2.1 Definition of ${\mathfrak {gl}\text {-}\mathbf {Web}}$

Let ${\mathbb {k}}$ be a commutative ring with identity. The definitions and results in this section are generally well known, and we record them for convenience.

Here and below, we will define combinatorial ${\mathbb {k}}$ -linear strict monoidal (super)categories by generators and relations. This method of construction is well known, and we only briefly describe how this works in our setting. See, for example, [Reference Kassel19, Section XII.1] or [Reference Turaev29, Section I.3] for a careful treatment in the classical case. The objects will be words from some set (e.g., ${\mathbb {Z}}_{\geq 1}$ or $\left\{\uparrow _{a}, \downarrow _{a} \mid a \in {\mathbb {Z}}_{\geq 1} \right\}$ ) with the monodial product given by concatenation of words. This set of objects will evidently generate the set of all objects under the monoidal product, and the empty word will be the monoidal unit object.

Morphisms will be given by providing a set of generating morphisms. A general morphism will be constructed from these generators (and identity morphisms) by a finite sequence of compositions, monoidal products, and ${\mathbb {k}}$ -linear combinations. Since composition is given by vertical concatenation and the monoidal product is given by horizontal concatenation, a general morphism will be a ${\mathbb {k}}$ -linear combination of diagrams with the same objects along the top and bottom, and where each diagram was obtained by a finite sequence of vertical and horizontal concatenations of generating morphisms and identities. To define the category, we impose relations on the morphisms. These relations are local in the sense that if two morphisms are identical other than in some small region where they differ by an imposed relation, then the morphisms are equal in the category. Finally, in the cases when we have a supercategory, the generating morphisms and defining relations will be homogenous in the ${\mathbb {Z}}_{2}$ -grading and this will provide the grading on the morphism spaces.

Definition 2.1.1 Let ${\mathfrak {gl}\text {-}\mathbf {Web}}$ denote the strict monoidal ${\mathbb {k}}$ -linear category given by generators and relations as follows. The objects are sequences of nonnegative integers. The morphisms are generated by the diagrams:

where $a,b \in {\mathbb {Z}}_{\geq 0}$ . We call these morphisms split and merge, respectively. The identity morphism of the object $(a_{1}, \dotsc , a_{k})$ will be depicted by k vertical strands labeled in order by $a_{1}, \dotsc , a_{k}$ .

On the morphisms in ${\mathfrak {gl}\text {-}\mathbf {Web}}$ , we impose the following relations for all $a,b,c \in {\mathbb {Z}}_{\geq 0}$ :

Web-associativity:

(2.1)

Rung swap:

(2.2)

Going forward, and in the relations defined above, we use the following conventions:

• • Strands labeled by “0” are to be deleted.

• • Diagrams containing a negatively labeled strand are to be read as zero.

• • We will sometimes choose to omit labels on strands when the label is clear from context.

For brevity, we also adopt the convention in calculations that when an equality follows from a previous result, this fact is indicated by placing the relevant equation number over the equals sign in question. We also adopt the convention that we sometimes write $0$ for the monoidal unit object.

2.2 Implied relations for ${\mathfrak {gl}\text {-}\mathbf {Web}}$

We first record a few relations which are implied by the defining relations of ${\mathfrak {gl}\text {-}\mathbf {Web}}$ . Many of the relations established in the remainder of Section 2 can be inferred from [Reference Brundan, Entova-Aizenbud, Etingof and Ostrik6, Reference Cautis, Kamnitzer and Morrison7] by using [Reference Brundan, Entova-Aizenbud, Etingof and Ostrik6, Remark 4.8 and Theorem 4.10], or may be viewed as analogues of those shown in [Reference Sartori and Tubbenhauer25, Section 2] in the case $q=1$ . Complete proofs of Lemmas 2.2.1, 2.2.2, 2.3.1, and 2.3.2 and Theorem 2.3.3 are also available in [Reference Davidson, Kujawa and Muth14].

Lemma 2.2.1 For all $a,b \in {\mathbb {Z}}_{\geq 0}$ , we have

Lemma 2.2.2 The following equalities hold in ${\mathfrak {gl}\text {-}\mathbf {Web}}$ :

for all admissible $a,b,c,r,r',r",s,s',s" \in {\mathbb {Z}}_{\geq 0}$ .

