element among that tail for which the number of elemenents $j<i$
in row $r-1$ equals the number of elements $j<i$ in row $r$. Remove the first
block of row $r$ and all blocks below it, shift all blocks in row $r$ filled
with $j<i$ one space to the left, and place a block filled with $i$ in the
newly formed opening. The maximality of $i$ will guarantee that the third
property of $T(n,d,r)$ is satisfied for elements larger than $i$, and the
shifting left will guarantee it is satisfied for elements smaller than $i$.
Call the resulting tableau $f(t)$.

To reverse this process, let $t^{\prime}$ be a tableaux in $T(n,d,r)$. If the
smallest element which does not appear in $t^{\prime}$ is larger than the
entry in the first box of row $r$, simply append all integers not already
appearing in the tableau in increasing order as the tail. Otherwise, let $i$
be the minimal filling in row $r$ which is smaller than the filling of the box
one space up and to the right, or the largest element of row $r$ if no such
element exists. Remove the box filled with $i$, shift all boxes to the left of
it one space right, and insert the remaining entries in increasing order to
form the tail. The right shift will preserve the standard yound tableau
property due to the minimality of $i$. Call the resulting tableau
$g(t^{\prime})$.

To verify that these two maps are indeed inverses, let $t\in
SYT(d^{r},1^{n-rd})$. If removing the tail of $t$ produces a tableau in
$T(n,d,r)$, then it is clear that $g(f(t))=t$. Otherwise, let $i$ be the
element inserted into row $r$ to obtain $f(t)$. Before this insertion, there
were the same number of elements less than $i$ in row $r-1$ and row $r$ of the
tableau, so the filling one space up and to the right of $i$ in $f(t)$ must be
larger than $i$. Additionally, all boxes $j$ to the left of $i$ were shifted
over, and since we started with a standard young tableau, the filling one
space up and to the right of them is smaller than $j$. Therefore, $i$ is the
filling removed by $g$, and $g(f(t))=t$.

A similar argument shows that $f\circ g$ is also the identity. Indeed, if $i$
was the element removed from row $r$ by $g$, then there must be the same
number of elements $j<i$ in row $r-1$ and $r$ of $g(t^{\prime})$, and no other
element larger than $i$ can have this propery as $t^{\prime}$ is in
$T(n,d,r)$.

######  Example 3.5.

An example of this bijection is given below for $n=16,d=4,r=3$. 12 is the
maximal filling of the tail for which the second and third rows contain the
same number of lesser fillings.

1&247

*(lightgray) 3*(lightgray) 5*(lightgray) 813

*(lightgray) 6 *(lightgray) 9*(lightgray) 1016

11

*(pink)12

14

15

$\rightarrow$ 1&247

35813

*(cyan)9*(cyan)1016

*(pink)12

$\rightarrow$ 1&247

35813

*(cyan)9*(cyan)1016

*(pink)12

$\rightarrow$ 1&247

35813

*(cyan)9*(cyan)10*(pink)1216

The bijection between $T(n,d,r)$ and $NT(n,d,r)$ is essentially repeated
applications of the standard Catalan bijection between two row rectangular
standard Young tableaux and noncrossing matchings. Given a tableau $t\in
T(n,d,r)$, for $i=1,\dots,r$, let $R_{i}(t)$ denote the entries in row $i$ of
$t$. Place the numbers 1 through $n$ in a line, and for each $i=1,\dots,r-1$,
draw $d$ arcs between elements of $R_{i}$ and $R_{i+1}$ such that

  1. (1)

Elements of $R_{i}$ are the left endpoints of arcs, and elements of $R_{i+1}$
are the right endpoints of arcs.

  2. (2)

There do not exist two arcs $(a,b)$ and $(c,d)$ such that $a<c<b<d$.

The standard Catalan bijection argument guarantees that this is uniquely
possible. Then, for each positive integer $m$ at most $n$ not appearing in
$t$, there is a unique shortest arc $(a,b)$ created at step $r-1$ such that
$a<m<b$. The third condition of $T(n,d,r)$ guarantees that such an arc exists,
and the noncrossing condition above guarantees it is unique. Draw the arc
$(a,m)$. Finally, create a set partition $\pi$ by placing all integers
connected by arcs into the same block. Then $\pi\in NT(n,d,r)$. To see that
the noncrossing condition is satisfied, note that for $i=1,\cdots,r-2$ if
$a_{i}$ and $a_{i+1}$ are the $i^{th}$ and $(i+1)^{th}$ smallest elements of a
block of $\pi$, then they must necessarily be connected by an arc created at
step $i$ in the above process.

The inverse of this bijection is simple, given a set partition $\pi\in
T(n,d,r)$ place the smallest element of each block in increasing order in row
1, the second smallest in row 2, and so on, up to row $r-1$. Finally, place
the largest element of each block in row $r$.

######  Example 3.6.

Consider the tableau shown below.

1&247

35811

9101216

We get the following arc diagram. Arcs created by matching the first two rows
are shown in red, arcs created by matching the second and third rows are shown
in green, and arcs created by connecting missing entries are shown in blue.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$

The resulting set partition is
$\\{\\{1,11,12\\},\\{2,3,13,14,15,16\\},\\{4,5,6,10\\},\\{7,8,9\\}\\}$.

∎

The second half of the proof of Proposition 3.2 also gives the following
corollary, which we will need later:

######  Corollary 3.7.

A $r$-weakly noncrossing set partition $\gamma$ is uniquely determined by the
$r-1$ sets

$\\{m\mid m\textrm{ is the }i^{th}\textrm{ smallest element of some block of
}\gamma\\}$

for $i=1,\dots,r-1$, along with the set

$\\{m\mid m\textrm{ is the largest element of some block of }\gamma\\}$

.

###### Proof.

The information in these sets determines the elements of each row of the
tableau in $T(n,d,r)$ as defined in the proof of Proposition 3.2. Placing
elements in increasing order within each row recovers the tableau, and thus
the set partition. ∎

We can now define and prove our basis.

######  Theorem 3.8.

Let $r\geq 2$. Order each weakly noncrossing set partition $\gamma\in
NT(n,d,r)$ to create an ordered set partition $\pi_{\gamma}$. Then the set
$\\{[\pi_{\gamma}]_{r}\mid\gamma\in NT(n,d,r)\\}$ is a basis for the flamingo
Specht module $S^{(d^{r},1^{n-rd})}$.

###### Proof.

By Proposition 3.2 and Theorem 2.2, it suffices to show that
$\\{[\pi_{\gamma}]_{r}\mid\gamma\in NT(n,d,r)\\}$ is linearly independent. To
do so, we introduce a monomial order and show that under this order, each
$[\pi_{\gamma}]_{r}$ has a unique leading term.

Recall that the $r$-jellyfish invariant is a polynomial in the $\nu\times n$
variables:

$\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\\
x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\\ \vdots&&\ddots&\vdots\\\
x_{\nu,1}&x_{\nu,2}&\cdots&x_{\nu,n}\end{bmatrix}$

We order these invariants in a somewhat unusual way. The indeterminate
$x_{i_{1},j_{1}}<x_{i_{2},j_{2}}$ if and only if one of the following holds:

  1. (1)

$i_{1}<i_{2}$

  2. (2)

$i_{1}=i_{2}\neq r\textrm{ and }j_{1}<j_{2}$

  3. (3)

$i_{1}=i_{2}=r\textrm{ and }j_{1}>j_{2}$

In other words, we order them in reading order except we read the $r^{th}$ row
backwards. We then take the lexicogrpahic monomial order with respect to this
ordering of variables. The unusual ordering is chosen to make use of Corollary
3.7. Without reversing the $r^{th}$ row, lexicogrpahic leading terms are not
unique.

Let the $i^{th}$ block of $\gamma$ be

$\gamma_{i}:=\\{\gamma_{i,1},\gamma_{i,2},\dots,\gamma_{i,|\gamma_{i}|}\\}$

Let $T$ be a jellyfish tableau associated to $\gamma$. Then the leading term
of $J(T)$ is straightforward to compute from the definition, we have

$\textrm{lt}(J(t))=\left(\prod_{i=1}^{d}\prod_{j=1}^{r-1}x_{j,\gamma_{i,j}}\right)\prod_{i=1}^{d}x_{r,\gamma_{i,|\gamma_{i}|}}\prod_{j=r+1}^{\nu}x_{j,u_{j}}$

where $u_{j}$ is the entry appearing in row $j$ of $J(T)$. In words, for
$i=1,\dots,r-1$, $x_{i,j}$ will appear if and only if $j$ is the $i^{th}$
smallest element of some block of $\gamma$, and $x_{r,j}$ will appear if and
only if $j$ is the largest element of some block of $\gamma$. The leading term
of $[\pi_{\gamma}]_{r}$ will be the leading term of one of these $J(T)$, and
we can see that the leading term contains the information of the sets
described in Corllary 3.7. Thus, the leading term of $[\pi_{\gamma}]_{r}$ is
unique. ∎

######  Remark 3.9.

The property of being $r$-weakly noncrossing is not preserved under rotation,
for example $\\{1,4,5\\},\\{2,3,6\\}$ is weakly 3-noncrossing, but
$\\{\\{1,3,4\\},\\{2,5,6\\}\\}$ is not. So the basis given in Theorem 3.8 is
not rotation invariant as desired of a web basis. The next section will give a
different basis which is rotation invariant in the case $r=3$.

##  4\. Rotation invariant webs for the $r=3$ case

For the rest of the paper, we specialize to the case $r=3$. We will introduce
a new basis for this case which is rotation and reflection invariant. To index
our basis, we introduce a subset normal plabic graphs which we call augmented
webs. We call them augmentd webs to allude to the fact that they are similar
to $SL_{3}$ webs, but potentially with vertices of higher degree.

######  Definition 4.1.

An augmented web is a normal plabic graph which contains no faces of degree
less than 6 and no black vertices of degree less than 3. The exceedance of an
augmented web is the number of black vertices minus the number of white
vertices. Let $A(n,d)$ denote the set of all augmented webs with $n$ boundary
vertices and exceedance $d$.

######  Remark 4.2.

The term exceedance is chosen because the exceedance of an augmented web is
also the number of exceedances in the trip permutation.

$1$$2$$3$$4$$5$$6$$7$$8$

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$

Figure 1. Examples of augmented in $A(8,2)$ and $A(10,3)$.

######  Remark 4.3.

When an augmented web has no interior white vertices, it contains exactly the
same information as a strongly noncrossing set partition, with the sets of all
boundary vertices connected to a particular interior vertex forming the
blocks.

###  4.1. Combinatorial properties of augmented webs

In this subsection, we develop combinatorial results for augmented webs. Our
first result is that all augmented webs are reduced plabic graphs.

######  Proposition 4.4.

Let $W\in A(n,d)$. Then $W$ is reduced.

###### Proof.

As no square faces or vertices of degree two are present, normal plabic graph
moves are not possible. Thus it suffices to check that $W$ itself has no
forbidden configurations and this is clear. ∎

Next, we show that augmented webs have an inductive structure we can exploit.

