# Webs and canonical bases in degree two

Chris Fraser  Michigan State University, Lansing, MI<EMAIL_ADDRESS>
###### Abstract.

We show that Lusztig’s canonical basis for the degree two part of the
Grassmannian coordinate ring is given by SL(k) web diagrams. Equivalently, we
show that every SL(2) web immanant of a plabic graph for Gr(k,n) is an SL(k)
web invariant.

###### Key words and phrases:

Grassmannians, webs, canonical basis, Catalan

Let $L_{n}(\lambda)$ denote the irreducible polynomial representation of ${\rm
GL}_{n}$ indexed by the weakly decreasing sequence $\lambda\in\mathbb{N}^{n}$.
Its character is the Schur polynomial $s_{\lambda}(x_{1},\dots,x_{n})$ and its
dimension is the cardinality of the set ${\rm SSYT}(\lambda,[n])$ of
semistandard Young tableaux whose shape is $\lambda$ and whose entries are
drawn from $[n]:=\\{1,\dots,n\\}$. One is interested in constructing bases of
$L_{n}(\lambda)$ with good symmetry and positivity properties and with basis
elements indexed by ${\rm SSYT}(\lambda,[n])$ in a natural way, see [12] for a
recent survey. Lusztig and Kashiwara introduced the dual canonical basis
(henceforth the “canonical” basis) as one solution to this problem. An ongoing
aim of combinatorial representation theory is to make this basis more
explicit.

Let $\omega_{k}$ denote the $k$th fundamental weight for ${\rm GL}_{n}$. In
the special case that $\lambda=d\omega_{k}$ for a positive integer $d$, one
can study the representation $L_{n}(\lambda)$ and its canonical basis via the
technology of ${\rm SL}_{k}$ web diagrams. These are planar diagrams drawn in
an $n$-gon and subject to a “$k$-valency condition.” Such a web $W$ determines
a web invariant

$[W]\in L_{n}(d\omega_{k})$

with $d$ determined by the fact that $W$ has $dk$ many boundary edges. Web
invariants span the representation $L_{n}(d\omega_{k})$. They are not linearly
independent, but all linear relations between them are a consequence of known
diagrammatical relations, the skein relations [4]. We gave an alternative
proof of this fact in [6] using plabic graphs.

The special case of the representations $L_{n}(d\omega_{k})$ is of some
interest because of the following connection with the Grassmannian ${\rm
Gr}(k,n)$ of $k$-subspaces in $\mathbb{C}^{n}$. Let $\mathbb{C}[{\rm
Gr}(k,n)]$ denote the homogeneous coordinate ring of ${\rm Gr}(k,n)$ in the
Plücker embedding and let $\mathbb{C}[{\rm Gr}(k,n)]_{(d)}$ denote the
subspace spanned by degree $d$ monomials in Plücker coordinates. The action of
${\rm GL}_{n}$ on $\mathbb{C}^{n}$ induces an action on the Grassmannian
coordinate ring and one has

$\mathbb{C}[{\rm Gr}(k,n)]_{(d)}\cong L_{n}(d\omega_{k})$

as ${\rm GL}_{n}$-representations. Thus the coordinate ring inherits a
canonical basis degree by degree.

In this realization, the canonical basis when $d=1$ is identified with the set
of Plücker coordinates on ${\rm Gr}(k,n)$. Our main result addresses the case
$d=2$:

######  Theorem A.

Each element of the dual canonical basis for $L_{n}(2\omega_{k})$ is an ${\rm
SL}_{k}$ web invariant, for any $k$ and $n$.

1 & 4
2 7
3 9
5 11
6 12
8 15
10 16
13 17
14 18
$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$622111413267$\bullet$$\bullet$$\bullet$$\bullet$765432118171615141312111098
Figure 1. A rectangular semistandard Young tableau with 9 rows and 2 columns
and its corresponding ${\rm SL}_{9}$ web diagram on 18 boundary vertices. The
main construction of this paper formulates such a recipe for rectangular
tableau with $k$ rows and $2$ columns for any $k$.

We prove our theorem constructively: we set up a multivalued tableau-to-web
map

${\rm SSYT}(2^{k},[n])\to\text{ ${\rm SL}_{k}$ web diagrams on $n$ vertices
with $2k$ boundary edges}$

which becomes well-defined once we pass from web diagrams to web invariants.
See Figure 1 for an instance of this map in the case $k=9$ which serves as our
running example.

Our tableau-to-web map passes through several intermediate Catalan-style
objects, see Figure 2 for a schematic of these intermediate steps and Figure 3
for an illustration of these intermediate objects. To produce a web diagram
from a tableau using our recipe, one first constructs a dissection of a
certain polygon and then extends this dissection to a triangulation. The
latter step involves choices but we give two different proofs that the
resulting web invariant is independent of such choice: a direct proof which
interprets the flip move on triangulations as a skein relation between web
diagrams (see Proposition 1.13) and a conceptual proof which shows that our
web diagrams satisfy a duality property which specifies their web invariants
uniquely (see Theorem 3.5).

The cluster algebra structure on $\mathbb{C}[{\rm Gr}(k,n)]$ suggests that the
notions of web invariant and canonical basis element should be intimately
related: every cluster monomial in $\mathbb{C}[{\rm Gr}(k,n)]$ is expected to
be both a web invariant [5] and a canonical basis element (see [8; 24] for
closely related results). One knows, however, that there are canonical basis
elements which are not web invariants and also that there are canonical basis
elements which are not cluster monomials (the latter happens even in degree
$d=2$).

Expressing a canonical basis element $b$ as a web invariant is a useful thing
to do. Due to the Plücker relations, there are manifold ways of expressing $b$
as a polynomial in Plücker coordinates. One can remove this ambiguity by
expressing $b$ as a $\mathbb{Z}$-linear combination of standard monomials in
Plücker coordinates, but this expression does not respect the cyclic symmetry
of the $n$-gon and can be a bit tedious to work out. By contrast, one can
“see” many properties of $b$ (e.g., its degree of cyclic symmetry, the
descents of its corresponding tableau, and so on) directly from a web diagram
whose invariant is $b$.

Previous work. There has been much previous work comparing canonical bases and
web invariants [5; 7; 14; 15; 18]. For fixed $d,k$ the complexity of the
underlying combinatorics stabilizes once $n\geq dk$. We can analyze the
problem by fixing either $k$ or $d$, letting the other parameter vary and
assuming that $n$ is in the stable range.

When $k=2$, every canonical basis element is an ${\rm SL}_{2}$ web invariant.
The underlying combinatorics is that of noncrossing matchings and the
Temperley-Lieb algebra. Once $k>2$, web invariants are linearly dependent and
it becomes an interesting problem to find a basis of web invariants with good
properties. This problem has a nice solution when $k=3$ [15] but remains
elusive when $k>3$. Our main theorem solves this problem for all $k$ in the
special case $d=2$.

The degree $d$ part of the Grassmannian coordinate ring when $d$ is either 2
or 3 has been actively studied, mostly through the lens of deciding which
canonical basis elements are cluster variables [1; 2; 3; 19]. Our approach
complements these papers by studying the whole basis (when $d=2$), not only
the cluster variables.

Cyclic symmetry. The canonical basis of $L_{n}(d\omega_{k})$ was used to
establish the cyclic sieving phenomenon for the action of promotion on ${\rm
SSYT}(d^{k},[n])$ [22]. The crucial property is that the canonical basis is a
weight basis whose elements are permuted by the automorphism of
$\mathbb{C}[{\rm Gr}(k,n)]$ induced by rotation of the $n$-gon. This action on
basis elements implements promotion on tableaux. It remains an interesting
open problem to find a basis of web invariants with the same properties.
Again, this problem is solved when $k=2,3$ and now holds when $d=2$ using our
Theorem A. We hope our construction might be a useful input towards solving
the problem when $d=3$, and perhaps for all $d$.

SYTs$>$$>$web diagrams$>$$>$web invariantsdissectionstriangulations Figure 2.
The tableau-to-web map sends a standard tableau $T$ to a weighted dissection
$\mathfrak{d}_{T}$ of a certain polygon. One makes a choice of triangulation
$\mathfrak{t}_{T}$ extending $\mathfrak{d}_{T}$ which determines a web diagram
$W(\mathfrak{t}_{T})$. The vertical maps are reversible but the dashed
horizontal maps are multivalued. Nonetheless the invariant
$[W(\mathfrak{t}_{T})]$ is well-defined.

Web immanants. The well-studied relationship between ${\rm Gr}(k,n)$ and
plabic graphs [21] yields a different perspective on the canonical basis for
$L_{n}(2\omega_{k})$. In this viewpoint, the canonical basis for
$L_{n}(2\omega_{k})$ coincides with the set of ${\rm SL}_{2}$ web immanants
[16; 17]. The web immanant is constructed somewhat indirectly using the double
dimer model on a plabic network $N$. Roughly speaking, web immanants are
indexed by noncrossing matchings $m$, and the web immanant $F_{m}$ is defined
as the generating function for double dimer covers of $N$ which match boundary
vertices according to $m$. It is far from obvious that the “${\rm
SL}_{2}$-style” combinatorics of noncrossing matchings should have any
relationship with the combinatorics of ${\rm SL}_{k}$ web diagrams.
Nonetheless, since ${\rm SL}_{2}$ web immanants and degree-two canonical basis
elements coincide, Theorem A can be reformulated as follows:

######  Theorem B.

Every SL2 web immanant for Gr(k,n) is an SLk web invariant, for any $k,n$.