2.3 Braiding for ${\mathfrak {gl}\text {-}\mathbf {Web}}$

We next establish the category ${\mathfrak {gl}\text {-}\mathbf {Web}}$ admits a symmetric braiding.

For any $a,b \in {\mathbb {Z}}_{\geq 0}$ , we define the crossing morphism:

(2.3)

Lemma 2.3.1 For all $a,b \in {\mathbb {Z}}_{\geq 0}$ , we have

Using the crossing, we record an identity in ${\mathfrak {gl}\text {-}\mathbf {Web}}$ which will be useful in later calculations. This may be viewed as a special case of the Schur product rule (see [Reference Green17, equation (2.3b)], [Reference Brundan, Entova-Aizenbud, Etingof and Ostrik6, Theorem 4.10]).

Lemma 2.3.2 For all $a,b,c,d \in {\mathbb {Z}}_{\geq 0}$ such that $a+b = c+d$ , we have

The following theorem describes the basic relations involving the crossing morphism.

Theorem 2.3.3 For all $a,b,c \in {\mathbb {Z}}_{\geq 0}$ , we have

(2.4)

(2.5)

(2.6)

We define a crossing morphism $\boldsymbol {a} \otimes \boldsymbol {b}~{\rightarrow }~\boldsymbol {b} \otimes \boldsymbol {a}$ for objects $\boldsymbol {a}$ and $\boldsymbol {b}$ by the following diagram:

(2.7)

The following follows from Theorem 2.3.3.

3.1 Definition of ${\mathfrak {p}\text {-}\mathbf {Web}}$

From now on, we assume ${\mathbb {k}}$ is an integral domain where $2$ is invertible. For example, ${\mathbb {k}}$ could be ${\mathbb {Z}}[\frac {1}{2}]$ . We again define a diagrammatic ${\mathbb {k}}$ -linear monoidal supercategories by generators and relations as discussed in Section 2.1.

Definition 3.1.1 Let ${\mathfrak {p}\text {-}\mathbf {Web}}$ be the strict ${\mathbb {k}}$ -linear monoidal supercategory given by generators and relations as follows. The objects are all sequences of non-negative integers.

The generating morphisms:

for $a,b \in {\mathbb {Z}}_{\geq 0}$ . We call these morphisms split, merge, cap, and cup, respectively. The ${\mathbb {Z}}_2$ -grading is given by declaring splits and merges to have parity $\bar 0$ , and caps and cups to have parity $\bar 1$ . The identity morphism of the object $(a_{1}, \dotsc , a_{k})$ will be depicted by k vertical strands labeled in order by $a_{1}, \dotsc , a_{k}$ .

To describe the imposed relations, it will be convenient to first define an additional odd morphism,

(3.1)

which we call the antenna. Here and below, when we scale a diagram by an element of ${\mathbb {k}}$ , we often write the scalar in parentheses to make clear it is not an edge label.

The defining relations of ${\mathfrak {p}\text {-}\mathbf {Web}}$ are (2.1) and (2.2) along with the following relations for all $a,b \in {\mathbb {Z}}_{\geq 0}$ :

Straightening:

(3.2)

Antenna retraction:

(3.3)

Cap/rung swap:

(3.4)

(3.5)

Cup/rung swap:

(3.6)

(3.7)

3.2 Implied relations for ${\mathfrak {p}\text {-}\mathbf {Web}}$

We first record a few additional relations which are implied by the defining relations of ${\mathfrak {p}\text {-}\mathbf {Web}}$ .

Lemma 3.2.1 For all $a \in {\mathbb {Z}}_{\geq 2}$ , we have

Proof We have

as desired.

Lemma 3.2.2 For all $a \in {\mathbb {Z}}_{\geq 2}$ , we have

Proof Applying cap-rung swap relations (3.4) and (3.5) to push caps to the bottom of diagrams, we have

(3.8)

Therefore, we have

proving the first claim. The second claim follows using analogous arguments.

3.3 Braiding for ${\mathfrak {p}\text {-}\mathbf {Web}}$

For $a,b \in {\mathbb {Z}}_{\geq 0}$ , we define the crossing morphism $(a,b) {\rightarrow } (b,a)$ in ${\mathfrak {p}\text {-}\mathbf {Web}}$ as in (2.3). The goal of this section is to prove that the crossing morphism can be used to define a symmetric braiding on ${\mathfrak {p}\text {-}\mathbf {Web}}$ .