######  Lemma 4.5.

Let $W\in A(n,d)$, and suppose $W$ has at least one white vertex. Then for
each connected component of $W$, there exists at least two black vertices each
connected to exactly one white vertex.

###### Proof.

Let $C$ be a maximal cycle in $W$, i.e. a cycle with no edge incident to an
interior face of $W$. Since $W$ is reduced by Proposition 4.4, it contains no
round trips or essential self-intersections. Therefore there are at least two
black vertices of $W$ exterior to $C$ which connect to a white vertex in $C$.
Create a graph $G$ with two types of vertices: a vertex for every vertex of
$W$ which is on the exterior of every cycle in $W$, and a vertex for every
maximal cycle. Add an edge between a vertex $v$ and a maximal cycle $C$
whenever $v$ is adjacent to a vertex in $C$. Then $G$ is a forest, and since
$W$ had at least one white vertex, $G$ has at least one edge. So $G$ has two
leaves, and these two leaves must be black vertices connected to exactly one
white vertex. ∎

The use of Lemma 4.5 is that every augmented web with at least one white
vertex can be built out of an augmented web with one fewer white vertices in
the following way. Let $u$ and $v$ be the black vertex and its white neighbor
identified by Lemma 4.5. If we remove vertices $u$ and $v$, then connect the
neighbors of $v$ to the boundary by at least one edge each in a planar way, we
get an augmented web $W^{\prime}$.

We will need a notion of depth of a face, which we now define.

######  Definition 4.6.

Let $W\in A(n,d)$. Let $f$ be a face of $W$, and let $f_{0}$ be the face
connected to the section of boundary between 1 and $n$. The depth of $f$ is
the number of exceedances which separate $f$ from $f_{0}$.

Let $e$ be an edge of $W$. We say that $e$ is a depth boundary edge if the
depth of the faces incident to $e$ are not equal. Equivalently, $e$ is a depth
boundary edge if exactly one of the trips passing through $e$ is an
exceedance. We say $e$ is a left-to-right depth boundary edge if, when
oriented towads its black vertex endpoint, the depth of the face on the right
is higher than the depth of the face on the left. Equivalently, $e$ is a left-
to-right depth boundary if only the trip passing through $e$ from towards its
black vertex is an exceedance. We define right-to-left deth boundary edges
similarly.

######  Lemma 4.7.

Let $W\in A(n,d)$. Let $v$ be an interior vertex of $W$. Then exactly two
edges incident to $v$ are depth boundary edges.

###### Proof.

Consider the set of all trips $t_{1},\dots,t_{k}$ passing through $v$, ordered
cyclically. Since $W$ is reduced, the starts of all these trips must appear in
the same cyclic order around the boundary of $W$, since otherwise we would
introduce a bad double corssing. Similarly, the ends of these trips appear in
the same cyclic order around $W$. Therefore, the set of exceedances passing
through $v$ is a proper nonempty subset of these trips which is cyclically
consecutive around $v$. The first and last of these trips will contribute a
depth boundary edge. ∎

######  Lemma 4.8.

Let $W\in A(n,d)$. Let $u$ and $v$ be two adjacent vertices of $W$. Let
$t_{u}$ be a ny trip which passes through $u$ but does not use edge $(u,v)$,
and let $t_{v}$ be any trip which passes through $v$ but does not use edge
$(u,v)$. Then trips $t_{u}$ and $t_{v}$ do not share any vertices.

###### Proof.

This follows from Euler’s formula for planar graphs. Assume the contrary, that
trips $t_{u}$ and $t_{v}$ meet at some vertex $x$. Let $C$ be the cycle formed
from $t_{u}$, $t_{v}$ and edge $(u,v)$, and suppose it is of length $k$.
Consider the graph $G$ containing all vertices and edges of $W$ that are part
of $C$ or in its interior. Let $V_{\textrm{int}}$ denote the number of
vertices strictly in the interior of $C$, and let $\alpha$ be the average
degree of these interior vertices. Then we have

$|E(G)|\geq\frac{5}{4}k+\frac{\alpha}{2}V_{\textrm{int}}-1$

and thus by Euler’s formula the number of faces of $G$ not including the
external face is at least $\frac{1}{4}k+\frac{\alpha-2}{2}V_{\textrm{int}}$.
The total degree of these faces is

$\frac{3}{2}k-2+\frac{\alpha}{V}_{\textrm{int}}$

and thus their average degree is strictly less than 6, a contradiction. ∎

###  4.2. A bijection from tableaux to augmented webs

We can now show that augmented webs are in bijection with weakly 3-noncrossing
set partitions, $NT(n,d,3)$. To define our bijection, we first formally define
the arc diagram used in the proof of Proposition 3.2. We call these
$m$-diagrams, based on the objects of the same name developed by J. Tymoczko
in [21].

######  Definition 4.9.

Let $\pi\in NT(n,d,3)$. To form the $m$-diagram for $\pi$, place the vertices
1 through $n$ equally spaced in a line, then for each block
$B=\\{b_{1}<b_{2}<\cdots<b_{k}\\}$, draw a semicircular arc in the lower half
plane from $b_{1}$ to $b_{2}$, and from $b_{2}$ to all other elements of $B$.
We call the arc between $b_{1}$ and $b_{2}$ a first arc, and all other arcs
second arcs. Note that the definition of weakly $3$-noncrossing guarantees
that first arcs do not cross first arcs, and second arcs do not cross second
arcs. For visual clarity, we will often color first arcs in red and second
arcs in black. The name $m$-diagram is due to the fact that in Tymoczko’s
definition, blocks were always size three and thus had a unique second arc, so
the diagram appeared visually as a number of intersecting $m$ shapes.

The collection of first arcs and maximal second arcs of each block dividde the
lower half plane into a number of regions. We define the depth of each region
to be the number of first arcs and maximal second arcs which contain it.

######  Example 4.10.

Let $\pi\in NT(13,3,3)$ be the weakly 3-noncrossing set partition
$\\{1,4,6,7,8\\},\\{2,3,9,10\\},\\{5,11,12,13\\}$. The $m$-diagram associated
to $\pi$ appears below.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$

We can now define our bijection.

######  Definition 4.11.

The function $\varphi:NT(n,d,3)\rightarrow A(n,d)$ is defined as follows:

Let $\pi\in NT(n,d,3)$, and let $M$ be its $m$ diagram. For each block
$B=\\{b_{1}<b_{2}<\cdots<b_{k}\\}$, introduce a black vertex $v_{B}$ slightly
above $b_{2}$, connected to $b_{2}$ by an edge. In a small region around
$b_{2}$, modify the arcs connecting to $b_{2}$ to instead connect to $v_{B}$.
Then, for every pair of blocks, if the first arc of one crosses some of the
second arcs of the other, replace a small region containg all intersections as
shown in figure 2.

$\rightarrow$ Figure 2. The replacement operation used in the definition of
$\varphi$. The first arc is depicted in red, and the second arcs are depicted
in black.

######  Example 4.12.

Continuing our prior example of
$\pi=\\{1,4,6,7,8\\},\\{2,3,9,10\\},\\{5,11,12,13\\}$, the resulting web is
depicted below.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$

######  Proposition 4.13.

The function $\varphi$ is well defined, i.e. if $\pi\in NT(n,d,3)$, then
$\varphi(\pi)$ is indeed in $A(n,d)$

###### Proof.

We need to check that the resulting graph does not have a cycle of length 4,
the other properties are clear. A 4 cycle would necessarily have two edges
coming from first arcs and two edge coming from second arcs, such that no new
edges are created. Orienting all edges in the $m$-diagram away from $v_{B}$
for each block $B$, a 4 cycle would require that the arcs intersect with
opposite orientations at each corner. But the the two first arcs would have to
have different orientations, and this is not possible. ∎

######  Lemma 4.14.

Let $\pi\in NT(n,d,3)$. The first arcs and maximal second arcs of the $m$
diagram for $\pi$ divide the half plane into a number of regions. There is a
depth preserving correspondence between faces of $\varphi(\pi)$ and these
regions.

###### Proof.

Each replacement can be made so that edges coming from first arcs and maximal
second arcs stay the same except in a $\epsilon$ radius region around each
intersection. Each face of $\varphi(\pi)$ is thus contained (except for an
$\epsilon$-small portion) in a unique region. The trip starting at the first
arc of each $m$ will be an exceedance of $\varphi(\pi)$, which travels left to
right along first arcs, maximal second arcs, and edges introduced by
intersection replacement steps. These trips either cross at each intersection,
using the new edge introduced at that intersection twice, or turn at from
eachother at each intersection, using the new edge introduced at that
intersection 0 times. Thus, the depth boundary paths consist exactly of those
edges which come from first arcs and maximal second arcs, and thus the depth
of each face matches the depth of the region it is contained in. ∎

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$Depth 1Depth 2Depth 3

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$Depth 1Depth 2Depth 3

Figure 3. An example of the correspondence between $m$-diagram depth and
augmented web depth. Above, an $m$-diagram with non-maximal second arcs
removed and regions shaded by depth. Below, the corresponding augmented web
with first and maximal second arcs mostly preserved, and faces shaded by
depth.

######  Theorem 4.15.

The function $\varphi:NT(n,d,3)\rightarrow A(n,d)$ is a bijection.

###### Proof.

To show that $\varphi$ is invertible, we introduce the following definition.
The idea is that we will record extra information in the process of applying
$\varphi$ by way of coloring the edges. This extra information will allow us
to invert the $\varphi$ map. We will then show that the extra information was
redundant, so $\varphi$ is invertible.

######  Definition 4.16.

Given an augmented web $W\in AW(n,k)$, a valid coloring of $W$ is a coloring
of the edges of $W$ with three colors, red, blue, and black such that the
following conditions are satisfied:

  1. (1)

Every interior vertex is incident to exactly one red edge, exactly one blue
edge, and at least one black edge. Additionally, at each interior vertex, the
incident red edge shares a face with the incident blue edge.

  2. (2)

Right-to-left depth boundaries are colored red.

  3. (3)

No face has three consecutive edges colored red-blue-red.

Note that we do not require this coloring to be proper, a vertex may have
multiple black edges incident.

Given a weakly 3-noncrossing set partition $\pi\in NT(n,d,3)$, we can create a
valid coloring of $\varphi(\pi)$ by initially coloring first arcs red, second
arcs black, and for each block $b$, the connection between $v_{b}$ and $b_{2}$
blue. Then, at each replacement step, color the newly introduced edge blue and
preserve all other colors. To see that the coloring obtained is a valid
coloring, first note that initially and at each replacement step, the created
vertices satisfy property 1 of a valid coloring. Property 2 of valid colorings
holds by Lemma 4.14. To see that the third property of valid colorings holds,
note that initially no such face exists and at no replacement step could such
a face be created.

The interior blue edges contain the information of exactly which replacement
steps have been performed, so given an augmented web $\varphi(\pi)$ and the
valid coloring obtained through $\varphi$, we can recover $\pi$. Therefore, it
suffices to show that every augmented web $w$ admits exactly one valid
coloring.