This is how we will approach Theorem A. Theorem B was anticipated in [6,
Observation 8.2], where we observed that it held when $k=2,3,4,5$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$765432118171615141312111098Matching
$M$$\bullet$3$\bullet$21654$\bullet$$\bullet$2$\bullet$$\bullet$2211111dissection
$\mathfrak{d}$$\bullet$3$\bullet$21654$\bullet$$\bullet$2$\bullet$$\bullet$221111100triangulation
$\mathfrak{t}$ Figure 3. Intermediate objects appearing in the tableau-to-web
map. A tableaux $T$ affords a noncrossing matching $M$ by a standard Catalan
bijection. The boundary vertices of such an $M$ are broken into color sets
incdicated in the figure. Gluing vertices of the same color and keeping track
of the number of edges between each interval, the matching $M$ begets a
weighted dissection $\mathfrak{d}$. The triangulation $\mathfrak{t}$ extends
this dissection $\mathfrak{d}$ by adding two weight zero diagonals (drawn
dashed). By a simple recipe, $\mathfrak{t}$ begets a web diagram, in this case
the web from Figure 1.

One also has a good notion of ${\rm SL}_{3}$ web immanants and these form a
rotation-invariant basis for the degree three part of the Grassmannian
coordinate ring. We conjecture that the analogue of Theorem B holds and hope
to pursue this in future work. We verified this expectation when $k=2,3$ in
loc. cit.

Duality of symmetric group representations. Canonical basis elements are
indexed by semistandard tableaux. We prove our main result by first
considering the weight subspace

$L_{dk}(d\omega_{k})^{\rm std}\subset L_{n}(d\omega_{k})$

spanned by the canonical basis elements whose tableau is standard. The
symmetric group $\mathfrak{S}_{n}$ acts on $L_{n}(d\omega_{k})$ and the
subspace $L_{dk}(d\omega_{k})^{\rm std}$ appears as an
$\mathfrak{S}_{n}$-irrep. Invoking Schur’s lemma, one obtains a uniquely
defined (up to homothety) perfect pairing

$\langle,\rangle\colon L_{dk}(d\omega_{k})^{\rm std}\otimes
L_{dk}(k\omega_{d})^{\rm std}\to\varepsilon$

where $\varepsilon$ is the sign representation of $\mathfrak{S}_{n}$.

Thus, a choice of basis for $L_{dk}(k\omega_{d})^{\rm std}$ induces a dual
basis for $L_{dk}(d\omega_{k})^{\rm std}$ with respect to $\langle,\rangle$.
It seems likely that this notion of duality respects canonical basis vectors
up to a predictable sign. We verified this expectation in [6] when $d=2$ using
${\rm SL}_{2}$ web immanants. Thus, to prove Theorem A on the “standard part”
of the representation $L_{2k}(2\omega_{k})$, we merely need to identify a set
of web diagrams whose corresponding invariants are dual to the canonical
“noncrossing matching” basis for $L_{2k}(k\omega_{2})^{\rm std}$. This result
is the heart of our paper, formulated as Theorem 3.5. For instance, the ${\rm
SL}_{9}$ web in Figure 1 is dual to the ${\rm SL}_{2}$ web given by the
noncrossing matching $M$ in Figure 3.

We extend the construction from $L_{2k}(2\omega_{k})^{\rm std}$ to the whole
representation $L_{n}(2\omega_{k})$ using routine “semistandardization”
constructions.

Organization. The new results and constructions are in the first three
sections. Section 1 formulates the map from standard tableaux to web diagrams
in purely combinatorial terms and proves the invariance under change of
triangulation using skein relations. Section 2 makes a connection between the
tableau-to-web map and plabic graphs. Each ${\rm SL}_{k}$ web diagram is in
particular a plabic graph, hence encodes a positroid, and we identify the
positroids which arise via the tableau-to-positroid map. Section 3 states and
proves the web-matching duality Theorem 3.5. Sections 4 and 5 give quick
definitions of web invariants and web immanants respectively. Section 6
extends all constructions from standard to semistandard tableaux. We then
deduce Theorem B, hence also Theorem A, from Theorem 3.5 by a quick argument.

We use the standard combinatorial notation $\binom{[n]}{k}$ for the set of
$k$-subsets of $[n]$.

##  1\. Tableau-to-web map: standard case

We formulate the combinatorics of our “degree two” tableau-to-web map and
provide a running example. We follow the conventions in [6] and refer to that
paper for a more leisurely tour through the definitions.

######  Definition 1.1 (Web diagram).

An ${\rm SL}_{k}$ web diagram $W$ is a planar bipartite graph $W=(V(W),E(W))$
embedded in a disk. The vertex set $V(W)$ has $n$ boundary vertices, each of
which is on the boundary of the disk and is colored black in the bipartition.
The boundary vertices are numbered $1,\dots,n$ in cyclic order. In addition,
$V(W)$ has interior vertices drawn in the interior of the disk and colored
either white or black. Finally, $W$ comes with a multiplicity function ${\rm
mult}\colon E\to[0,k]$ with the property that the multiplicity sum around
every interior vertex $v$ equals $k$:

$\sum_{e\sim v}{\rm mult}(e)=k.$

We require that every edge incident to a boundary vertex has multiplicity one.
We consider such $W$ up to isotopy of the disk fixing the boundary vertices.

The content of an ${\rm SL}_{k}$ web on $n$ boundary vertices is the sequence
$(d_{1},\dots,d_{n})$ where $d_{i}$ is the number of boundary edges incident
to boundary vertex $i$. By elementary arguments, one knows always that
$\sum_{i\in[n]}d_{i}=dk$ for some $d\in\mathbb{N}$ called the degree of $W$.
We call $W$ standard if it has content $(1,1,\dots,1)$. Most arguments
involving webs can first be carried out for standard webs and then transferred
to arbitrary webs by routine operations.

######  Example 1.2.

The right diagram in Figure 1 is a degree two standard ${\rm SL}_{9}$ web. The
black numbers on the outside of the diagram label are the boundary vertex
labels and those on the inside indicate edge multiplicities. Edge
multiplicities of boundary edges equal one and are omitted.

By a known recipe which we defer to Definition 4.1, an ${\rm SL}_{k}$ web $W$
of degree $d$ with $n$ many boundary vertices determines an element

(1.1) $[W]\in L_{n}(d\omega_{k})\cong\mathbb{C}[{\rm Gr}(k,n)]_{(d)}.$

To understand the constructions in this section, the reader need only know
that there are certain diagrammatic moves on web diagrams which induce linear
relations between web invariants [4; 6]. In our set of conventions, these
moves are contraction of bivalent vertices, bigon removal, dipole removal, and
the square move, see [6, Section 6] for a discussion.

######  Example 1.3.

The simplest example of an ${\rm SL}_{k}$ web is a claw graph with one
interior white vertex joined to $k$ distinct boundary vertices each by an edge
of multiplicity one. Letting $I$ denote the subset of boundary vertices which
are used in this claw graph, the corresponding web invariant is the Plücker
coordinate $\Delta_{I}\in\mathbb{C}[{\rm Gr}(k,n)]$.

######  Remark 1.4.

Unlike some other treatments of webs, we do not require that internal vertices
are trivalent or that boundary vertices have degree one. (In this paper,
because we study the $d=2$ case, the numbers $d_{i}$ appearing in the content
of a web will always equal either 0,1 or 2.) We do require that each boundary
edge has multiplicity one, i.e. its label is $1\in\mathbb{N}$. We also
circumvent the formalism of tags from [4] using instead the results from [6]
and an extra result on signs (see Lemma 4.3).

We associate a web diagram to a tableau via three steps, indicated
schematically in Figure 2. We now describe each of these these steps while
reviewing the intermediate combinatorial gadgets which appear along the way.

###  1.1. From SYT’s to dissections

We construct a map $T\mapsto\mathfrak{d}_{T}$ from the set ${\rm SYT}(2^{k})$
of standard Young tableaux with $k$ rows and $2$ columns to the set of
weighted dissections of polygons of weight $k$, i.e. we formulate the
downwards arrow in Figure 2.

We assume familiarity with the concept of noncrossing matching of a $2k$-gon.
Such matchings are equinumerous with ${\rm SYT}(2^{k})$ and these two sets are
related by a well-known Catalan bijection described as follows. If $ij$ is an
an arc in a noncrossing matching $M$ and if $i<j$ then we say $i$ is the left
endpoint of $ij$ and $j$ is the right endpoint. Given $T\in{\rm SYT}(2^{k})$,
there exists a unique noncrossing matching $M(T)$ of the $2k$-gon with the
property that the left endpoints of $M(T)$ (resp. right endpoints) are exactly
the entries of the first (resp. second) column of $T$.

######  Example 1.5.

The noncrossing matching $M$ of the 18-gon appearing in Figure 3 is related to
the tableau $T\in{\rm SYT}(2^{9})$ from Figure 1 by the Catalan bijection.

######  Definition 1.6 (Short arcs).

A short arc in a noncrossing matching $M$ is an arc joining adjacent boundary
vertices of the $2k$-gon. We let $s(M)$ denote the number of short arcs in
$M$.

Observe that every noncrossing matching has at least two short arcs (provided
$k\geq 2$).