Theorem 3.3.1 For all $a \in {\mathbb {Z}}_{\geq 0} $ , we have

(3.9)

Proof We have

and

By Lemma 3.2.2 and (2.1), we have

which implies that

proving the first equality in (3.9). Now, using this equality, we may precompose with two cup morphisms to arrive at the equality

Applying the straightening relation (3.2) to both sides of this equation gives the second equality in (3.9).

Theorem 3.3.1 demonstrates that cups and caps are natural with respect to the crossing. Because the relations in ${\mathfrak {gl}\text {-}\mathbf {Web}}$ also hold in ${\mathfrak {p}\text {-}\mathbf {Web}}$ , the following is an immediate consequence of the above and Theorem 2.3.3.

3.4 Additional relations in ${\mathfrak {p}\text {-}\mathbf {Web}}$

The crossing morphisms allow for the following additional relations on ${\mathfrak {p}\text {-}\mathbf {Web}}$ .

Lemma 3.4.1 For every $a \in {\mathbb {Z}}_{\geq 0}$ , we have

(3.10)

Lemma 3.4.2 We have

Proof The first two relations are easily seen to hold by direct computation and applications of (3.4) to (3.7). The third is obtained by combining the first two:

where the first equality is deduced by putting a cap on top of the first relation, and the second equality is deduced by putting a cup under the second relation. Because $2$ is invertible in the ground ring ${\mathbb {k}}$ , the third relation follows.

Lemma 3.4.3 For all $a,b \in {\mathbb {Z}}_{\geq 0}$ , we have

Proof First, note that

and similarly,

Therefore, we have

as desired.

Lemma 3.4.4 For all $a,b,c \in {\mathbb {Z}}_{\geq 0}$ , we have

Proof For the first equality, we have

as desired. The second equality is proved in a similar fashion.

Lemma 3.4.5 The following relation holds in ${\mathfrak {p}\text {-}\mathbf {Web}}$ :

Proof We have

proving the first claim. The second claim is similar. For the third, we have

completing the proof.

3.5 A basis for morphism spaces in ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$

In this section, we construct ${\mathbb {k}}$ -spanning sets for the morphism spaces in ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$ . In Section 6.3, we will show that these are in fact bases. The bases themselves are diagrammatically analogous to bases defined in a different setting in [Reference Stroppel and Webster27, Section 5].

We will write multiple splits and merges in the form

(3.11)

where the diagram should be interpreted as n vertically composed splits, or merges, respectively. By (2.1), the resulting morphism is independent of the split (or merge) order. It will also be convenient to define

(3.12)

Given a matrix A, let us write $A^{T}$ for the transpose. For any $\boldsymbol {a} \in {\mathbb {Z}}_{\geq 0}^t$ , $\boldsymbol {b} \in {\mathbb {Z}}_{\geq 0}^u$ , let $\chi (\boldsymbol {a}, \boldsymbol {b})$ be the set of tuples $(A,B,C,D)$ , such that

$$ \begin{align*} A \in \text{Mat}_{t \times t}(\{0,1\}), \qquad B \in \text{Mat}_{u \times u}(\{0,1\}), \qquad C \in \text{Mat}_{t \times u}({\mathbb{Z}}_{\geq 0}), \qquad D \in \{0,1\}^t, \end{align*} $$

$$ \begin{align*} A^{T} = A, \qquad B^{T} = B, \qquad A_{ii} = 0\,\, \text{for all } i=1, \ldots, t, \qquad B_{ii} = 0\,\,\text{for all } i=1, \ldots, u, \end{align*} $$

$$ \begin{align*} 2D_{i} + \sum_{j=1}^tA_{ij} + \sum_{j=1}^u C_{ij} = a_i \qquad \text{for all } i =1, \ldots, t, \end{align*} $$

$$ \begin{align*} \sum_{i=1}^uB_{ij} + \sum_{i=1}^t C_{ij} = b_j \qquad \text{for all } j =1, \ldots, u. \end{align*} $$

For any $(A,B,C,D) \in \chi (\boldsymbol {a}, \boldsymbol {b})$ , we define an associated element $\xi ^{(A,B,C,D)} \in {\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(\boldsymbol {a}, \boldsymbol {b})$ via

where X is any diagram composed only of crossings, cups, and caps, where no cup occurs below any cap, and in which:

• • the strands labeled by $A_{ij}$ and $A_{ji}$ meet in a cap in X;

• • the strands labeled by $B_{ij}$ and $B_{ji}$ meet in a cup in X; and

• • the strand labeled by $C_{ij}$ at the bottom of X meets the strand labeled by $C_{ij}$ at the top of X.