To do so, we will give an algorithm for finding a valid coloring and show that
each step is forced. Let $W$ be an augmented web.

Step 1: Consider the set of all right-to-left depth boundaries. In order to
satisfy condition 2 of a valid coloring, we must color all such faces red. By
Lemma 4.7, every vertex now has exactly one red edge.

Step 2: Every interior vertex is now incident to a red edge, so all remaining
edges must be blue or black. Edges which do not share a face with the red edge
at each of their vertices must be black by condition 1 of valid colorings, so
color all such edges black.

Step 3: Consider the set of yet uncolored edges adjacent to two red edges on
the same face. By condition 3 of valid colorings, these edges must be black,
so color all such edges black.

Step 4: Consider the set of yet uncolored edges. Every interior vertex is
incident to at most two of these edges, so their union consists of a set of
disjoint paths and cycles. We claim that their union is in fact a disjoint
union of paths with exactly one interior endpoint and odd length paths with
two interior endpoints. Given this claim, there is exactly one way to satisfy
condition 1 of valid colorings by coloring some of these edges blue and the
rest black, that is, by coloring edges of a path alternating blue and black,
starting at an interior endpoint of the path. So, if the claim holds, we are
done.

Proof of Claim: Suppose towards a contradiction there is a cycle or boundary-
boundary path $P$. Let $D_{k}$ be the maximal $k$ depth boundary path sharing
a vertex with $P$. Then $D_{k}$ shares exactly one edge with $P$, as the
uncolored edges incident to $v$ cannot be on the same side of the depth
boundary path passing through $v$. But this is a contradiction as that edge
would have been colored black at step 3. Now suppose there is an even length
path whose endpoints are both interior vertices. The path must be as shown
below, though possibly longer or with the colors of vertices and trips
reversed.

For the red edges to have been colored red in step 1, the trips drawn in
orange must be exceedances, and the trip drawn in blue must not be. The two
rightmost blue trips cannot cross more than once as $W$ is reduced, so the
face between $n$ and $1$ must lie to the right of the second rightmost blue
trip. By Lemma 4.8, the leftmost orange trip and the three leftmost blue trips
cannot cross eachother, and since $W$ is reduced, the leftmost orange trip
cannot cross itself. Thus, the leftmost orange trip cannot be an exceedance,
but this is a contradiction.

Thus, every augmented web has exactly one valid coloring, and $\varphi$ is a
bijection.

∎

##  5\. $SL_{3}$ invariants for normal plabic graphs

In this section we introduce an invariant associated to each perfectly
orientable normal plabic graph in such a way that the invariants associated to
augmented webs form a basis of $S^{(d^{3},1^{n-3d})}$. To do so, we first need
to give an orientation to each plabic graph, which will determine the sign of
our invariant.

###  5.1. Perfect orientations

A key idea in defining our augmented web invariants is that of a perfect
orientation. Perfect orientations were introduced by A. Postnikov in [11]. Our
definition will be slightly different in that our sink vertices will be
interior vertices rather than boundary vertices, and we also include the
information of a total order on the sinks.

######  Definition 5.1.

Let $W$ be a normal plabic graph. A perfect orientation $\mathcal{O}$ of $W$
is a choice of two things. First, an orientation of each edge of $W$ such that
each boundary edge is oriented away from the boundary, each interior white
vertex has exactly one ingoing edge, each interior black vertex has at most
one outgoing edge. There will then be a set of $d$ black vertices with no
outward edges, we refer to these as the sinks of $\mathcal{O}$ and denote them
by $S_{\mathcal{O}}$. We also require that for every vertex $v$ in $W$ there
is a directed path from $v$ to a sink vertex. A perfect orientation also
includes the information of a total order on the sinks, i.e. a bijection
$f_{\mathcal{O}}:S_{\mathcal{O}}\rightarrow\\{1,\dots,d\\}$. For each perfect
orientation, we call the set of edges which are oriented away from black
vertices the independent set of $\mathcal{O}$ and denote it $I(\mathcal{O})$.

If there exists at least one perfect orientation of $W$, we say that $W$ is
perfectly orientable.

######  Remark 5.2.

The set of perfectly orientable plabic under our definition differs slightly
from the set of perfectly orientable plabic graphs under Postnikov’s
definition. For example, a white vertex connected by three edges to a single
black vertex which is also connected to the boundary is perfectly orientable
under Postnikov’s definition but not ours. However, every plabic graph with an
acyclic perfect orientation per Postnikov’s definition will be perfectly
orientable per our definition, and Postnikov, Speyer, and Williams show that
all reduced plabic graphs have an acyclic perfect orientation [PSW, Lemma
3.2].

We use this modified definition in order to allow for perfect orientations to
be obtained from eachother via a sequence of small changes.

######  Definition 5.3.

Let $W$ be an augmented web with perfect orientation $\mathcal{O}$. Let $v$ be
a white vertex of $W$. A swivel move is a change in orientation of exactly one
ingoing edge and one outgoing edge at $v$ which connect to distinct black
vertices. We have necessarily removed one sink vertex and added one sink
vertex, let the new sink be in the same position in the total order as the old
sink. We call this a swivel move due to the fact that the set $I(\mathcal{O})$
is being rotated around this white vertex.

######  Proposition 5.4.

Any two perfect orientations can be transformed into each other via a sequence
of swivel moves and a reordering of the sink vertices.

###### Proof.

Let $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ be two perfect orientations.
Consider the symmetric difference of independent sets of the two perfect
orientations, $I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2})$ it is necessarily
a union of disjoint cycles and paths between sinks of $\mathcal{O}_{1}$ and
sinks of $\mathcal{O}_{2}$. We induct on the number of cycles present in
$I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2})$. If there are no cycles,
performing a swivel move at each white vertex along the paths from sinks of
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ using the edges of this path will
transform $\mathcal{O}_{1}$ into a perfect orientation the same as
$\mathcal{O}_{2}$ up to a reordering of its sinks. If there is a cycle in
$I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2})$, find a walk in $W$ which starts
at a sink vertex of $\mathcal{O}_{1}$, travels to a white vertex of the cycle,
travels around the cycle, then returns to the starting sink via the same path
such that every other edge of this walk is in $I(\mathcal{O}_{1})$. Performing
a swivel move at each white vertex of this walk using the edges along this
walk will result in a perfect orientation $\mathcal{O}_{3}$ for which
$I(\mathcal{O}_{3})\Delta I(\mathcal{O}_{2})$ has one fewer cycles. ∎

We first check that this is a sensible definition of sign.

######  Proposition 5.5.

Let $\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}$ be three perfect
orientations of a web $W\in A(n,d)$. Then we have

$\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{3})=\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})\textrm{sign}(\mathcal{O}_{2},\mathcal{O}_{3})$

###### Proof.

By Proposition 5.4, it suffices to check this holds when $\mathcal{O}_{2}$ and
$\mathcal{O}_{3}$ differ by a single swivel move. Note that

$I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{3})=(I(\mathcal{O}_{1})\Delta
I(\mathcal{O}_{2}))\Delta(I(\mathcal{O}_{2})\Delta I(\mathcal{O}_{3})),$

and $(I(\mathcal{O}_{2})\Delta I(\mathcal{O}_{3}))$ is size two. We split into
three cases.

  * •

Case 1: $(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ and
$(I(\mathcal{O}_{2})\Delta I(\mathcal{O}_{3}))$ do not intersect. Then
$(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ and $(I(\mathcal{O}_{1})\Delta
I(\mathcal{O}_{2}))$ are the same except one path is two edges longer, and
$\sigma(\mathcal{O}_{1},\mathcal{O}_{2})=\sigma(\mathcal{O}_{1},\mathcal{O}_{3})$.
Thus,
$\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{3})=-\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})$

  * •

Case 2: $(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ and
$(I(\mathcal{O}_{2})\Delta I(\mathcal{O}_{3}))$ intersect in a single edge.
Then $(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ and
$(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ are the same except for two
paths. In this case, $\sigma(\mathcal{O}_{1},\mathcal{O}_{2})$ and
$\sigma(\mathcal{O}_{1},\mathcal{O}_{3})$ differ by a single transposition and
$\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{3})=-\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})$.

  * •

Case 3: $(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ and
$(I(\mathcal{O}_{2})\Delta I(\mathcal{O}_{3}))$ interesect in two edges. Then
$(I(\mathcal{O}_{1})\Delta I(\mathcal{O}_{2}))$ and $(I(\mathcal{O}_{1})\Delta
I(\mathcal{O}_{2}))$ are the same except one path is two edges shorter, and
$\sigma(\mathcal{O}_{1},\mathcal{O}_{2})=\sigma(\mathcal{O}_{1},\mathcal{O}_{3})$.
Thus,
$\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{3})=-\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})$

∎

Proposition 5.5 allows for an alternative definition of relative sign, which
we find more intuitive. The relative sign between two perfect orientations is
given by the parity of the number of swivel moves and the sign of the
permutation of the sinks in any sequence of swivel moves and a permutation of
the sinks which transforms one orientation into the other. Proposition 5.5
guarantees this is well defined. In this sense, swivel moves can be thought of
as playing a similar role to adjacent transpositions in determing the sign of
a permutation.

###  5.2. Consistent Labellings

Our invariants will be defined in terms of consistent labellings, which we now
define.

######  Definition 5.6.

Let $W$ be a perfectly orientable normal plabic graph. A consistent labelling
$\ell$ of $W$ is a choice of a possibly empty subset $\ell(e)$ of
$\\{1,\dots,\nu\\}$ for each edge $e$ of $W$, such that the following hold:

  1. (1)

At each interior white vertex, incident edge labels are disjoint and their
union is $\\{1,2,3\\}$.

  2. (2)

At each black vertex, incident edge labels are disjoint and their union
contains $\\{1,2,3\\}$.

  3. (3)

The label at each boundary edge has size 1, and for each
$i\in\\{4,\dots,\nu\\}$, $\\{i\\}$ appears exactly once among boundary labels.