For a noncrossing matching $M$ we can always define uniquely numbers
$i_{1}<\cdots<i_{s}$ with the property that $i_{j}$ and $i_{j}+1\mod 2k$ are
joined by an arc in $M$ and these are a complete list of the short arcs in
$M$. (These typically coincide with the set of left endpoints of the short
arcs of $M$ except in the special case that $1$ and $2k$ are joined by a short
arc in $M$.)

######  Definition 1.7 (Color sets of $M$).

With $i_{1},\dots,i_{s}$ as just defined, the color sets of a noncrossing
matching $M$ are the cyclic intervals $C_{j}:=[i_{j},i_{j+1})$.

The color sets solve the following coloring problem: color, with as few colors
as possible, the boundary vertices of the $2k$-gon so that no arc joins two
vertices of the same color and all of the color sets are cyclic intervals.

######  Example 1.8.

The matching $M$ from Figure 3 has six short arcs whose left endpoints are
$1$,$3$,$6$,$8$,$10$,$14$. The color sets are the intervals are
$[1,3]$,$[4,6]$, $[7,8]$, $[9,10]$, $[11,14]$, $[15,18]$ indicated by the
colors in the figure.

######  Remark 1.9.

Directly from the tableau, one can see that entries $i$ and $i+1$ will be in
the same color set of $M(T)$ whenever $i$ appears in a strictly higher row of
$T$ than $i+1$ does. When this happens, $i$ is often called a descent of $T$.
One also needs to decide under what circumstances the pair $\\{1,2k\\}$ form a
descent. Let $T\setminus\\{1,2k\\}$ denote the tableau obtained by deleting
entries 1 and 2k and sliding the entries in the first column up one box. Then
$\\{1,2k\\}$ form a descent if and only if $T\setminus\\{1,2k\\}$ is not a
standard tableau on the ground set $[2,2k-1]$, i.e. if and only
$T\setminus\\{1,2k\\}$ has a decreasing row.

######  Definition 1.10 (Weighted dissections).

A dissection of an $s$-gon is a choice of diagonals in the $s$-gon which are
mutually noncrossing. (For our purposes, the sides of the $s$-gon are also
considered diagonals.) A weighted dissection $\mathfrak{d}$ is a dissection
together with an $\mathbb{N}$-weighting of its diagonals. The weight of
$\mathfrak{d}$ is the sum of the weights of its various diagonals. The content
$d_{i}$ of the vertex $i$ in $\mathfrak{d}$ is the sum of the weights of its
incident diagonals.

One has always

(1.2) $\sum_{i\in[s]}d_{i}=2{\rm wt}(\mathfrak{d})$

since each diagonal is counted twice in the above sum.

A triangulation is a dissection which is maximal by inclusion.

Construction. Given a noncrossing matching $M=M(T)$ with $s$ many short arcs
and with color sets $C_{1},\dots,C_{s}$, we obtain a weighted dissection
$\mathfrak{d}_{T}$ of the $s$-gon by merging the boundary vertices in each
color set, replacing multiple edges by a single edge of the same weight.

This construction is illustrated in the passage from $M$ to $\mathfrak{d}$ in
Figure 3.

Observe that the total weight of $\mathfrak{d}_{T}$ is the number of arcs in
$M$, which is $k$. Similarly, the weight of $\mathfrak{d}_{T}$ at boundary
vertex $i$ in the $s$-gon equals the cardinality of the color set $C_{i}$ of
$M$.

######  Remark 1.11.

The recipe $T\mapsto\mathfrak{d}_{T}$ is reversible provided we also keep
track of the value of the color set $C_{1}\subset[2k]$, not merely its
cardinality. Otherwise, the recipe is only reversible up to rotation of the
$n$-gon.

###  1.2. From dissections to triangulations

We now discuss the passage from weighted dissections of weight $k$ to weighted
triangulations of weight $k$, i.e. the bottom dashed arrow in Figure 2.

Given a weighted dissection $\mathfrak{d}$ of the $s$-gon, we say that a
weighted triangulation $\mathfrak{t}$ extends $\mathfrak{d}$ if the underlying
triangulation for $\mathfrak{t}$ extends the underlying dissection for
$\mathfrak{d}$ and if moreover, the weight function for $\mathfrak{t}$ agrees
with that for $\mathfrak{d}$ on the diagonals they share and assigns zero
weight to the diagonals not present in $\mathfrak{d}$.

For a tableau $T\in{\rm SYT}(2^{k})$, we will use the notation
$\mathfrak{t}_{T}$ to indicate any choice of weighted triangulation which
extends the weighted dissection $\mathfrak{d}_{T}$ defined in the previous
section. Observe that if $\mathfrak{t}$ and $\mathfrak{t}^{\prime}$ are two
such extensions, then their underlying triangulations are related by a
sequence of flip moves at diagonals not present in $\mathfrak{d}$.

######  Example 1.12.

Continuing our running example, there are four choices of triangulations
$\mathfrak{t}$ extending the dissection $\mathfrak{d}$ from Figure 3 because
we can flip either of the dashed diagonals.

###  1.3. From triangulations to web diagrams

We now construct a map $\mathfrak{t}\mapsto W(\mathfrak{t})$ from weight $k$
weighted triangulations of an $s$-gon to standard ${\rm SL}_{k}$ webs of
degree 2. That is, we formulate the upwards arrow in Figure 2.

Construction. Let $(d_{1},\dots,d_{s})$ be the content of $\mathfrak{t}$. We
first describe $W(\mathfrak{t})$ as a graph and then address its edge
multiplicities. The graph $W(\mathfrak{t})$ has 2k many boundary vertices, has
$s$ many interior white vertices $w_{1},\dots,w_{s}$ and has interior
trivalent black vertices $b(t)$ indexed by the triangles $t$ in
$\mathfrak{t}$. If $t$ has has boundary vertices $a,b,c$ in $\mathfrak{t}$,
then the trivalent vertex $b(t)$ is joined to the white vertices
$w_{a},w_{b},w_{c}$ by edges. Finally, each interior white vertex $w_{i}$ is
incident to $d_{i}$ many distinct boundary vertices by a single edge. This
completes the description of $W(\mathfrak{t})$ as a graph.

The multiplicity of each boundary edge equals one by definition. The
multiplicities of the edges $b(t)w_{a}$, $b(t)w_{b}$, and $b(t)w_{c}$ around a
trivalent vertex are specified as follows. Note that
$\mathfrak{t}\setminus{\rm int}(t)$ is a union of three weighted
triangulations, one for each side of $t$. We let $A$ denote the triangulation
which contains the side $w_{b}w_{c}$, i.e. the triangulation which is
“opposite” the edge $b(t)w_{a}$. Then we set

${\rm mult}_{W(\mathfrak{t})}(b(t)w_{a})=\sum_{e\in\text{ polygon $A$ of
$\mathfrak{t}$}}{\rm wt}_{T}(e).$

The multiplicities of the edges $b(t)w_{b}$ and $b(t)w_{c}$ are defined
analogously using the triangles $B,C$ opposite the sides $b(t)w_{b}$ and
$b(t)w_{c}$. This completes the definition of $W(\mathfrak{t})$.

Let us check that this above construction indeed produces an ${\rm SL}_{k}$
web of degree two. Since each diagonal $e\in\mathfrak{t}$ appears in exactly
one of the triangulations $A,B,C$, the multiplicity sum around each $b(t)$
equals $k$ as required. We must also argue that the multiplicity sum around
every white vertex equals $k$. Indeed, summing the multiplicities of the edges
from $w_{i}$ to interior black vertices counts the weight of all of the
diagonals of $\mathfrak{t}$ except those which are incident to boundary vertex
$i$, i.e. this sum equals $k-d_{i}$. But we have added exactly $d_{i}$
additional edges of multiplicity one from $w_{i}$ to boundary vertices so that
the total multiplicity sum equals $k$ indeed. Finally, the total number of
boundary vertices used in $W(\mathfrak{t})$ is $\sum_{i\in[s]}d_{i}$ which
equals $2k$ by (1.2).

The above recipe determines the web $W(\mathfrak{t})$ up to rotation in the
$2k$-gon and this ambiguity is removed in our setting by specifying that
boundary vertex $w_{i}$ connects to the boundary vertices in the cyclic
interval $C_{i}\subset[2k]$, where $C_{1},\dots,C_{s}$ are the color sets of
the matching $M(T)$.

Summary. The tableau-to-web map is the map

(1.3) $T\mapsto W(\mathfrak{t}_{T})$

where $\mathfrak{t}_{T}$ is any choice of triangulation extending the
dissection $\mathfrak{d}_{T}$.

Our next lemma says that the web invariant $[W(\mathfrak{t})]$ is independent
of the choice of triangulation $\mathfrak{t}_{T}$ extending the dissection
$\mathfrak{d}_{T}$.