All such choices for X are equivalent up to sign because of Theorem 2.3.3.

It should be noted that the method of using splits and merges to “explode” or “collapse” the bottom and top of morphisms as done here can be found elsewhere in the literature (see, e.g., [Reference Rose and Tubbenhauer24]).

Example 3.5.1 Let $\boldsymbol {a} = (9,4,8)$ , $\boldsymbol {b} = (9,6)$ , and set

$$ \begin{align*} A= \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ 1 & 1 & 0 \end{bmatrix} \qquad B= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \qquad C= \begin{bmatrix} 2& 4 \\ 3 & 0 \\ 3 & 1 \end{bmatrix} \qquad D= \begin{bmatrix} 1 & 0 & 1 \end{bmatrix}. \end{align*} $$

Then $(A,B,C,D) \in \chi (\boldsymbol {a}, \boldsymbol {b})$ , and

Proposition 3.5.2 The set

$$ \begin{align*} \mathscr{B}:=\left\{ \left. \xi^{(0,0,C,0)}\; \right| (0,0,C,0) \in \chi(\boldsymbol{a}, \boldsymbol{b})\right\} \end{align*} $$

is a ${\mathbb {k}}$ -spanning set for ${\mathrm { Hom}}_{{\mathfrak {gl}\text {-}\mathbf {Web}}}(\boldsymbol {a},\boldsymbol {b})$ .

The set

$$ \begin{align*} \mathscr{B}:=\left\{ \left. \xi^{(A,B,C,D)}\; \right| (A,B,C,D) \in \chi(\boldsymbol{a}, \boldsymbol{b}) \right\} \end{align*} $$

is a ${\mathbb {k}}$ -spanning set for ${\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(\boldsymbol {a},\boldsymbol {b})$ .

Proof We will focus primarily on the statement for the spanning set of morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}$ . The statement for ${\mathfrak {gl}\text {-}\mathbf {Web}}$ is simpler due to the lack of cups and caps, and should be considered known (see, e.g., [Reference Stroppel and Webster27, Theorem 3.11] or [Reference Brundan, Entova-Aizenbud, Etingof and Ostrik6, Lemma 4.9]).

Let f be a diagram in ${\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(\boldsymbol {a},\boldsymbol {b})$ . By inducting on the number of “out of place” parts, we may apply the defining relations (2.1) and (3.7) of ${\mathfrak {p}\text {-}\mathbf {Web}}$ , Theorem 2.3.3, and Lemma 2.3.2, to rewrite f as a linear combination of diagrams consisting of splits, merges, cups, caps, antennas, and crossings, where:

• • no merge occurs below any split;

• • no cup occurs below any cap;

• • no crossing occurs above any merge or below any split;

• • no antenna occurs above any merge, cup, cap, or crossing.

Any diagram that satisfies all of the above is equivalent to a constant multiple of some diagram with the following form:

(3.13)

for some labels $d_i, x_j^{(i)}, y_j^{(i)} \in {\mathbb {Z}}_{\geq 0}$ , and X is a diagram composed only of crossings, cups, and caps, where no cup occurs below any cap.

Now, we consider (3.13). Note the following:

1. (1) If two strands which split from $a_i$ meet in a cap in X, then by (3.4), the diagram can be rewritten by adding one to $d_i$ , deleting the strand, and multiplying by 2.

2. (2) If two strands which merge in $b_i$ meet in a cup in X, then by (3.7), the diagram is zero.

3. (3) If for some $i \neq j$ , there is more than one instance of a strand which splits from $a_i$ and a strand which splits from $a_j$ meeting in a cap in X, then the diagram is zero by Lemma 3.4.5.