The boundary word of $\ell$ is the word given by reading off the labels at
boundary edges in order, denoted

$\textrm{bd}(\ell)=\textrm{bd}(\ell)_{1}\cdots\textrm{bd}(\ell)_{n}$

The boundary monomial of $\ell$ is the monomial

$\mathbf{x}_{\textrm{bd}(\ell)}=x_{\textrm{bd}(\ell)_{1},1}\cdots
x_{\textrm{bd}(\ell)_{n},n}$

The weight of a consistent labelling is

$\textrm{wt}(\ell)=\left(-\frac{1}{2}\right)^{\\#\textrm{edge labels of size
two}}$

To each consistent labelling we also associate a sign, made up of a number of
factors. Firstly, for each $1\leq i\leq 3$, let $E_{i}$ denote the set of
edges of $W$ whose label contains $i$. Consider the symmetric difference
$E_{i}\Delta\mathcal{O}$. This will be a union of disjoint cycles and disjoint
paths from sinks of $\mathcal{O}$ to boundary vertices whose incident edge is
labelled $i$. For each boundary vertex $b$ with label $i$, let the origin of
$b$ be the sink vertex it connects to, denoted $\textrm{origin}(b)$. An origin
inversion $of$ $w$ is a pair of boundary vertices $b_{1}<b_{2}$ with
$\textrm{origin}(b_{1})>\textrm{origin}(b_{2})$. For each $i$, we get a
contribution to the sign of the labelling of

$\textrm{sign}(-1)^{\\#\textrm{origin inversions between vertices labelled
}i+\\#\textrm{cycles of length 2 modulo 4 in }E_{i}\Delta\mathcal{O}}$

We also have contributions to the sign of a consistent labelling coming from
the number of edges of $W$ with labels of size 2, the number of edges of
$I(\mathcal{O})$ with an even size label, and inversions in the boundary word
of $\ell$. The sign of $\ell$ with respect to orientation $\mathcal{O}$ is
given by

$\left(\prod_{i=1}^{3}(-1)^{\\#\textrm{cycles of length 2 modulo 4 in
}E_{i}\Delta\mathcal{O}}\right)(-1)^{\\#\textrm{origin
inversions}+\textrm{inv}(\textrm{bd}(\ell))+\\#\\{e\in
I(\mathcal{O})\mid|\ell(e)|\textrm{ is even}\\}}.$

We can also think of the sign contribution coming from origin inversions in a
different way. Let the decorated boundary word of $\ell$,
$\tilde{\textrm{bd}}(\ell)$, be the boundary word of $\ell$ with a subscript
for the origin attached to each letter $1\leq i\leq 3$. We can consider the
decorated boundary word to be a permutation under the order

$1_{1}<1_{2}<\dots<1_{d}<2_{1}<\dots<3_{d}<4<\dots<\nu$

Then the sign of $\ell$ is

$\left(\prod_{i=1}^{3}(-1)^{\\#\textrm{cycles of length 2 modulo 4 in
}E_{i}\Delta\mathcal{O}}\right)\textrm{sign}(\tilde{\textrm{bd}}(\ell))(-1)^{\\#\\{e\in
I(\mathcal{O})\mid|\ell(e)|\textrm{ is even}\\}}.$

times the sign of the decorated boundary word.

######  Remark 5.7.

The definition of weight of a labelling is a bit mysterious to us. It is
chosen to make the Skein relations upcoming in Section 6 hold, and we lack any
further explanation beyond that. The definition of sign of a labelling is
chosen so that a change in orientation introduces a consistent change in sign
among all possible consistent labellings.

######  Example 5.8.

Consider the augmented web and consistent labelling shown below. Edges in
$\mathcal{I}(O)$ are highlighted in yellow, and the three sinks are labelled
with their position in the total order on sinks.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$211$\emptyset$12$\emptyset$12$\emptyset$233142333321231

The boundary word of this labelling is $4332121312$, with 28 inversions. The
origins of the boundary vertices for each label are:

boundary vertices labelled 1: | 5 | 7 | 9
---|---|---|---
origins: | 2 | 1 | 3
boundary vertices labelled 2: | 4 | 6 | 10
---|---|---|---
origins: | 3 | 1 | 2
boundary vertices labelled 3: | 2 | 3 | 8
---|---|---|---
origins: | 2 | 3 | 1

The total number of origin inversions of $\ell$ is 5, one from label 1 and two
each from labels 2 and 3. The decorated boundary word of $\ell$ is
$43_{2}3_{3}2_{3}1_{2}2_{1}1_{1}3_{1}1_{3}2_{2}$, with 33 inversions. Our
orientation is acyclic, so there are no cycles to consider, and two of the
highlighted edges have an even size label. The sign of this labelling is
therefore

$(-1)^{33}(-1)^{2}=-1$

There are three edges with labels of size two, so the weight of this labelling
is $(-\frac{1}{2})^{3}=-\frac{1}{8}$.

######  Proposition 5.9.

Let $W\in A(n,d)$ with consistent labelling $\ell$. Let $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$ be two distinct perfect orientations for $\ell$. Then the
sign of $\ell$ with respect to these orientations is related by

$\textrm{sign}(\ell,\mathcal{O}_{1})=\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})\textrm{sign}(\ell,\mathcal{O}_{2})$

###### Proof.

It suffices to show that this holds when $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$ differ by a swivel move at a white vertex $v$. Let $u_{1}$
and $u_{2}$ be the sinks which vary between $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$, and let $u_{3}$ denote the third neighbor of $v$. The
origin of each boundary vertex is the same in $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$ except for the the boundary vertices whose origin path
travels along edge $(v,u_{3})$, which have swapped origins in
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. Either we have $|\ell(v,u_{3})|$ is
odd, in which case $|\ell(v,u_{2})|$ and $|\ell(v,u_{1})|$ are the same
parity, or $|\ell(v,u_{3})|$ is even, in which case $|\ell(v,u_{2})|$ and
$|\ell(v,u_{1})|$ have opposite parity. In either case, we have

$\textrm{sign}(\ell,\mathcal{O}_{1})=-\textrm{sign}(\ell,\mathcal{O}_{2})$

as desired. ∎

We can now define our invariants for normal plabic graphs.

######  Definition 5.10.

Let $W$ be a perfectly orientable normal plabic graph with perfect orientation
$\mathcal{O}$. Let $CL(W)$ denote the set of all consistent labellings of $W$.
Define an $SL_{3}$ invariant attached to $W$, denoted $[W,\mathcal{O}]$ by:

$[W,\mathcal{O}]=\sum_{\ell\in
CL(W)}\textrm{sign}(\ell,\mathcal{O})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}$

######  Example 5.11.

Consider the augmented web $W\in AW(7,2)$ and perfect orientation
$\mathcal{O}$ shown below.

$1$$2$$3$$4$$5$$6$$7$12

There are 288 consistent labellings in total, but only 2 up to graph
automorphism (not necessarily boundary preserving) and permutation of
$\\{1,2,3\\}$, shown below:

$1$$2$$3$$4$$5$$6$$7$213$\emptyset$123213412$1$$2$$3$$4$$5$$6$$7$214312132312

The left labelling has combined sign and weight of $-\frac{1}{2}$, and the
right labelling has combined sign and weight $1$. Let
$\textrm{Aut}(W)\subset\mathfrak{S}_{n}$ denote the group of automorphisms of
$W$ identified with the corrseponding permutation of boundary vertices, which
has size $6\cdot 2\cdot 2\cdot 2=48$. Then we have

$\displaystyle[W,\mathcal{O}]=\sum_{\sigma\in\textrm{Aut}(W)}\sum_{\omega\in
A_{3}}\textrm{sign}(\sigma)($
$\displaystyle-\frac{1}{2}x_{\omega(3),\sigma(1)}x_{\omega(2),\sigma(2)}x_{\omega(1),\sigma(3)}x_{\omega(3),\sigma(4)}x_{4,\sigma(5)}x_{\omega(1),\sigma(6)}x_{\omega(2),\sigma(7)}$
$\displaystyle+x_{4,\sigma(1)}x_{\omega(1),\sigma(2)}x_{\omega(3),\sigma(3)}x_{\omega(2),\sigma(4)}x_{\omega(3),\sigma(5)}x_{\omega(1),\sigma(6)}x_{\omega(2),\sigma(7)})$

where $A_{3}$ is the alternating group on $\\{1,2,3\\}$.

To verify that this is a sensible definition, we first check that changing the
orientation only introduces a change of sign.

######  Proposition 5.12.

Let $W$ be a perfectly orientable normal plabic graph with perfect
orientations $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. Then

$[W,\mathcal{O}_{1}]=\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})[W,\mathcal{O}_{2}]$

###### Proof.

By Proposition 5.5, we have

$\displaystyle[W,\mathcal{O}_{1}]$ $\displaystyle=\sum_{\ell\in
CL(W)}\textrm{sign}(\ell,\mathcal{O}_{1})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}$
$\displaystyle=\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})\sum_{\ell\in
CL(W)}\textrm{sign}(\ell,\mathcal{O}_{2})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}$
$\displaystyle=\textrm{sign}(\mathcal{O}_{1},\mathcal{O}_{2})[W,\mathcal{O}_{2}]$

∎

We call these invariants because the resulting polynomials will be invariant
under a certain action of $SL_{3}$. Define an action of $SL_{3}$ which acts on
the matrix of $3n$ variables

$\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\\
x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\\ x_{3,1}&x_{3,2}&\cdots&x_{3,n}\end{bmatrix}$

via left multiplication and leaves all other variables fixed. Then we have the
following:

######  Proposition 5.13.

Let $X\in SL_{3}$ and $W\in A(n,d)$ with perfect orientation $\mathcal{O}$.
Then

$X\cdot[W,\mathcal{O}]=[W,\mathcal{O}]$

This is clear for normal plabic graphs without interior white vertices and
with only vertices of degree at least 3, i.e. jellyfish invariants. We will
defer the proof for all augmented webs until Section 6, which will show that
augmented web invariants live in $\mathfrak{S}_{n}$ closure of jellyfish
invariants. The action of $\mathfrak{S}_{n}$ commutes with the action of
$SL_{3}$, so the result will follow.

We next show that this definition generalizes jellyfish invariants.

######  Proposition 5.14.

Let $W\in A(n,d)$ have no white vertices, let $\mathcal{O}$ be a perfect
orientation of $W$, and let $\pi$ be the corresponding ordered set partition.
Then $[W,\mathcal{O}]=[\pi]_{3}$.

###### Proof.

Let $v_{1},\dots,v_{d}$ denote the interior vertices of $W$ in order. We claim
that a consistent labelling of $W$ corresponds to a choice of jellyfish
tableau for $\pi$ as well as a choice of permutation for the elements of each
block of $\pi$. Indeed, we can create a jellyfish tableau $T_{\ell}$ for $\pi$
in the following manner. For each boundary vertex $1\leq b\leq n$, if $b$ has
label $i$ and is connected to interior vertex $v_{j}$, fill box $i,j$ with the
entry $b$. Then, for each interior vertex $v_{j}$, let $R_{j}(\ell)$ be the
set of boundary vertices connected to $v_{j}$. Let
$\sigma(v_{j})\in\mathfrak{S}_{n}$ denote the permutation which reorders the
elements of $R_{j}$ to have increasing labels. We claim that

(1)
$\textrm{sign}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=\textrm{sign}(J(T_{\ell}))\prod_{j=1}^{d}\textrm{sign}(\sigma(v_{j}))\left(\prod_{i\in
R_{j}}x_{\textrm{bd}(\ell)_{i},i}\right)$

The right side here represents one term in the monomial expansion of the
product of determinants defining $J(T_{\ell})$. From the definition of
$\mathbf{X}_{\textrm{bd}(\ell)}$ we see that the variables appearing on both
sides of (1) agree, so the content of this claim is that the signs match. We
show this in two parts. First, we claim that

(2) $\\#\textrm{origin inversions of }\ell=\\#\textrm{inversions within rows
of }J(T_{\ell})$

Suppose $(b_{1},b_{2})$ is an origin inversion of $\ell$, with label $i$. Then
$b_{1}$ appears in box $(i,\textrm{origin}(b_{1}))$ and $b_{2}$ appears in box
$(i,\textrm{origin}(b_{2}))$. So $(b_{1},b_{2})$ is also an origin inversion
of $T_{\ell}$. Next, we claim that

(3) $(-1)^{\textrm{inv}(\textrm{bd}(\ell))}=(-1)^{\\#\textrm{inversions
between rows of }J(T_{\ell})}\prod_{j=1}^{d}\textrm{sign}(\sigma(v_{j}))$

or equivalently,

(4)
$(-1)^{\textrm{inv}(\sigma(v_{1})\sigma(v_{2})\cdots\sigma(v_{d})\cdot\textrm{bd}(\ell))}=(-1)^{\\#\textrm{inversions
between rows of }J(T_{\ell})}.$

If the $k^{th}$ letter of
$(\sigma(v_{1})\sigma(v_{2})\cdots\sigma(v_{d})\cdot\textrm{bd}(\ell))$ is
$i$, then a $k$ appears in the $i^{th}$ row of $J(T)$, so (4) holds. We
therefore have

$[W,\mathcal{O}]=[\pi]_{3}$

as desired. ∎

Augmented web invariants satisfy the rotation and reflection invariance
properties laid out in [4], as well as something slightly stronger.