######  Proposition 1.13.

Let $\mathfrak{d}$ be a weighted dissection of weight $k$ and let
$\mathfrak{t},\mathfrak{t}^{\prime}$ be two extensions of $\mathfrak{d}$ to
weighted triangulations. Then the corresponding ${\rm SL}_{k}$ web invariants
coincide:

$[W(\mathfrak{t})]=[W(\mathfrak{t}^{\prime})]\in
L_{2k}(2\omega_{k})\cong\mathbb{C}[{\rm Gr}(k,2k)]_{(2)}.$

###### Proof.

It suffices to show the claim when $\mathfrak{t}$ and $\mathfrak{t}^{\prime}$
differ by a flip in a quadrilateral since triangulations which extend
$\mathfrak{d}$ are connected by such moves. The diagonal of such quadrilateral
is not in $\mathfrak{d}$ and therefore has zero weight in both
$\mathfrak{t},\mathfrak{t}^{\prime}$. The local picture around the
quadrilateral is illustrated in Figure 4 where $\alpha$, $\beta$, $\gamma$,
$\delta$ are the weights of the four regions outside the quadrilateral. The
equality $[W(\mathfrak{t})]=[W(\mathfrak{t}^{\prime})]$ is an instance of the
“square switch” skein relation [FLL (eq 6.2)] with parameters $r:=k$,
$j:=\alpha+\beta$,$s:=\alpha$, $v:=\gamma$, $\ell:=\beta+\gamma$. Note that
$j-\ell+v-s=0$, so the only term on the right hand side of [6, Equation (6.2)]
is the case $t=0$ with coefficient $\binom{0}{0}=1$. ∎

$\bullet$$\bullet$$\bullet$$\bullet$$\alpha$$\beta$$\delta$$\gamma$$\circ$$\circ$$\circ$$\circ$$\bullet$$\bullet$$\alpha$$\beta$$\gamma$$\delta$$\gamma+\delta$$\alpha+\beta$$\circ$$\circ$$\circ$$\circ$$\bullet$$\bullet$$\gamma$$\delta$$\alpha$$\beta$$\alpha+\delta$$\beta+\gamma$
Figure 4. The left picture indicates schematically a diagonal of
$\mathfrak{t}\setminus\mathfrak{d}$ (drawn dashed) which can be flipped inside
its quadrilateral to obtain a new triangulation $\mathfrak{t}^{\prime}$ . The
numbers $\alpha,\dots,\delta$ indicate the weights of the four polygons
outside this quadrilateral. The middle picture and right diagrams are the
local picture for the corresponding web diagrams $W(\mathfrak{t})$ and
$W(\mathfrak{t}^{\prime})$ inside the quadrilateral. One has
$[W(\mathfrak{t})]=[W(\mathfrak{t}^{\prime})]$ by a skein relation.

######  Remark 1.14.

When the dissection $\mathfrak{d}_{T}$ happens to be a triangulation there is
no freedom in our constructions. In particular, this happens whenever the
number of short arcs of $M$ equals either two or three. Such $M$ are called
tripartite pairings and have recently arisen in a seemingly different
mathematical context [10; 11]. We think it should be possible to state the
main identity from [10] as a quadratic relation between ${\rm SL}_{2}$ web
immanants.

When $M$ has exactly two short arcs the corresponding web invariant is a
cluster monomial $\Delta_{I}\Delta_{[2k]\setminus I}$ for a cyclic interval
$I\in\binom{[2k]}{k}$. Its web diagram is a disjoint union of two claw graphs.
When $M$ has exactly three short arcs, the web diagram is a tree with one
trivalent black vertex connected to three interior white vertices. This
special case of our construction was previously formulated by I. Le and E.
Yildirim who prove moreover that these web invariants are cluster variables.

##  2\. Webs as plabic graphs

Forgetting the data of the edge multiplicities, any standard web is in
particular a plabic graph in the sense of [21] and it is interesting to
investigate how tight the connection between webs and plabic graphs is, see
also [6; 9]. Here, we explore the tableau-to-web map as a tableau-to-plabic-
graph map. This is a bit of a detour: we use the language of plabic graphs and
dimer covers to a small extent in the proof of Theorem 3.5 but it would be not
be hard to forego this language.

We assume familiarity with the notions of positroids
$\mathcal{M}\subset\binom{[n]}{k}$ and their encodings by decorated
permutations $\pi$, Grassmann necklaces $\vec{\mathcal{I}}$, or reduced plabic
graphs $G$ modulo move equivalence. We work always with graphs which are
bipartite (not just bicolored) with boundary vertices colored black.

Let $G$ be a reduced plabic graph in the $n$-gon. A dimer cover $\pi$ of $G$
is a subset $\pi\subset E(G)$ which covers each interior vertex once. We
denote by $\partial(\pi)\subset[n]$ the subset of boundary vertices used in
$\pi$. Observe that

(2.1) $\\#\partial(\pi)=\\#\text{ of white vertices of $G$}-\\#\text{ of
interior black vertices of $G$}.$

In particular this number depends only on $G$ not on $\pi$. We denote this
number by $k$ and say that $G$ has type $(k,n)$.

######  Remark 2.1.

When $W=W(\mathfrak{t}_{T})$ is a tableau-to-web image of some $T\in{\rm
SYT}(2^{k})$, the number of white vertices of $W$ is the number $s=s(M(T))$ of
short arcs while the number of interior black vertices is the number of
triangles in a triangulation of the $s$-gon. It follows that the number (2.1)
equals $2$ for $W$, i.e. that $W$ is a plabic graph of type $(2,2k)$.

######  Definition 2.2 (Cyclic interval positroids).

Let $C_{1},\dots,C_{s}$ be a decomposition of $[2k]$ into disjoint cyclic
intervals. We obtain a positroid
$\mathcal{M}_{C_{1},\dots,C_{s}}\subset\binom{[2k]}{2}$ whose bases are
exactly the pairs $\\{i,j\\}$ with the property that $i\in C_{a}$ and $j\in
C_{b}$ with $a\neq b$.

The positroid subvariety of ${\rm Gr}(2,2k)$ corresponding to
$\mathcal{M}_{C_{1},\dots,C_{s}}\subset\binom{[2k]}{2}$ imposes the cyclic
rank condition

(2.2) ${\rm rank}(v_{i}\colon i\in C_{a})\leq 1$

with one such rank condition for each $a=1,\dots,s$, where
$v_{1},\dots,v_{2k}$ are the column vectors of $2\times 2k$ matrix
representatives for points in ${\rm Gr}(2,2k)$.

######  Proposition 2.3.

Let $W(\mathfrak{t}_{T})$ be an ${\rm SL}_{k}$ web diagram arising by
evaluating the tableau-to-web map on $T\in{\rm SYT}(2^{k})$. Let
$C_{1},\dots,C_{S}$ be the color sets of the noncrossing matching $M(T)$. Then
$W(\mathfrak{t}_{T})$ is a reduced plabic graph whose corresponding positroid
is $\mathcal{M}_{C_{1},\dots,C_{s}}$.

###### Proof.

Abbreviate $W=W(\mathfrak{t}_{T})$. Let $C_{j}=[i_{j},i_{j+1})$ be a color set
of $M(T)$. Then the boundary vertices in $C_{j}$ are incident to the same
white vertex $w_{j}$ of $W$. For $b=i_{j},\dots,i_{j+1}-2$, the zizag path
starting at $b$ (also known as the trip starting at $b$) proceeds to $w_{j}$
and then ends at vertex $b+1$. It remains to understand the zigzag paths
starting at $i_{j}-1$. The “interior” of $W$ is the plabic graph dual to a
triangulation of the $s$-gon and the trips starting at $i_{j}-1$’s are
identified with trips in this smaller plabic graph. The zigzag paths of this
smaller graph encode the top cell for ${\rm Gr}(2,s)$, i.e. each such zigzag
path proceeds two units clockwise. Using this reasoning, one can see that the
zigzag path of starting at $i_{j}-1$ ends at $i_{j+2}$. We now see that $G$ is
reduced and moreover we have computed its trip permutation. Using the standard
recipes to translate from trip permutations to rank conditions, we see that
the rank conditions (2.2) hold; on the other hand, by comparing with the top
cell for ${\rm Gr}(2,s)$, we see that these are the only rank conditions which
hold, completing the proof. ∎

We conclude this section with two remarks which explore further properties of
the tableau-to-plabic-graph map.

######  Remark 2.4.

The data of $W$ as a plabic graph does not specify the corresponding canonical
basis element $[W]$. For example, the pair of weighted triangulations

(2.3)
$\bullet$$\bullet$$\bullet$$\bullet$22221$\mathfrak{t}$$\bullet$$\bullet$$\bullet$$\bullet$11331$\mathfrak{t}^{\prime}$

determine ${\rm SL}_{9}$ webs $W(\mathfrak{t})$ and $W(\mathfrak{t}^{\prime})$
with the same underlying plabic graph but with different edge multiplicities.
As follows from Theorem 3.5, these two triangulations are dual to distinct
matchings of the $18$-gon, hence $[W(\mathfrak{t})$ and
$[W(\mathfrak{t}^{\prime})]$ correspond to distinct canonical basis elements.

######  Remark 2.5.

Not every decomposition $[2k]=C_{1}\coprod\cdots\coprod C_{s}$ into cyclic
intervals arises from a noncrossing matching. For example, it is necessary
that each $|C_{i}|\geq 2$. More substantively, there is no noncrossing
matching of the octagon whose color sets have cardinalities $4$, $2$, and $2$.
Even if we consider a set of cyclic intervals $C_{1}\coprod\cdots\coprod
C_{s}$ which do come from a matching, not every reduced plabic graph for the
positroid $\mathcal{M}_{C_{1},\dots,C_{s}}$ can be assigned edge
multiplicities in such a way that the corresponding web invariant is a
canonical basis element.

For instance, the plabic graph

(2.4)
$\bullet$$5$$\bullet$$4$$\bullet$$3$$\bullet$$2$$\bullet$$1$$\bullet$$10$$\bullet$$9$$\bullet$$8$$\bullet$$7$$\bullet$$6$$\circ$$\circ$$\circ$$\circ$$\bullet$$\bullet$

can only be made into an ${\rm SL}_{5}$ web diagram in a unique way up to
reflection in the 10-gon. One can check that the resulting web diagram has
nontrivial pairing with (at least) two different noncrossing matchings in the
sense of Definition 3.2 and is therefore not a canonical basis element.

We can explain “what goes wrong” with this plabic graph as follows. There is a
unique matching $M$ of the 10-gon whose color sets are $[1,2]$, $[3,4,5]$,
$[6,7]$, and $[8,9,10]$. The corresponding dissection $\mathfrak{d}$ is a
triangulated quadrilateral and the above plabic graph (2.4) corresponds to the
“other” triangulation of this quadrilateral, which is forbidden.

##  3\. Web-matching duality

We start by reviewing some classical facts and some additional results from
[6].

We denote by $S(\lambda)$ the irreducible representation of the symmetric
group $\mathfrak{S}_{n}$ indexed by a partition $\lambda$ of $n$. (Thus, the
character $\chi_{\lambda}$ of $S(\lambda)$ satisfies
$\frac{1}{n!}\sum_{w\in\mathfrak{S}_{n}}\chi_{\lambda}(w)p_{\alpha(w)}=s_{\lambda}$
where $s_{\lambda}$ is the schur function, $\alpha(w)$ is the cycle type of
$w$ written as a partition, and $p_{\alpha(w)}$ is the corresponding monomial
in power sum symmetric functions.) Recall that each $\mathfrak{S}_{n}$-irrep
is self-dual: $S(\lambda)\cong S(\lambda)^{*}$ where asterisk denotes duality.

Let $\varepsilon=S(1^{n})$ denote the sign representation which is
1-dimensional. If $\lambda$ and $\mu$ are conjugate partitions one knows that
$S(\lambda)\cong S(\mu)\otimes\varepsilon$. By Schur’s lemma, there is
therefore a unique $\mathfrak{S}_{n}$-equivariant map $S(\lambda)^{*}\to
S(\mu)\otimes\varepsilon$, or equivalently, a unique equivariant pairing
$S(\lambda)\otimes S(\mu)\to\varepsilon$.

The action of $\mathfrak{S}_{n}$ on $\mathbb{C}^{n}$ by coordinate permutation
induces an action on the coordinate ring $\mathbb{C}[{\rm Gr}(k,n)]$. Let
$\mathbb{C}[{\rm Gr}(k,dk)]^{\rm std}$ denote the subspace spanned by
canonical basis vectors indexed by standard Young tableaux. One knows that

$\mathbb{C}[{\rm Gr}(k,dk)]^{\rm std}\cong S(d^{k})$

as $\mathfrak{S}_{dk}$-modules. From the previous paragraph, one has a
canonical pairing

(3.1) $\langle,\rangle\colon\,\mathbb{C}[{\rm Gr}(k,dk)]^{\rm
std}\otimes\mathbb{C}[{\rm Gr}(d,dk)]^{\rm std}\to\varepsilon.$

We will use this pairing to show that our tableau-to-web map is “the right
one.”

Our next definition allows for explicit computations with this pairing.

######  Definition 3.1.

Let $W$ be an ${\rm SL}_{k}$ web diagram. A consistent labeling $\ell$ of $W$
is a choice of subset $\ell(e)\in\binom{[k]}{{\rm mult}(e)}$ for each edge $e$
of $W$ subject to the requirement that $\ell(e)$ and $\ell(e^{\prime})$ are
disjoint whenever $e,e^{\prime}$ are incident edges of $W$.

Since the multiplicities around every vertex sum to $k$, the union of the
subsets around every interior vertex equals $[k]$.

When $W$ is a standard ${\rm SL}_{k}$ web and $\ell$ is a consistent labeling,
the labels of the boundary edges $e_{i}$ for $i=1,\dots,dk$ determine a
sequence of singleton subsets $\ell(e_{1}),\dots,\ell(e_{n})$ of $[k]$. One
can show that each number from $1,\dots,k$ appears exactly $d$ many times in
this sequence. We call this the boundary word of $\ell$ and denote it by ${\bf
w}(\ell)$.

We now introduce several combinatorial gadgets which one can attach to such a
word. Consider the set

$\mathcal{A}_{k,d}=\\{1_{1},\dots,1_{d},2_{1},\dots,2_{d},\cdots,k_{1},\dots,k_{d}\\}$

with its symbols linearly ordered in the way we have just written them.

######  Definition 3.2.

A $(k,d)$ balanced word w is a word of length $dk$ in which each symbol from
$1,\dots,k$ appears exactly $d$ times.

We upgrade such a w to a permutation of the ordered set $\mathcal{A}_{k,d}$ by
adding subscripts to the appearances of symbol $i$ so that subscripts increase
from left to right in ${\bf w}$. Denote by sign(w) the sign of this
permutation.

Given a standard web $W$ and a balanced word ${\bf w}$, we let

$a(W,{\bf w})=\\#\\{\ell\colon\,{\bf w}(\ell)={\bf w}\\}$

be the number of consistent labelings of $W$ with boundary word ${\bf w}$.

We let

$\Delta({\bf w}):=\prod_{j=1}^{k}\Delta_{\\{i\in[dk]\colon{\bf
w}_{i}=j\\}}\in\mathbb{C}[{\rm Gr}(d,dk)]^{\rm std}$

denote the corresponding monomial in Plücker coordinates.

$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$456789123456712347895612348956789123$\bullet$$\bullet$$\bullet$$\bullet$554332112899847766$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$554332112899847766
Figure 5. A consistent labeling $\ell$ of the ${\rm SL}_{9}$ web $W$ from
Figure 1. We write e.g. $1234789$ to indicate $\ell(e)=\\{1,2,3,4,7,8,9\\}$.
One sees each number from $1,\dots,9$ around every interior vertex and the
cardinality of the set $\ell(e)$ matches ${\rm mult}_{W}(e)$. Each number
$i=1,\dots,9$ appears exactly twice as on a boundary edge of this labeling;
joining these two copies by an arc yields the noncrossing matching $M$ from
Figure 3 (redrawn at right for convenience). Web-matching duality asserts that
$\ell$ is the unique labeling whose boundary word is a noncrossing matching.

######  Example 3.3.

Figure 5 gives a consistent labeling $\ell$ of the ${\rm SL}_{9}$ web $W$ from
Figure 1. The boundary word of this labeling is

(3.2) ${\bf w}(\ell)=123345566774899821\text{ of type (9,2)}.$

The corresponding monomial

$\Delta({\bf
w}(\ell))=\Delta_{1,18}\Delta_{2,17}\Delta_{34}\Delta_{5,12}\Delta_{67}\Delta_{89}\Delta_{10,11}\Delta_{13,16}\Delta_{14,15}$

is a noncrossing monomial for ${\rm Gr}(2,n)$.

By [6, Equation (5.16) and Theorem 8.1], the unique pairing $\langle,\rangle$
(3.1) can be understood in terms of the numbers $a(W,{\bf w})$:

######  Lemma 3.4.

For a standard ${\rm SL}_{k}$ web $W$ of degree $d$ and a $(k,d)$ balanced
word ${\bf w}$, one has

$\langle[W],\Delta({\bf w})\rangle=a(W,{\bf w}).$

We now specialize to the case $d=2$. A noncrossing matching $M$ of the
$2k$-gon gives us a $(k,2)$ balanced word ${\bf w}(M)$ with the property that
the two locations of the symbol $i$ in ${\bf w}(M)$ are exactly the two
endpoints of the $i$th arc of $M$ in the $2k$-gon. The word (3.2) is this word
when $M$ is the matching from Figures 3 and 5. We abbreviate

$\Delta(M):=\Delta({\bf w}(M)).$

Such monomials are called noncrossing monomials and these are the canonical
basis for $\mathbb{C}[{\rm Gr}(2,2k)]^{\rm std}$.

We arrive at our main combinatorial assertion.

######  Theorem 3.5 (Web-matching duality).

For $T\in{\rm SYT}(2^{k})$, let $W=W(\mathfrak{t}_{T})$ be a web diagram
constructed via our tableau-to-web map. Let $M=T(M)$ be the noncrossing
matching corresponding to $T$ under the Catalan bijection.

Then $a(W,{\bf w}(M))=1$ and $a(W,{\bf w}(M^{\prime}))=0$ for noncrossing
matchings $M^{\prime}\neq M$.

That is, the web invariant $[W(\mathfrak{t}_{T})]$ is dual to the noncrossing
monomial $\Delta(M)$ under the pairing (3.1). Since this is true for any
triangulation $\mathfrak{t}_{T}$ extending the dissection $\mathfrak{d}_{T}$,
we get an alternative proof that the tableau-to-web-invariant map is well-
defined.

###### Proof.

By induction on $k$ with the base cases $k=1,2$ easily checked by hand.
Suppose that $\ell$ is a consistent labeling of $W$ whose boundary word ${\bf
w}(\ell)$ is the word ${\bf w}(M^{\prime})$ of a noncrossing matching
$M^{\prime}$ of the $2k$-gon. We will argue that $M^{\prime}=M$. (The
uniqueness of the labeling $\ell$ will also follow.)

Certainly, $M^{\prime}$ has at least one short arc $i,i+1$. Suppose this is
the $j$th arc of $M^{\prime}$ when arcs are ordered by their left endpoints.
This means that $\ell(e_{i})=\ell(e_{i+1})=j$. Thus, boundary vertices $i,i+1$
are incident to different white vertices of $W$, thus $i$ and $i+1$ reside in
different color sets of $W$, i.e. $i,i+1$ is also a short arc of $M$.