######  Proposition 5.15.

Let $W\in A(n,d)$ with perfect orientation $\mathcal{O}$ and let
$\sigma\in\mathfrak{S}_{n}$. Let $\sigma\cdot W$ be the graph obtained by
permuting the boundary vertices of $W$ according to $\sigma$, i.e. if $b$ is a
boundary vertex and $(v,b)$ is an edge of $W$, then $(v,\sigma(b))$ will be an
edge of $\sigma\cdot W$. Suppose that $\sigma\cdot W$ is planar, so it is also
in $A(n,d)$. Then, abusively letting $\mathcal{O}$ also be a perfect
orientation of $\sigma\cdot W$, we have

$\sigma\cdot[W,\mathcal{O}]=\textrm{sign}(\sigma)[\sigma\cdot W,\mathcal{O}]$

###### Proof.

For each consistent labelling $\ell$ of $W$, we get a corresponding consistent
labelling $\sigma\circ\ell$ of $\sigma\circ W$. The decorated boundary word of
$\sigma\cdot\ell$ is obtained by applying $\sigma$ to the decorated boundary
word of $\ell$, so the result follows. ∎

######  Corollary 5.16.

Let $c$ be the long cycle in $\mathfrak{S}_{n}$ and let $w_{0}$ be the long
element. Given an augmented web $W\in AW(n,d)$ with orientation
$\mathcal{O}_{W}$, let rot(W) and $\textrm{rot}(\mathcal{O}_{W})$ be the web
obtained by rotating $W$ and $\mathcal{O}_{W}$ clockwise by $\frac{2\pi}{n}$.
Let $\textrm{refl}(W)$ and $\textrm{refl}(\mathcal{O}_{W})$ be the web and
orientation obtained by reflecting $W$ and $\mathcal{O}_{w}$ across the
perpendicular bisector between boundary vertices 1 and n. We have

$c\cdot[W,\mathcal{O}]=(-1)^{n-1}[\textrm{rot}(W),\textrm{rot}(\mathcal{O}_{W})]$

and

$w_{0}\cdot[W,\mathcal{O}]=(-1)^{n-1}[\textrm{refl}(W),\textrm{refl}(\mathcal{O}_{W})]$

######  Remark 5.17.

In the definition of $\sigma\cdot W$, we are implicitly using the fact that if
a web is planar and the positions of its boundary vertices are fixed, it has a
unique planar embedding in the disk up to boundary-preserving homeomorphism.
This is due to a classical theorem of Whitney [23].

Our main theorem regarding augmented web invariants is that they form a basis
for the flamingo Specht module. We state it now but defer its proof until the
end of the next section.

######  Theorem 5.18.

Choose a perfect orientation $\mathcal{O}_{W}$ for each augmented web $W\in
A(n,d)$. Then the set $\\{[W,\mathcal{O}_{W}]\mid W\in A(n,d)\\}$ is a basis
for the flamingo Specht module $S^{(d^{3},1^{n-3d})}$.

##  6\. Skein relations for augmented webs

This section will introduce skein relations for normal plabic graph
invariants, showing that they satsify property (5) of web bases. Furthermore,
these skein relations will demonstrate that the span of augmented web
invariants is an $\mathfrak{S}_{n}$ invariant module containing
$S^{(d^{3},n-3d)}$. Along with our combinatorial bijection from standard young
tableaux, Proposition 4.15, we will thus obtain a proof of Theorem 5.18.

We first give a diagrammatic representation of these relations. In each image
below, shaded gray areas represent an unknown number of edges connecting to
other vertices of the graph, and in the perfect orientation, edges are assumed
to be oriented towards black vertices unless shown otherwise.

Crossing Reduction Rule

$i$$i+1$$x$$y$$=$$s_{i}\cdot$$i$$i+1$$x$$y$$-$$i$$i+1$$x$$y$$v$$u$$-\frac{1}{2}$$i$$i+1$$x$$y$$-\frac{1}{2}$$i$$i+1$$x$$y$

Square Reduction Rule

$=\frac{1}{2}$$+\frac{1}{2}$$+\frac{1}{2}$$+\frac{1}{2}$$-\frac{1}{4}$$-\frac{1}{4}$$-\frac{1}{4}$$-\frac{1}{4}$

Double Edge Reduction Rule

$=$

Bivalent Vertex Reduction Rule

$=-\frac{1}{2}$$-\frac{1}{2}$$-\frac{1}{2}$$+\frac{1}{2}$$-\frac{1}{4}$$-\frac{1}{4}$$-\frac{1}{4}$$-\frac{1}{4}$

Leaf Reduction Rule

$=$

Boundary-Adjacent Bivalent and Leaf Reduction Rules

$=$$=0$$=\frac{1}{2}$$+\frac{1}{2}$

######  Proposition 6.1 (Crossing reduction rule).

Let $W$ be a perfectly orientable normal plabic graph. Suppose $W$ has two
adjacent boundary vertices $i,i+1$ which connect to distinct interior vertices
$x$ and $y$ respectively. Let $W_{I}$ denote the web obtained from $W$ by
removing edges $(x,i)$ and $(y,i+1)$, and attaching an “I” shape, i.e. adding
an interior black vertex $u$ and an interior white vertex $v$, then adding in
edges $(x,v),(y,v)(v,u),(u,i)$ and $(u,i+1)$. We have the following relation:

(5) $s_{i}\cdot[W]=[W]-[W_{I}]--\frac{1}{2}[W_{x}]-\frac{1}{2}[W_{y}]$

###### Proof.

We divide consistent labellings for our webs into classes based on a fixed
choice $C$ of labels among edges in these webs other than those between
$x,y,u,v,i,$ and $i+1$, then show that within each class, equation 5 holds. Up
to symmetry, there are five possible cases, we will explain the first in
detail and give a table for the rest.

Case 1: Among the fixed labels of edges incident to $x$, $2$ and $3$ are
present. Among the fixed labels of edges incident to $y$, $1,2,$ and $3$ are
present. The missing labels around the boundary are $1$ and $4$. Then there is
exactly one way to label the remaining edges of $W$, edge $(x,i)$ must have
label 1 and edge $(y,i+1)$ must have label 4. Call this labelling $\ell$.
There are two ways to label $W_{I}$, both with weight $\frac{1}{2}$: edge
$(x,u)$ must have label 1, edge $(y,u)$ must be unlabelled, edge $(u,v)$ must
have label $\\{2,3\\}$, and edges $(v,i)$ and $(v,i+1)$ must have labels 1 and
4 in either order. Call these labellings $\ell_{I,1}$ and $\ell_{I,2}$. There
are two ways to label $W_{x}$. There are no ways to label $W_{y}$. Note that
origin inversions do not change between these labellings, and the chosen
orientation of each web is compatible with the specified labelling, so the
relative sign is given only by the relative change in the boundary word. Thus,
there exist a fixed monomial $m$ such that

$s_{i}\cdot\sum_{\begin{subarray}{c}\ell\in CL(W)\\\ \ell\textrm{ extends
}C\end{subarray}}\textrm{sign}(\mathcal{O},\mathcal{O}_{\ell})\textrm{sign}(\ell,\mathcal{O}_{\ell})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=(x_{4,i}x_{1,i+1})m$
$\sum_{\begin{subarray}{c}\ell\in CL(W)\\\ \ell\textrm{ extends
}C\end{subarray}}\textrm{sign}(\mathcal{O},\mathcal{O}_{\ell})\textrm{sign}(\ell,\mathcal{O}_{\ell})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=(x_{1,i}x_{4,i+1})m$
$\sum_{\begin{subarray}{c}\ell\in CL(W_{I})\\\ \ell\textrm{ extends
}C\end{subarray}}\textrm{sign}(\mathcal{O},\mathcal{O}_{\ell})\textrm{sign}(\ell,\mathcal{O}_{\ell})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=\frac{1}{2}(x_{1,i}x_{4,i+1}-x_{4,i}x_{1,i+1})m$
$\sum_{\begin{subarray}{c}\ell\in CL(W_{x})\\\ \ell\textrm{ extends
}C\end{subarray}}\textrm{sign}(\mathcal{O},\mathcal{O}_{\ell})\textrm{sign}(\ell,\mathcal{O}_{\ell})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=(x_{1,i}x_{4,i+1}-x_{4,i}x_{1,i+1})m$
$\sum_{\begin{subarray}{c}\ell\in CL(W_{y})\\\ \ell\textrm{ extends
}C\end{subarray}}\textrm{sign}(\mathcal{O},\mathcal{O}_{\ell})\textrm{sign}(\ell,\mathcal{O}_{\ell})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=0$

Therefore, since

$(x_{4,i}x_{1,i+1})=(x_{1,i}x_{4,i+1})-\frac{1}{2}(x_{1,i}x_{4,i+1}-x_{4,i}x_{1,i+1})-\frac{1}{2}(x_{1,i}x_{4,i+1}-x_{4,i}x_{1,i+1})-0,$

among classes of labellings which fit into this case, equation 5 holds.

The remaining cases are as follows. To read the following table, first note
that to condense information, we have replaced the monomial $x_{a,i}x_{b,i+1}$
with the word $ab$. Then let $C$ denote a fixed way to label the edges of $W$,
$W_{I}$, $W_{x}$ and $W_{y}$ other than those between $x,y,u,v,i$, and $i+1$.
The fixed labels of $C$ at $x$, $y$, and the missing labels of $C$ around the
boundary fit into one of the cases listed in the rows of this table, up to a
permutation of $\\{1,2,3\\}$ and $\\{x,y\\}$. Then there is a fixed monomial
$m$ such that for each column headed by a web, if the entry in that row and
column headed by $s_{i}\cdot W$ is $a$, then there exists a monomial $m$ such
that

$\sum_{\begin{subarray}{c}\ell\in CL(W)\\\ \ell\textrm{ extends
}C\end{subarray}}\textrm{sign}(\mathcal{O},\mathcal{O}_{\ell})\textrm{sign}(\ell,\mathcal{O}_{\ell})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=am$

The first row is Case 1.

Labels at $x$ | Labels at $y$ | Boundary | $s_{i}\cdot W$ | $W$ | $W_{I}$ | $W_{x}$ | $W_{y}$
---|---|---|---|---|---|---|---
$\\{2,3\\}$ | $\\{1,2,3\\}$ | $\\{1,4\\}$ | 41 | 14 | $\frac{1}{2}(14-41)$ | $14-41$ | 0
$\\{2,3\\}$ | $\\{2,3\\}$ | $\\{1,1\\}$ | 11 | 11 | 0 | 0 | 0
$\\{2,3\\}$ | $\\{1,3\\}$ | $\\{1,2\\}$ | 21 | 12 | $12-21$ | 0 | 0
$\\{1,2,3\\}$ | $\\{1,2,3\\}$ | $\\{4,5\\}$ | $54-45$ | $45-54$ | $45-54$ | $45-54$ | $45-54$
$\\{1\\}$ | $\\{1,2,3\\}$ | $\\{2,3\\}$ | 0 | 0 | $\frac{1}{2}(21-12)$ | $12-21$ | 0

∎

######  Proposition 6.2 (Square reduction rule).

Let $W$ be a perfectly orientable normal plabic graph. Suppose $W$ has a face
of degree 4. Let $v_{1}$, $v_{2}$ denote the white vertices of this face, and
let $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$ denote the neighbors of $v_{1},v_{2}$,
connected by edges $(v_{1},u_{1}),(v_{1},u_{2}),(v_{1},u_{4})$ and
$(v_{2},u_{2}),(v_{2},u_{3}),(v_{3},u_{4})$. Let $\pi$ be a noncrossing set
partition of $\\{1,2,3,4\\}$ in which no block contains both $u_{2}$ and
$u_{4}$. Let $W_{\pi}$ be the web obtained from $W$ by first deleting $u_{1}$
or $u_{2}$ if they connect to two vertices $v_{i},v_{j}$ whose indices lie in
the same block of $\pi$, then identifying all vertices among
$v_{1},\dots,v_{4}$ whose indices are in the same block in $\pi$. We will
write these set partitions without brackets and with vertical bars between
blocks, e.g. $W_{1|23|4}$. Then we have the following relation:

(6) $\sum_{\pi}\left(-\frac{1}{2}\right)^{4-\\#\textrm{ blocks of
}\pi}[W_{\pi}]=0$

###### Proof.

The proof is similar to that of the crossing rule, but there are many more
cases. Again, we explain one case in detail and give a table for the rest.

Let $C$ be a fixed way to label edges of the $W_{\pi}$ other than those
incident to $v_{1}$ and $v_{2}$. Then the boundary monomial of any consistent
labelling extending $C$ is fixed, so we need to check that the coefficients
satisfy equation 6. We proceed casewise, based on the fixed labels of $C$
present at each of the four black vertices $u_{1}$, $u_{2}$, $u_{3}$, and
$u_{4}$.

Case 1: The fixed labels at $v_{1}$ contain $1,2,$ and 3. The fixed labels at
$v_{2}$ contain $1$ and 2 but not $3$. The fixed labels at $v_{3}$ contain 3
but not 1 or 2. The fixed labels at $v_{4}$ do not contain 1, 2, or 3.

There are two consistent labellings of $W_{1|2|3|4}$ which extend $C$, as
shown below.

$\emptyset$$\emptyset$$3$$123$$12$$\emptyset$$123$$12$$\emptyset$$3$12$\emptyset$$3$$\emptyset$$12$$12$$3$$123$$12$$\emptyset$$3$12

Relative to eachother, the left labelling has weight and sign $-\frac{1}{2}$,
and the right labelling has weight and sign $-\frac{1}{4}$.

There is one consistent labelling of each of $W_{1|23|4}$, $W_{1|2|34}$,
$W_{14|2|3}$, and $W_{14|23}$ which extends $C$, and none for the remaining
$W_{\pi}$.

$\emptyset$$\emptyset$$123$$123$$12$$\emptyset$$3$12$\emptyset$$3$$12$$123$$12$$\emptyset$$3$12$3$$12$$\emptyset$$123$$12$$\emptyset$$3$12$123$$12$$\emptyset$$3$12

Relative to our earlier labellings, these carry weights and sign $-1$,
$-\frac{1}{2}$, $\frac{1}{2}$ and $1$, respectively. We check that these
satsify equation 6:

$0=(-\frac{1}{2}-\frac{1}{4})+(-\frac{1}{2})(-1+\frac{1}{2}-\frac{1}{2})+(\frac{1}{4})(1)$

The remaining cases our included in the following table, case 1 appears in row
4. Cases which can be obtained by a permutation of $\\{1,2,3\\}$ or a symmetry
of the square are not included.

Labels at $u_{1}$ | $u_{2}$ | $u_{3}$ | $u_{4}$ | $W_{1|2|3|4}$ | $W_{1|23|4}$ | $W_{1|2|34}$ | $W_{12|3|4}$ | $W_{14|2|3}$ | $W_{14|23}$ | $W_{12|34}$ | $W_{123|4}$ | $W_{134|2}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\{1,2,3\\}$ | $\\{1,2,3\\}$ | $\emptyset$ | $\emptyset$ | $-1$ | $-1$ | $-1$ | 0 | 1 | 1 | 0 | 0 | 1
$\\{1,2,3\\}$ | $\\{1,2\\}$ | $\\{3\\}$ | $\emptyset$ | $-\frac{3}{4}$ | $-1$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | 1 | 0 | 0 | 0
$\\{1,2,3\\}$ | $\\{1,2\\}$ | $\emptyset$ | $\\{3\\}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | 0 | 0 | 0 | 0 | 0
$\\{1,2,3\\}$ | $\\{1\\}$ | $\\{2,3\\}$ | $\emptyset$ | $-\frac{3}{4}$ | $-1$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | 1 | 0 | 0 | 0
$\\{1,2,3\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | 0 | 0 | 0 | 0 | 0
$\\{1,2\\}$ | $\\{1,2,3\\}$ | $\\{3\\}$ | $\emptyset$ | $-\frac{1}{4}$ | $0$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | 0 | 0 | 0 | 1
$\\{1,2\\}$ | $\\{1,2,3\\}$ | $\emptyset$ | $\\{3\\}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $1$ | 1 | 0 | 0 | 1
$\\{1,2\\}$ | $\\{1,2\\}$ | $\\{3\\}$ | $\\{3\\}$ | $-\frac{1}{4}$ | $-\frac{1}{2}$ | 0 | 0 | $\frac{1}{2}$ | 1 | 0 | 0 | 0
$\\{1,2\\}$ | $\\{1,3\\}$ | $\\{2,3\\}$ | $\emptyset$ | $0$ | 0 | $-1$ | 0 | 1 | 0 | 0 | 0 | 0
$\\{1,2\\}$ | $\\{1,3\\}$ | $\\{2\\}$ | $\\{3\\}$ | $-\frac{3}{4}$ | $-\frac{1}{2}$ | $-1$ | 0 | $\frac{1}{2}$ | 1 | 0 | 0 | 0
$\\{1,2\\}$ | $\\{1,3\\}$ | $\\{3\\}$ | $\\{2\\}$ | $\frac{1}{2}$ | 0 | $1$ | 0 | 0 | 0 | 0 | 0 | 0
$\\{1,2\\}$ | $\\{1\\}$ | $\\{2,3\\}$ | $\\{3\\}$ | $-\frac{3}{4}$ | $-\frac{1}{2}$ | $0$ | 0 | $-\frac{1}{2}$ | 1 | 0 | 0 | 0
$\\{1,2\\}$ | $\\{3\\}$ | $\\{1,2\\}$ | $\\{3\\}$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | 1 | -1 | 0 | 0
$\\{1,2\\}$ | $\\{3\\}$ | $\\{1,2\\}$ | $\\{3\\}$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | 1 | -1 | 0 | 0
$\\{1\\}$ | $\\{1,2,3\\}$ | $\\{1\\}$ | $\\{2,3\\}$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | 1 | -1 | 0 | 0
$\\{1\\}$ | $\\{1,2,3\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\frac{1}{4}$ | $0$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | 0 | 0 | 0 | 1
$\\{1\\}$ | $\\{1,2,3\\}$ | $\emptyset$ | $\\{2,3\\}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $1$ | 1 | 0 | 0 | 1
$\\{1\\}$ | $\\{1,2\\}$ | $\\{3\\}$ | $\\{2,3\\}$ | $\frac{1}{4}$ | $\frac{1}{2}$ | 0 | 0 | $\frac{1}{2}$ | 1 | 0 | 0 | 0
$\\{1\\}$ | $\\{2,3\\}$ | $\\{1\\}$ | $\\{2,3\\}$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | 1 | -1 | 0 | 0
$\emptyset$ | $\\{1,2,3\\}$ | $\emptyset$ | $\\{1,2,3\\}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1

∎

######  Proposition 6.3 (Double edge reduction rule).

Let $W$ be a perfectly orientable normal plabic graph with a white vertex $v$
with exactly two neighbors $u_{1}$, connected by 1 edge, and $u_{2}$,
connected by 2 edges. Let $\mathcal{O}$ be a perfect orientation such that
$I(\mathcal{O})$ contains $(v,u_{1})$. Let $W^{\prime}$ and
$\mathcal{O}^{\prime}$ be the plabic graph and orientation obtained by
contracting $v,u_{1},$ and $u_{2}$. Then

$[W,\mathcal{O}]=[W^{\prime},\mathcal{O}^{\prime}]$

###### Proof.

Fix a labelling of edges other than those incident to $v$. We have 4 cases up
to a permutation of $\\{1,2,3\\}$

  * •

Case 1: Among the fixed labels at $u_{1}$, $\\{1,2,3\\}$ appear. Then the
union of the labels of the two edges between $v$ and $u_{1}$ is $\\{1,2,3\\}$.
There are two ways to label these edges with one of size 3, and these come
with each with relative sign and weight $-1$. There are 6 ways to label the
edges with one label of size 2 and the other of size 1, each with relative
sign and weight $frac{1}{2}$, for a total weight and sign of $1$.

  * •

Case 2: Among the fixed labels at $u_{2}$, $\\{1,2\\}$ appear. Then
$(v,u_{1})$ has label $3$ and the edges between $v$ and $u_{2}$ have labels
$1$ and $2$ split between them. There are two ways to have a label of size 2,
these appear with relative sign and weight $-\frac{1}{2}$. There are two ways
to have two labels of size 1, these appear with relative sign and weight 1,
for a total weight and sign of $1$.