The edges $e\colon j\in\ell(e)$ form a dimer cover $\pi$ of $W$ whose boundary
subset is given by $\partial(\pi)=\\{i,i+1\\}$. On the other hand,
$\\{i,i+1\\}$ is a term in the Grassmann necklace $\vec{\mathcal{I}}$ for the
positroid $\mathcal{M}_{C_{1},\dots,C_{s}}$ determined by the color sets
$C_{1},\dots,C_{s}$ of the matching $M$. Thus $\pi$ is the unique such dimer
cover of $W$.

Given a dimer cover $\pi$ of a web $W$, we obtain an ${\rm SL}_{k-1}$ web
$W\setminus\pi$ from $W$ by decrementing the multiplicity of each edge used in
$\pi$. That is, $W\setminus\pi$ is the same underlying graph but has
multiplicity function

${\rm mult}_{W\setminus\pi}(e)=\begin{cases}&{\rm mult}_{W}(e)-1\text{ if
$e\in\pi$}\\\ &{\rm mult}_{W}(e)\text{ if $e\not\in\pi$.}\end{cases}$

If ${\rm mult}_{W}(e)=1$ and $e\in\pi$, one can interpret this operation as
deleting the edge $e$ from $W$. In the case of the dimer cover $\pi$ from the
previous paragraph, this means that the boundary edges incident to vertices
$i,i+1$ are deleted, and $W\setminus\pi$ is naturally a degree two standard
web on the ground set $[2k]\setminus\\{i,i+1\\}$.

It follows that

(3.3) $a(W,M^{\prime})=a(W\setminus\pi,M^{\prime}\setminus\\{i,i+1\\})$

where the notation $M^{\prime}\setminus\\{i,i+1\\}$ stands for the noncrossing
matching of the $2(k-1)$-gon on the ground set $[2k]\setminus\\{i,i+1\\}$
obtained by removing the arc $i,i+1$ from $M^{\prime}$. In particular,
$a(W\setminus\\{i,i+1\\},M^{\prime}\setminus\\{i,i+1\\})\neq 0$.

To complete the proof, it would suffice to show that the ${\rm SL}_{k-1}$ web
$W\setminus\pi$ is an image of the tableau-to-web map for the matching
$M\setminus\\{i,i+1\\}$, since by induction we would be able to conclude that
$M^{\prime}\setminus\\{i,i+1\\}=M\setminus\\{i,i+1\\}$. As we will see,
$W\setminus\pi$ is not always in the image of the tableau-to-web map, but this
is true up to a minor fix.

Let $\mathfrak{d}$, $\mathfrak{t}$ be the dissection and chosen triangulation
corresponding to $T$ and $W$. We analyze what happens when passing from $M$ to
$M\setminus\\{i,i+1\\}$.

The numbers $i,i+1$ are in adjacent color sets of $M$, hence determine a side
$S$ of the dissection $\mathfrak{d}$ and the triangulation $\mathfrak{t}$. If
the weight of this side is at least two in $\mathfrak{t}$, then we decrement
the weight of $S$ by one in when passing from $M$ to $M\setminus\\{i,i+1\\}$.
The two white vertices of $W$ corresponding to the side $S$ have one smaller
degree, hence each is joined to one fewer boundary vertex (indeed the boundary
edges incident to $i,i+1$ are deleted). We can choose the same triangulation
$\mathfrak{t}$ (decrementing the weight of the corresponding side by one).
Thus the underlying plabic graph does not change. We decrement the
multiplicity of exactly one edge around each trivalent vertex (namely, we
decrement the edge whose corresponding region contains the side $S$).
Altogether, in passing from $M$ to $M\setminus M\setminus\\{i,i+1\\}$ we
delete the edges of a dimer cover of $W$ with boundary $i,i+1$. By the
uniqueness, we must have deleted the edges in $\pi$, so that $W\setminus\pi$
indeed is an instance of the tableau-to-web map in this case.

To complete the proof, we need to analyze what happens in passing from $M$ to
$M\setminus\\{i,i+1\\}$ in the special case that the side $S$ has weight one
in $\mathfrak{d}$. The color sets of $M\setminus\\{i,i+1\\}$ are obtained by
deleting $i,i+1$ from the ground set and merging the two color sets containing
these numbers. At the level of weighted dissections, we contract the side $S$
to a point and replace parallel weighted edges by a single edge while adding
their weights. We can perform the same contraction operation to the weighted
triangulation $\mathfrak{t}$ extending $\mathfrak{d}$. We delete exactly one
triangle during this process, namely, the triangle containing the side $S$. At
the level of webs, we delete a trivalent black vertex and merge the two white
vertices which correspond to the deleted side $S$ while deleting their
boundary legs at $i,i+1$. Let $W^{\prime}$ be the web which results from these
steps.

The webs $W_{i}$ and $W^{\prime}$ are related to each other by the following
simple operation. Let $t$ be the triangle containing the side $S$ in
$\mathfrak{t}$. The corresponding trivalent black vertex $b(t)\in W$ has three
edges, one of which is dual to the side $S$ and has multiplicity one. Any
dimer cover of $W$ with boundary $i,i+1$ must use this edge. Thus, we delete
this edge in the web $W\setminus\pi$ so that $b(t)$ becomes a bivalent vertex.
It is not hard to see that $W^{\prime}$ is obtained from $W\setminus\pi$ by
contracting this bivalent vertex. Contracting bivalent vertices does not
effect the number of consistent labelings so that
$a(W^{\prime},M^{\prime}\setminus\\{i,i+1\\})=a(W\setminus\pi,M^{\prime}\setminus\\{i,i+1\\})\neq
0$. By the induction hypothesis applied to $W^{\prime}$ we conclude that
$M^{\prime}\setminus\\{i,i+1\\}=M\setminus\\{i,i+1\\}$ completing the proof. ∎

##  4\. Web invariants

We briefly review the definition of web invariant here.

We identify the coordinate ring $\mathbb{C}[{\rm Gr}(k,n)]$ with the algebra
of ${\rm SL}_{k}$-invariant polynomial functions of $n$-tuples of vectors in
$\mathbb{C}^{k}$. For example, the Plücker coordinate $\Delta_{I}$ is the
${\rm SL}_{k}$-invariant polynomial

$(v_{1},\dots,v_{n})\mapsto{\rm det}(v_{i_{1}},\dots,v_{i_{k}})$

where $i_{1},\dots,i_{k}$ are the elements of $I$ written in ascending order
and $(v_{1},\dots,v_{n})$ is an $n$-tuple of vectors in $\mathbb{C}^{k}$.

As a special case, we have the identification

$\mathbb{C}[{\rm Gr}(k,dk)]^{\rm std}={\rm Hom}_{{\rm
SL}_{k}}(\otimes_{i=1}^{dk}\mathbb{C}^{k},\mathbb{C}).$

By multilinearity, an element of the above hom-space is determined by how it
evaluates on tensor products $e_{i_{1}}\otimes\cdots\otimes
e_{i_{dk}}\in\otimes^{dk}\mathbb{C}^{k}$ where each $e_{i_{j}}$ is a standard
basis vector for $\mathbb{C}^{k}$.

######  Definition 4.1 (Web invariant).

Let $W$ be a standard ${\rm SL}_{k}$ web diagram of degree $d$. Then the web
invariant $[W]\in\mathbb{C}[{\rm Gr}(k,dk)]_{(d)}$ is the polynomial function
defined by its evaluation on tensor products of basis vectors as follows:

(4.1) $[W](e_{i_{1}},\dots,e_{i_{dk}})={\rm sign}({\bf w})a(W,{\bf w})$

for all words $w=i_{1},\dots,i_{dk}$ drawn from $[k]$.

######  Remark 4.2.

We have only defined ${\rm sign}({\bf w})$ when $w$ is a balanced word, but
this is a necessary condition for $a(W,{\bf w})$ to be nonzero, so the above
definition parses.

It follows from [6, Lemma 5.4] that the function defined in this way indeed
determines an element of $\mathbb{C}[{\rm Gr}(k,dk)]^{\rm std}$.

When $W$ is the claw graph on vertices $1,\dots,k$ (see Example 1.3), one has
$[W]=\Delta_{1,\dots,k}$ and (4.1) encodes the Liebnitz formula for the
determinant.

Warning: Definition 4.1 is a slick way of defining $[W]$ for all standard
${\rm SL}_{k}$ webs $W$, but it sometimes is the “wrong” definition up to a
sign. For example, this definition can clash with the more familiar way of
turning an ${\rm SL}_{2}$ or ${\rm SL}_{3}$ web into a web invariant up to a
multiplicative factor of $-1$. Thus, we refer to both $[W]$ and $-[W]$ as web
invariants when stating our two main theorems. Our next lemma identifies this
sign discrepancy when $k=2$ and allows us to identify the “correct” sign for
the webs $W(\mathfrak{t}_{T})$ in Section 6.

A noncrossing matching $M$ of the $2k$-gon naturally becomes a standard ${\rm
SL}_{2}$ web $W_{M}$ by replacing each arc $ij\in M$ by a bivalent interior
white vertex joined to vertices $i$ and $j$ by an edge of multiplicity one.
The reader should not confuse the ${\rm SL}_{2}$ web $W_{M}$ with the ${\rm
SL}_{k}$ webs $W(\mathfrak{t}_{T})$ which we have introduced in Section 1.

Our next result compares the web invariant $[W_{m}]$ to the noncrossing
monomial $\Delta(M)$.