  * •

Case 3: Among the fixed labels at $u_{1}$, only $\\{1\\}$ appears. Then
$(v,u_{1})$ has label $\\{2,3\\}$, and one of the edges to $u_{2}$ has label
$\\{1\\}$. There are two ways to do this, each with relative sign and weight
1.

  * •

Case 4: The fixed labels at $u_{1}$ are all empty. Then there is only one way
to label the edges incident to $v$, with weight one.

In all cases $W^{\prime}$ has no choices to be made, and thus has relative
sign and weight 1, so the result holds. ∎

######  Proposition 6.4 (Leaf vertex removal).

Let $W$ be a perfectly orientable plabic graph with a black vertex $u$ of
degree 1 connected to white vertex $v$. Let $\mathcal{O}$ be a perfect
orientation wth the edge $(u,v)$ oriented towards $u$. Let $W^{\prime}$ be the
plabic graph obtained by removing vertices $u$ and $v$ and
$\mathcal{O}^{\prime}$ be the resulting perfect orientation. Then
$[W,\mathcal{O}]=[W^{\prime},\mathcal{O}^{\prime}]$.

###### Proof.

If we take a consistent labelling of $W$, and remove vertices $u$ and $v$, we
get a consistent labelling of $W^{\prime}$ with the same sign and weight, so
the result follows. ∎

######  Proposition 6.5 (Boundary adjacent leaf removal).

Let $W$ be a perfectly orientable normal plabic graph with a black vertex of
degree one or two only connected to the boundary. Then $[W]=0$.

###### Proof.

There are no consistent labellings of $W$, so the result follows. ∎

######  Proposition 6.6 (Boundary adjacent bivalent vertex removal).

Let $W$ be a perfectly orientable normal plabic graph with a degree 2 black
vertex $u$ connected to one boundary vertex and one white vertex $v$ Let the
other neighbors of $v$ be $x$ and $y$. Let $\mathcal{O}$ be a perfect
orientation of $W$ in which $(u,v)$ is oriented towards $v$. Let $W_{x}$ be
the graph obtained by removing $u$ and $v$ and connecting $x$ to the boundary,
and let $\mathcal{O}_{x}$ be the orientation obtained from $\mathcal{O}_{x}$
in the same fashion. Let $W_{y}$ and $\mathcal{O}_{y}$ be analogous. Then we
have

$[W,\mathcal{O}]=\frac{1}{2}[W_{x},\mathcal{O}_{x}]+\frac{1}{2}[W_{y},\mathcal{O}_{y}]$

###### Proof.

A consistent labelling of $W$ must have an edge label of size one on the
boundary edge incident to $u$, and edge label of size 2 on the edge $(u,v)$,
and an edge label of size one on exactly one of the edges $(u,x)$ and $(u,y)$.
Thus, consistent labellings of $W$ are in bijection with the disjoint union of
consistent labellings of $W_{x}$ and $W_{y}$, and each carries relative sign
and weight $\frac{1}{2}$. Thus

$[W,\mathcal{O}]=\frac{1}{2}[W_{x},\mathcal{O}_{x}]+\frac{1}{2}[W_{y},\mathcal{O}_{y}]$

as desired. ∎

######  Proposition 6.7 (Bivalent vertex reduction rule).

Let $W$ be a normal plabic graph with a black vertex $u$ of degree 2 connected
to two white vertices $v_{1}$ and $v_{2}$. Let $u_{1},u_{2},u_{3},u_{4}$ be
the other neighbors of $v_{1}$ and $v_{2}$. Let $\mathcal{O}$ be a perfect
orientation such that $(u_{1},v_{1})$ and $(u,v_{2})$ are in $I(\mathcal{O})$.
For $1\leq i\leq 8$, let $W_{i}$ denote the plabic graphs with orientation
$\mathcal{O}_{i}$ as shown in the bivalent vertex reduction rule above. We
have

$[W,\mathcal{O}]=-\frac{1}{2}[W_{1},\mathcal{O}_{1}]-\frac{1}{2}[W_{2},\mathcal{O}_{2}]-\frac{1}{2}[W_{3},\mathcal{O}_{3}]+\frac{1}{2}[W_{4},\mathcal{O}_{4}]-\frac{1}{4}[W_{5},\mathcal{O}_{5}]-\frac{1}{4}[W_{6},\mathcal{O}_{6}]-\frac{1}{4}[W_{7},\mathcal{O}_{7}]-\frac{1}{4}[W_{8},\mathcal{O}_{8}]$

###### Proof.

As per the proof of the square rule, we have four cases to consider as shown
in the following table

Labels at $u_{1}$ | $u_{2}$ | $u_{3}$ | $u_{4}$ | $W$ | $W_{1}$ | $W_{2}$ | $W_{3}$ | $W_{4}$ | $W_{5}$ | $W_{6}$ | $W_{7}$ | $W_{8}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\{1,2,3\\}$ | $\\{1,2,3\\}$ | $\\{1,2,3\\}$ | $\emptyset$ | 1 | 0 | $-1$ | $-1$ | 1 | 0 | 1 | 1 | 0
$\\{1,2,3\\}$ | $\\{1,2,3\\}$ | $\\{1,2\\}$ | $\\{3\\}$ | $\frac{1}{2}$ | 0 | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | 0 | 0 | 0
$\\{1,2,3\\}$ | $\\{1,2\\}$ | $\\{1,3\\}$ | $\\{2,3\\}$ | $-\frac{1}{2}$ | 0 | 0 | 0 | $-1$ | 0 | 0 | 0 | 0
$\\{1,2,3\\}$ | $\\{1,2\\}$ | $\\{1,2,3\\}$ | $\\{3\\}$ | $-\frac{1}{4}$ | $0$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | 0 | 0 | -1 | 0

∎

We can now give a proof of Theorem 5.18, which states the set
$\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)\\}$ is a basis of
$S^{(d^{3}1^{n-3d})}$.

###### Proof.

Let $W\in AW(n,d)$. Apply the crossing reduction rule to rewrite
$s_{i}\cdot[W,\mathcal{O}_{w}]$ as a sum of invariants for normal plabic
graphs $G_{i}$ with coefficients $c_{i}$, i.e.

$s_{i}\cdot[W,\mathcal{O}_{w}]=\sum_{i}c_{i}[G_{i},\mathcal{O}_{G_{i}}]$

If any of the $G_{i}$ are not augmented webs, i.e. they have a face of degree
four or a black vertex of degree less than 3, we can apply one of the other
skein relations to rewrite $[G_{i},\mathcal{O}_{G_{i}}]$. Each time we do so,
we replace a plabic graph with $k$ white vertices by a linear combination of
plabic graphs with strictly fewer than $k$ white vertices, so by iterating
this process we eventually rewrite $s_{i}\cdot[W,\mathcal{O}_{w}]$ as a linear
combination of augmented web invariants. Consequently,

$\textrm{span}(\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)\\})$

is closed under the action of $\mathfrak{S}_{n}$. By Theorem 2.2 and
Proposition 5.14, $\textrm{span}(\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)\\})$
has a nonzero intersection with $S^{(d^{3},1^{n-3d})}$. By Theorem 4.15, the
dimension of $\textrm{span}(\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)\\})$ is at
most the dimension of $S^{(d^{3},1^{n-3d})}$. Since $S^{(d^{3},1^{n-3d})}$ is
irreducible, we have

$\textrm{span}(\\{[W,\mathcal{O}_{W}]\mid W\in
AW(n,d)\\})=S^{(d^{3},1^{n-3d})}$

and Theorem 4.15 shows that $\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)\\}$ is
indeed a basis. ∎

##  7\. Augmented web invariants via weblike subgraphs

In this section we explain how to interpret our augmented web invariants in
terms of the weblike subgraphs introduced by T. Lam in [8]. Let
$V_{0}\cong\mathbb{C}^{\nu}$ be an $n$ dimensional vector space with basis
$\\{e_{1},\dots,e_{\nu}\\}$, and let
$V=\textrm{span}(\\{e_{1},e_{2},e_{3}\\})$. Recall that our augmented web
invariants live in the polynomial ring generated by the $n\times\nu$ variables

$\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\\
x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\\ \vdots&&\ddots&\vdots\\\
x_{\nu,1}&x_{\nu,2}&\cdots&x_{\nu,n}\end{bmatrix}$

Furthermore, they live in the span of monomials which contain exactly one
variable from each column, each with degree one. There is a natural
identification of the span of such monomials with $V_{0}^{\otimes n}$, given
by

$x_{i_{1},1}x_{i_{2},2}\cdots x_{i_{n},n}\mapsto e_{i_{1}}\otimes
e_{i_{2}}\otimes\cdots\otimes e_{i_{n}}$

In each augmneted web invariant, the tensor factors corresponding to basis
vectors $e_{4},\dots,e_{n}$ are alternating. We can thus replace
$\textrm{span}(\\{e_{4},\dots,e_{\nu}\\})$ with a one-dimensional vector space
which we choose to be $\bigwedge^{0}V$. Denote the resulting tensor by
$\widetilde{[W,\mathcal{O}]}$. Thus, our augmented web invariants can be
considered to live in the space

$(\bigwedge^{0}V\oplus V)^{\otimes n}$

Note that the action of $\mathfrak{S}_{n}$ is not just simply permuting tensor
factors. An adjacent transposition $s_{i}$ acts on simple basis tensors by

$s_{i}\cdot(v_{1}\otimes\cdots\otimes v_{i}\otimes v_{i+1}\otimes\cdots\otimes
v_{n})=\begin{cases}-s_{i}\cdot(v_{1}\otimes\cdots\otimes v_{i}\otimes
v_{i+1}\otimes\cdots\otimes v_{n})&v_{1},v_{i+1}\in\bigwedge^{0}V\\\
s_{i}\cdot(v_{1}\otimes\cdots\otimes v_{i}\otimes v_{i+1}\otimes\cdots\otimes
v_{n})&\textrm{otherwise}\end{cases},$

i.e. it picks up a sign if both tensor factors come from $\bigwedge^{0}V$.

$SL_{3}$ acts on $V$ via its defining representation, and this gives rise to
an action of $SL_{3}$ on $(\bigwedge^{0}V\oplus V)\oplus(\bigwedge V^{*}\oplus
V^{*})$. We then let $SL_{3}$ act diagonally on

$\sum_{k=1}^{\infty}((\bigwedge^{0}V\oplus V)\oplus(\bigwedge^{0}V^{*}\oplus
V^{*})^{\otimes k}$

If we distribute the tensor power across the direct sums, we can write this as
a large direct sum of $k$-fold tensors, and Kuperberg described the $SL_{3}$
invariant subspace of each summand in terms of clasped webs [7].

As introduced by Lam, given a normal plabic graph $W$ a 3-weblike subgraph is
an assignment of a nonnegative integer to each edge such that the sum of edges
around each interior vertex is 3. A weblike subgraph can be identified with an
$SL_{3}$ web with $0$-clasps (i.e. unused boundary vertices) by deleting each
edge assigned 0 or 3, and contracting each path of edges alternately assigned
1’s and 2’s to a single edge. A consistent labelling $\ell$ gives rise to a
weblike subgraph $W^{\prime}(\ell)$ via the sizes of its edge labels. We have
the following

######  Proposition 7.1.

Let $W$ be a normal plabic graph with perfect orientation $\mathcal{O}$. Let
$W^{\prime}$ be a 3-weblike subgraph of $W$ with $d(W^{\prime})$ edges of
multiplicity $2$. Then

$\sum_{\begin{subarray}{c}\ell\in CL(G)\\\
W^{\prime}(\ell)=W^{\prime}\end{subarray}}\textrm{sign}(\ell,\mathcal{O})\textrm{wt}(\ell)\mathbf{x}_{\textrm{bd}(\ell)}=\pm(-\frac{1}{2})^{d(W^{\prime})}[W^{\prime}]_{SL_{3}}$

where $[W^{\prime}]_{SL_{3}}$ denotes the usual $SL_{3}$ web invariant.

###### Proof.

Consistent labellings $\ell$ with $W^{\prime}(\ell)=W^{\prime}$ are in
bijection with proper edge labellings of $W^{\prime}$. So it suffices to check
that the difference in the definition of sign for consistent labellings and
proper edge labellings is the same among all such labellings. Any two proper
edge labellings of $W^{\prime}$ can be transformed into eachother via swapping
the labels of any path alternately labelled $i,j,\dots,i,j$ for $1\leq i\leq
j$. Swapping such a path will introduce a sign change both in the definition
of sign for proper edge labellings and for consistent labellings. ∎

######  Corollary 7.2.

We thus have

$\widetilde{[W,\mathcal{O}]}=\sum_{\textrm{weblike subgraphs
}W^{\prime}}\pm(\frac{1}{2})^{d(W^{\prime})}[W^{\prime}]_{SL_{3}}$

######  Remark 7.3.

Note that when a normal plabic graph is an $SL_{3}$ web, i.e. all vertices are
degree 3. Our invariants do not match the usual $SL_{3}$ invariants. Instead,
we get a sum over all $SL_{3}$ webs which appear as a weblike subgraph in $W$.

##  8\. Cyclic sieving

The basis $\\{[W,\mathcal{O}_{W}]\mid W\in A(n,d)\\}$ given in Theorem 3.8 has
all the necessary properties to obtain a cyclic sieving result via Springer’s
theorem of regular elements. The only detail left is that we need to be
careful about the orientations. To address orientation, we need the following
Lemma:

######  Lemma 8.1.

Let $W\in AW(n,d)$ with perfect orientation $\mathcal{O}$. Suppose $W$ is
fixed by rotation by $i\geq 2$, i.e. $\textrm{rot}^{i}(W)=W$. Then $i$ divides
$n$ and exactly one of the following holds:

  * •

$\frac{n}{i}\mid d$

  * •

$\frac{n}{i}\mid d-1$

  * •

$\frac{n}{i}=3\textrm{ and }\frac{n}{i}\mid d+1$

Let $k=\frac{di}{n},\frac{(d-1)i}{n},\frac{(d+1)i}{n}$ depending on which of
the cases above holds. Then we have

$\textrm{sign}(\mathcal{O},\textrm{rot}^{i}(\mathcal{O}))=(-1)^{(\frac{n}{i}-1)k}$

The relevance here is that this sign depends only on $n,d,$ and $i$, not on
$W$ itself.

###### Proof.

Note that by Lemma 5.5 it suffices to prove this for some orientation
$\mathcal{O}$. We proceed by induction on the number of interior white
vertices. If there are no interior white vertices, then a perfect orientation
is simply a total order on the black vertices. Rotation by $i$ induces a
permutation on the black vertices with at most one cycle of size 1 and cycles
of size $\frac{n}{i}$, and thus $\frac{n}{i}\mid d$ or $\frac{n}{i}\mid d-1$.
Thus,

$\textrm{sign}(\mathcal{O},\textrm{rot}^{i}(\mathcal{O}))=(-1)^{(\frac{n}{i}-1)k}$

If there is a single white vertex $v$, it is necessarily fixed by rotation by
$i$, and thus $\frac{n}{i}=3$. Rotation by $i$ induces a permutation of the
$d+1$ black vertices into cycles of size $3$, and thus $3\mid d+1$. The
orientation $\mathcal{O}_{W}$ differs from $\textrm{rot}^{i}(W)$ via a swivel
move at $v$, a transposition of the two sink vertices adjacent to $v$, and a
3-cycle applied to each other orbit of sinks. Thus,

$\textrm{sign}(\mathcal{O},\textrm{rot}^{i}(\mathcal{O}))=1=(-1)^{(\frac{n}{i}-1)k}$

If there is more than one white vertex, then by Lemma 4.5 and our rotation
invariance assumption, we can find three black vertices each connected to
exactly one interior white vertex such that these three white vertices are
distinct. Remove these 6 vertices and connect their neighbors to the boundary
in a planar and rotationally invariant way to get a web $W^{\prime}$. From any
perfect orientation $\mathcal{O}^{\prime}$ of $W^{\prime}$ we can build a
perfect orientation $\mathcal{O}$ of $W$ by orienting the removed edges from
whtie vertex to black vertex. We then have by inductive hypothesis

$\textrm{sign}(\mathcal{O},\textrm{rot}^{i}(\mathcal{O}))=\textrm{sign}(\mathcal{O^{\prime}},\textrm{rot}^{i}(\mathcal{O^{\prime}}))=(-1)^{(\frac{n}{i}-1)k}.$

So the result follows by induction. ∎

We can now state our cyclic sieving result.

######  Theorem 8.2.

Let $C=\mathbb{Z}/n\mathbb{Z}$ be the cyclic group with generator $c$ acting
on $AW(n,d)$ by rotation. Let $X_{n,d}(q)$ be the fake degree polynomial for
$S^{(}d^{3},1^{n-3d})$, i.e.

$X_{n,d}(q)=q^{3(d-1)+\binom{n-3(d-1)}{2}}\frac{[n]!_{q}}{\prod_{(i,j)\in\lambda}[h_{ij}]_{q}}$

If $n$ is odd, then the triple

$(AW(n,d),C,X_{n,d}(q))$

exhibits the cyclic sieving phenomenon.

If $n$ is even, then we have cyclic sieving up to sign, i.e.

$|AW(n,d)^{c^{i}}|=|X_{n,d}(\zeta^{i})|$

where $AW(n,d)^{c^{i}}$ dentoes the fixed point set of $AW(n,d)$ under the
action of $c^{i}$, and $\zeta$ is a primitive $n^{th}$ root of unity.

###### Proof.

If $n$ is odd, then we can choose orientations $\mathcal{O}_{W}$ for each web
$W\in AW(n,d)$ such that

$c\cdot[W,\mathcal{O}_{W}]=[\textrm{rot}(W),\mathcal{O}_{\textrm{rot}(W)}]$

To do so, select a web $W$ from each $C$-orbit and pick any orientation
$\mathcal{O}_{W}$ for it. For $1\leq i\leq n$, let
$\mathcal{O}_{\textrm{rot}^{i}(W)}=\textrm{rot}^{i}(\mathcal{O}_{W})$. Lemma
8.1 guarantees that this is possible even if $W$ has rotational symmetry, as
$(-1)^{(\frac{n}{i}-1)k}=1$.

Thus, $S^{(d^{3},1^{n-3d})}$, $\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)$ and
the rotation action of $C$ satisfy the hypotheses of Theorem 2.8 and the
result follows.

If $n$ is even, choose any orientation for each web $W\in AW(n,d)$. Then
$\\{[W,\mathcal{O}_{W}]\mid W\in AW(n,d)$ is not necessarily fixed by the
action of $c$, but $c$ will act via a signed permutation matrix. Lemma 8.1
shows that the diagonal of $c^{i}$ will either contain only $0$’s and $1$’s or
only $0$’s and $-1$’s. In either case, $|tr(c^{i})|=|AW(n,d)^{c^{i}}|$, and
the proof of Theorem 2.8 [18] shows that

$|AW(n,d)^{c^{i}}|=|X_{n,d}(\zeta^{i})|$

holds as desired. ∎

######  Example 8.3.

When $n=10$, $d=3$, then $X_{10,3}(q)$ has a rather nice form. The hook
lengths are

6&43

532

421

1

and thus $X_{10,3}(q)=q^{12}\begin{bmatrix}10\\\ 4\end{bmatrix}_{q}$. Since
$X_{10,3}(q)$ is a single $q$-binomial ($n=10,d=3$ is the only case for which
this is true), we can verify the cyclic sieving result in this case by
checking that the orbits of $AW(10,3)$ under rotation are in size-preserving
bijection with the orbits of size 4 subsets of $\\{1,\dots,10\\}$ under cyclic
permutation. There are two orbits of size 5 for each set, the orbits
containing sequences $\\{1,2,6,7\\}$ and $\\{1,3,6,8\\}$ and the orbits
containing webs shown below

The remaining 20 orbits are all size 10, one web from each is shown below

##  9\. Final Remarks

Most of the results in Section 5 as well as the crossing reduction rule in
Section 6 can be generalized to $r>3$ for $r$ odd (when $r$ is even, a
different definition treatment of signs is needed, as a change in orientation
will not give a global change in sign). We can thus obtain spanning set for an
$\mathfrak{S}_{n}$ invariant submodule containing $S^{d^{r},1^{n-rd}}$ and the
question remains as to how to prune it down to a basis, as the results of
section 4 do not seem to readily generalize. We expect this to likely be quite
difficult, as doing so would encompass constructing a rotationally invariant
basis of $S^{d^{r}}$, a question which was only recently answered in the case
$r=4$ by C. Gaetz, O Pechenik, S. Pfannerer, J. Striker, and J. Swanson [5]
and remains open for $r>4$. However, most investigation into this question has
been concerned with finding a subset of $SL_{n}$ webs which forms a basis.
Towards this end, it is perhaps a feature, rather than a bug of our
construction that it does not consist of genuine $SL_{r}$ webs, but rather
linear combinations of ones with the same underlying simple graph and its
minors, as it gives a new place to search.

If we do consider the difference between our augmented web invariants and
$SL_{3}$ web invariants to be something to be fixed, we can do so by
constructing a poset on $SL_{3}$ webs with $W\leq V$ whenever $W$ is a weblike
subgraph of $V$, which is equivalent to the graph minor poset restricted to
$SL_{3}$ webs. We can thus write

$[W]=\sum_{V\leq W}h(V,W)[V]$

for some $h$ in the incidence algebra for this poset. Inverting $h$ would give
an alternate basis of $S^{(d^{3},1^{n-3d})}$ which precisely generalizes
$SL_{3}$ webs. It may be interesting to try to understand this alternate basis
combinatorially.

##  10\. Acknowledgements

We thank Brendon Rhoades, Greg Kuperberg, Oliver Pechenik, and Pasha Pylyavksy
for helpful comments and conversations.

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