######  Lemma 4.3.

Let $T\in{\rm SYT}(2^{k})$ with entries $i_{1},i_{2},\dots,i_{k}$ in the first
column. Let $M=M(T)$. Then

$[W_{M}]=\prod_{j\in[k]}(-1)^{i_{j}-j}\Delta(M)\in\mathbb{C}[{\rm Gr}(2,2k)].$

###### Proof.

We get a balanced word ${\bf w}(T)$ of type $(2,k)$ by putting 1’s positions
$i_{1},\dots,i_{k}$ and putting $2$’s in positions
$[2k]\setminus\\{i_{1},\dots,i_{k}\\}$. Thus every left endpoint of $M$ is
marked with a $1$ and every right endpoint is marked with a $2$. If follows
that $\Delta(M)(e({\bf w}(T))=+1$.

On the other hand, $[W_{M(T)}](e({\bf w}(T))={\rm sign}({\bf w}(T))$. The
result follows by checking that

${\rm sign}({\bf w}(T))=\prod_{j\in[k]}(-1)^{i_{j}-j}$

which is easy to see. ∎

##  5\. Web immanants

We briefly review ${\rm SL}_{2}$ web immanants introduced by T. Lam [17] in
the spirit of [23].

A plabic network is a choice of reduced plabic graph $G$ of type (k,n)
together with an edge weighting ${\rm wt}_{N}\colon E(G)\to\mathbb{C}^{*}$.
For a dimer cover $\pi$ of $G$ let ${\rm wt}_{N}(\pi):=\prod_{e\in\pi}{\rm
wt}_{N}(e)$ be the product of the edge weights used in $\pi$. For
$I\in\binom{[n]}{k}$, one has the complex number

(5.1) $\Delta_{I}(N):=\sum_{\pi\colon\,\partial(\pi)=I}{\rm wt}_{N}(\pi),$

a weight-generating function for dimer covers $\pi$ of $N$ using boundary
vertices $I$.

For any plabic network $N$, the numbers $\Delta_{I}(N)\colon
I\in\binom{[N]}{k}$ satisfy the Plücker relations, i.e. they determine a point
in the Grassmannian ${\rm Gr}(k,n)$. That is, for any $N$, one can find an
$n$-tuple $(v_{1},\dots,v_{n})$ of vectors in $\mathbb{C}^{k}$ whose Plücker
coordinates match the $\Delta_{I}(N)$’s:

(5.2) ${\rm det}(v_{i_{1}},\dots,v_{i_{k}})=\Delta_{I}(N)\text{ for all
$I\in\binom{[n]}{k}$}$

where again $i_{1},\dots,i_{k}$ are the elements of $I$ in ascending order. It
is known that the set of points in ${\rm Gr}(k,n)$ which arise from a network
$N$ in this way is Zariski-dense.

The previous two paragraphs construct the canonical basis for $\mathbb{C}[{\rm
Gr}(k,n)]_{(1)}$ using networks. The ${\rm SL}_{2}$ web immanants do the same
for $\mathbb{C}[{\rm Gr}(k,n)]_{(2)}$.

######  Definition 5.1.

A partial noncrossing matching of type $(k,n)$ is a pair $(M,P)$ where $M$ is
a noncrossing matching of some subset $S\subset[n]$ and $P\subset[n]\setminus
S$ is a subset satisfying $\\#S+2\\#P=2k$. The content of a partial
noncrossing matching $(M,P)$ of type $(k,n)$ is the sequence
$(d_{1},\dots,d_{n})$ defined by $d_{i}=1$ when $d_{i}\in S$, $d_{i}=2$ when
$d_{i}\in P$, and $d_{i}$ equals zero otherwise.

We think of elements of $P$ as the result of merging two vertices in a short
arc to obtain a boundary vertex “matched with itself.”

Given a plabic graph network $N$ of type $(k,n)$, a $2$-like subgraph $W$ of
$G$ is an ${\rm SL}_{2}$ web whose underlying plabic graph is a subgraph of
$N$. Thus, $W$ is a disjoint union of edges of multiplicity two, cycles of
even length formed by multiplicity-one edges, and multiplicity-one paths
joining boundary vertices. Such $W$ naturally begets a type $(k,n)$ partial
noncrossing matching $(M(W),P(W))$ where $M$ is the noncrossing matching of
boundary vertices encoded by the paths of $W$ and $P$ is the subset of
boundary vertices whose boundary edge is doubled in $W$.

Let $c(W)$ denote the number of cycles in $W$. One defines

${\rm wt}_{N}(W)=2^{c(W)}\prod_{e\in E(N)}{\rm wt}_{N}(e)^{{\rm
mult}_{W}(e)},$

a degree two analogue of the numbers ${\rm wt}_{N}(\pi)$.

The web immanant $F_{(M,P)}$ indexed by a partial noncrossing matching $(M,P)$
of type $(k,n)$ is the element of $\mathbb{C}[{\rm Gr}(k,n)]_{(2)}$ which
evaluates on vectors $(v_{1},\dots,v_{n})$ as follows. Choose a plabic network
$N$ of type $(k,n)$ whose Plücker coordinates match those of
$(v_{1},\dots,v_{n})$ as in (5.2). Then

$F_{(M,P)}(v_{1},\dots,v_{n})=\sum_{M(W)=M,P(W)=P}{\rm wt}_{N}(W),$

where on the left hand side we evaluate the web immanant on
$(v_{1},\dots,v_{n})$ and on the right hand side we sum over $2$-like
subgraphs $W$ of $N$ whose connectivity is given by $(M,P)$. By the
aforementioned Zariski-denseness, this recipe indeed specifies $F_{(M,P)}$ as
an element of the Grassmannian coordinate ring.

By [16, Theorems 2.1, 5.5, and 5.13], the elements $F_{(M,P)}$ form a basis
for $\mathbb{C}[{\rm Gr}(k,n)]_{(2)}$. This basis coincides with Lusztig’s
canonical basis for $L_{n}(2\omega_{k})$ and inherits many good properties.
The basis is cyclically invariant and has good restriction properties to
positroid subvarieties of ${\rm Gr}(k,n)$. Moreover, each basis element is
nonnegative when evaluated on points in ${\rm
Gr}(k,n)\cap\mathbb{RP}^{\binom{n}{k}-1}_{\geq 0}$.

##  6\. Semistandardization

The constructions in Sections 1, 3, and 4 were formulated for standard
tableaux. We now extend the tableau-to-web map to the setting of semistandard
tableaux and prove Theorems A and B.

The following encoding step appears often in this section. Let
$(d_{1},\dots,d_{n})$ be a sequence drawn from $\\{0,1,2\\}$ and with
$\sum_{i}d_{i}=2k$. Then there is a unique weakly increasing sequence
$a_{1},\dots,a_{2k}$ in which each symbol $i=1,\dots,n$ appears exactly
$d_{i}$ many times. We say that $(d_{1},\dots,d_{n})$ encodes to the sequence
$a_{1},\dots,a_{2k}$.

######  Definition 6.1 (Standardization of $T$).

The standardization of $T\in{\rm SSYT}(2^{k},[n])$ is the standard tableau
$\hat{T}\in{\rm SYT}(2^{k})$ defined as follows. Let $(d_{1},\dots,d_{n})$ be
the content of $T$ encoded as the sequence $a_{1},\dots,a_{2k}$. One has a
sequence of tableaux $\emptyset=:T_{0}\subset\cdots\subset T_{2k}:=T$ via the
following requirements:

  * •

each $T_{i}$ is obtained from its predecessor $T_{i-1}$ by adding a single box
filled with the number $a_{i}$

  * •

if $a_{i}=a_{i+1}$, then the box $T_{i}\setminus T_{i-1}$ is in the first
column.

The tableau $\hat{T}$ is defined by instead filling the box $T_{i}\setminus
T_{i-1}$ with the number $i$ rather than the number $a_{i}$.

######  Remark 6.2.

Recall that the first column of $T\in{\rm SYT}(2^{k})$ encodes left endpoints
of arcs in the matching $M(T)$, i.e. the endpoint of the arc which is smaller
in the order $1<2<\cdots<2k$. Standardization says that when $T$ has two
copies of the same number, we should regard the entry that appears in the
first column as “smaller.”

######  Definition 6.3 (Semistandard tableau-to-web map).

Let $T\in{\rm SSYT}(2^{k},[n])$ with content encoded by the sequence
$a_{1},\dots,a_{2k}$. Let $\hat{W}$ be a tableau-to-web image of the
standardization $\hat{T}$. Thus $\hat{W}$ has boundary edges
$e_{1},\dots,e_{2k}$ and is drawn in the $2k$-gon. We construct a web $W$ in
the $n$-gon which has the same interior vertices and edges as $\hat{W}$ and
the same edge multiplicities, but whose edge $e_{i}$ is attached instead to
the boundary vertex $a_{i}$ in the $n$-gon. By definition, such a web $W$ is a
tableau-to-web image of $T$.

######  Example 6.4.

The semistandard tableau $T$ in Figure 6 standardizes to the standard tableau
in Figure 1 that has served as our running example. The content of $T$ is
encoded by the sequence

(6.1) $a_{1},\dots,a_{18}=1,2,3,3,4,5,5,7,8,8,9,10,11,12,14,14,15,16,18.$

To obtain a tableau-to-web image of $T$, we reattach the 18 boundary legs of
the web in Figure 1 according to this sequence as we have done in Figure 6.

1 & 3
2 7
3 8
5 10
8 11
8 14
9 15
12 16
14 18
$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\bullet$622111413267$\bullet$$\bullet$$\bullet$$\bullet$765432118171615141312111098
Figure 6. To obtain a tableau-to-web image of the semistandard tableau $T$ at
left, we standardize $T$ to obtain the tableau from Figure 1 and then reattach
the legs of the web from Figure 1 according to the sequence (6.1).

The invariant $[W]$ of a nonstandard web is defined in the following way:

(6.2) $[W](v_{1},\dots,v_{n})=[\hat{W}](v_{a_{1}},\dots,v_{a_{2k}})$

where $a_{1},\dots,a_{2k}$ is the sequence which encodes the content of $W$.
Thus, we evaluate $[W]$ on $(v_{1},\dots,v_{n})$ by “forgetting” the vectors
$v_{i}$ for which $d_{i}=0$ and using the vectors $v_{i}$ with $d_{i}=2$
“twice.”

Let $(M,P)$ be a partial noncrossing matching of type $(k,n)$ whose content is
encoded by $a_{1},\dots,a_{2k}$. We obtain a noncrossing matching
$\widehat{(M,P)}$ uniquely by joining $i$ and $j$ in the $2k$-gon if either
$a_{i}=a_{j}$ or $a_{i}\neq a_{j}$ and $a_{i}$ is joined to $a_{j}$ in $M$.

######  Remark 6.5.

To a pair $(M,P)$ as above, we associate a semistandard tableau $T(M,P)\in{\rm
SSYT}(2^{k},[n])$ by first computing the tableau
$\hat{T}:=T(\widehat{(M,P)})\in{\rm SYT}(2^{k})$ using the Catalan bijection,
and then replacing the symbol $i\in\hat{T}$ with the symbol $a_{i}\in T$. It
is simple to check that the map $(M,P)\mapsto T(M,P)$ is bijective. Thus,
${\rm SL}_{2}$ web immanants can be indexed by semistandard tableaux rather
than by matchings.

######  Lemma 6.6.

For a partial noncrossing matching $(M,P)$ as above, one has

(6.3)
$F_{(M,P)}(v_{1},\dots,v_{n})=F_{\widehat{(M,P)}}(v_{a_{1}},\dots,v_{a_{2k}}).$

###### Proof.

If $d_{i}=0$, one can argue that we can remove $i$ from the ground set using
the explicit formula [16, Theorem 3.3] for the immanant $F_{(M,P)}$ in terms
of Plücker coordinates: if $M$ is a matching of the vertices $S\subset[n]$,
then this formula only depends on the set $S\cup P$.

So we reduce to the case that all $d_{i}\in\\{1,2\\}$. We need to prove:

$F_{(M,P)}(v_{1},\dots,v_{n})=F_{\widehat{(M,P)}}(v_{a_{1}},\dots,v_{a_{2k}})$

where the tuple appearing in the parentheses on the right hand side is
obtained from that no the left hand side by putting two consecutive copies of
each vector $v_{i}\in i\in P$. The left hand side is a generating function
$\sum_{W}{\rm wt}_{N}(W)$ with the sum over 2-like subgraphs with doubled
edges at the boundary vertices indexed by $P$. Note that even if
$v_{1},\dots,v_{n}$ are generic vectors, the vectors
$v_{a_{1}},\dots,v_{a_{2k}}$ lie in a proper positroid subvariety $\Pi_{P}$.
We modify the plabic network $N$ to obtain a new plabic graph $N^{\prime}$ via
the following construction. Let $v_{i}$ be the unique interior vertex incident
to boundary vertex $i$ in $N$. We delete all such edges $iv_{i}$ in $N$ and
replace them by edges $a_{i}v_{i}$ in $N^{\prime}$. Using similar arguments to
the proof of Proposition 2.3, one may check that $N^{\prime}$ is a reduced
plabic graph for the positroid $\Pi_{P}$. Clearly, $2$-like graphs $W\subset
N$ with double edges at the edge $iv_{i}$ are in weight-preserving bijection
$2$-like subgraphs $W^{\prime}\subset N^{\prime}$ using the bijection which
decrements splits the multiplicity-two edge $iv_{i}$ into two multiplicity-one
edges $a_{j}v_{i}$ and $a_{j+1}v_{i}$ where $i=a_{j}=a_{j+1}$. ∎

We conclude by proving Theorems A and B.

###### Proof.

Let $i_{1},\dots,i_{k}$ be the entries in the first column of $T\in{\rm
SYT}(2^{k})$. By Theorem 3.5, $[W(\mathfrak{t}_{T})]$ is dual to the
noncrossing monomial $\Delta(M)\in\mathbb{C}[\rm Gr(2,2k)]^{\rm std}$. On the
other hand, by [6, Equation (5.17)], the web immanant $F_{M}$ is dual to
$[W_{M}]$, which equals $\prod_{j}(-1)^{i_{j}-1}\Delta(M)$ using Lemma 4.3.
(This is because ${\rm Web}_{2}(N)$ is a generating function for the
$[W_{M}]$’s rather than for the $\Delta(M)$’s, see [6, Definition 4.2 and
equation (5.6)].)

By comparing these, it follows that
$\prod_{j}(-1)^{i_{j}-1}[W(\mathfrak{t}_{T})]$ is a web immanant, i.e. a
canonical basis element by [16, Theorem 2.1 (2)].

The web invariant for a semistandard tableau is defined in terms of those for
standard tableaux by the evaluation equation (6.2). We have checked that the
same relationship holds between standard web immanants and semistandard ones
in Lemma 6.6. Thus each web invariant for semistandard tableaux is also a web
immanant and a a canonical basis element. This completes the proof of both of
our main theorems.

∎

Acknowledgments

This project began during at the 2014 Cluster Algebras AMS MRC where we
collaborated with Darlayne Addabbo, Eric Bucher, Sam Clearman, Laura Escobar,
Ian Le, Ningning Ma, Suho Oh, and Hannah Vogel on a problem proposed by Thomas
Lam and David Speyer. In particular, we had already conjectured Theorem B and
recognized the importance of Definition 1.7 at at that time, see [6,
Appendix]. We thank all of the above people for their contribution and thank
especially Ian Le for many conversations which influenced our approach to this
topic.

## References

  * [1] Cluster categories from Grassmannians and root combinatorics, K. Baur, D. Bogdanic, A. G. Elsener, Nagoya Mathematical Journal, 1–33, doi:10.1017/nmj.2019.14.
  * [2] K. Baur, D. Bogdanic, A. G. Elsener, J.-R. Li, Rigid Indecomposable Modules in Grassmannian Cluster Categories arXiv:2011.09227 (2020).
  * [3] K. Baur, D. Bogdanic, J.-R. Li, Construction of Rank 2 Indecomposable Modules in Grassmannian Cluster Categories, arXiv:2011.14176 (2021).
  * [4] S. Cautis, J. Kamnitzer, S. Morrison, Webs and Quantum Skew Howe Duality, Math. Ann. 360 (2014), no. 1–2, 351–390.
  * [5] S. Fomin, P. Pylyavskyy, Tensor diagrams and cluster algebras, Adv. Math. (2016) 717–787.
  * [6] C. Fraser, T. Lam, I. Le, From dimers to webs, Trans. Amer. Math. Soc.  371 (2019), no. 9, 6087–6124.
  * [7] I. B. Frenkel, M. G. Khovanov, Canonical bases in tensor products and graphical calculus for Uq(sl2), Duke Math. J.  87 (1997), no. 3, 409–480.
  * [8] C. Geiss, B. Leclerc, J. Schröer, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 825–876.
  * [9] S. Hopkins, M. Rubey, Promotion of Kreweras words, Selecta Math. (N.S.), 28 (1) (2022).
  * [10] H. Jenne, Combinatorics of the double-dimer model, Adv. Math. 392 (2021), Paper No. 107952, 85 pp.
  * [11] H. Jenne, G. Webb, B. Young, The combinatorial PT-DT correspondence, arXiv:2012.08484, (2020).
  * [12] J. Kamnitzer, Perfect bases in representation theory: three mountains and their springs, arXiv:2201.02289 (2022).
  * [13] M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485.
  * [14] M. Khovanov, G. Kuperberg, Web bases for sl(3) are not dual canonical, Pacific J. Math.  188 (1999), no. 1, 129–153.
  * [15] G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109–151.
  * [16] T. Lam, Cyclic Demazure modules and positroid varieties, Electron. J. Combin.  26 (2019), no. 2, Paper No. 2.28, 20 pp.
  * [17] , Dimers, webs, and positroids, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 633–656.
  * [18] L. Lamberti, Tensor diagrams and Chebyshev polynomials, Int. Math. Res. Not. 2020, no. 20, 7218–7278.
  * [19] I. Le and E. Yildirim, forthcoming.
  * [20] G. Lusztig, Introduction to quantum groups, Progress in Mathematics, 110. Birkhäuser Boston, Inc., Boston, MA, 1993. xii+341 pp.
  * [21] A. Postnikov, Total positivity, grassmannians, and networks, (2006), arXiv:math/0609764.
  * [22] B. Rhoades, Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A 117 (2010), no. 1, 38–76.
  * [23] B. Rhoades, M. Skandera, Temperley-Lieb immanants, Ann. Comb. 9 (2005), no. 4, 451–494.
  * [24] F. Qin, Dual canonical bases and quantum cluster algebras, arXiv:2003.13674, (2020).