4. (4) If for some $i \neq j$ , there is more than one instance of a strand which merges into $b_i$ and a strand which merges into $b_j$ meeting in a cup in X, then the diagram is zero by Lemma 3.4.5.

5. (5) If $d_i>1$ , then the diagram is zero by Lemma 3.4.5.

6. (6) If there is more than one strand in X which splits from $a_i$ and merges into $b_j$ , then by (2.2), the diagram can be written with a single strand which splits from $a_i$ and merges into $b_j$ , multiplied by some constant.

After rewriting as above, we have via Lemma 2.3.1 that (3.13) is equivalent to a constant multiple of some diagram of the form $\xi ^{(A,B,C,D)}$ , completing the proof for ${\mathfrak {p}\text {-}\mathbf {Web}}$ .

An entirely analogous argument applies for ${\mathfrak {gl}\text {-}\mathbf {Web}}$ . Since there are no cups, caps, or antennas, it is easier and we leave it to the reader.

3.6 Generating sets for the morphism spaces of ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$

In this section, we describe generating sets for the morphism spaces of ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$ using only the operations of composition and ${\mathbb {k}}$ -linear combinations, and not the monoidal product. These generators (and the relations among them, given in Lemma 3.7.1) are used to establish a Howe duality in [Reference Davidson, Kujawa and Muth13].

Let ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ be the full subcategory of ${\mathfrak {p}\text {-}\mathbf {Web}}$ consisting of objects ${\textbf {a}} \in {\mathbb {Z}}^m_{\geq 0}$ . We emphasize that the monoidal product in ${\mathfrak {p}\text {-}\mathbf {Web}}$ does not preserve the subcategory ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ . Hence, ${\mathfrak {p}\text {-}\mathbf {Web}}_{m}$ is a supercategory and does not inherit a monoidal structure. For $t \in {\mathbb {Z}}_{\geq 0}$ , ${\textbf {a}} \in {\mathbb {Z}}_{\geq 0}^m$ , $1 \leq r < s \leq m$ , we define the following morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ :

For $1 \leq u \leq m$ , we define the additional morphism

where again the dot denotes a combination of a split and antenna, as in (3.12).

To remove some of the clutter in the calculations which follow, we will sometimes write products for compositions (e.g., $fg = f \circ g$ ) and will occasionally omit the label ${\textbf {a}}$ when the domain is clear from context (e.g., writing $e_{[r,s]}^{(t)}$ instead of $e_{[r,s],{\textbf {a}}}^{(t)}$ ). Let us write $1_{{\textbf {a}}}$ for the identity morphism of ${\textbf {a}}$ . Pre- or post-composing by these gives a convenient alternate method for specifying the domain or range of a morphism. For example, $e_{[r,s],{\textbf {a}}}^{(t)}=e_{[r,s]}^{(t)} 1_{{\textbf {a}}}$ .

In order to establish a generating set for the morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ , we need the following technical lemma. Because we are no longer in a monoidal supercategory, we only use composition and ${\mathbb {k}}$ -linear combinations when generating morphisms in this category.

Lemma 3.6.1 Label the set of morphisms:

$$ \begin{align*} Y_{\{e,f\}}(m):= \left\{ \left. e_{[r,s],{\textbf{a}}}^{(t)}, f_{[r,s],{\textbf{a}}}^{(t)} \; \right| t \in {\mathbb{Z}}_{\geq 0}, 1 \leq r < s \leq m, {\textbf{a}} \in {\mathbb{Z}}_{\geq 0}^m \right\}. \end{align*} $$

Let $S_{\{e,f\}}(m)$ represent the ${\mathbb {k}}$ -linear subcategory of ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ consisting of all objects in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ , with morphisms generated by $Y_{\{e,f\}}(m)$ . Then, for all ${\textbf {a}}, {\textbf {b}} \in {\mathbb {Z}}_{\geq 0}^m$ , $k \in {\mathbb {Z}}_{\geq 0}$ , and $i,j \in [1,m]$ , and morphisms $y \in S_{\{e,f\}}(m)$ , the morphism

is also in $S_{\{e,f\}}(m)$ .

Proof We may assume that y is itself a composition of u many morphisms in $Y_{\{e,f\}}(m)$ . Our argument will go by nested induction on k and u. First, note that if $k=0$ , the claim holds trivially, so we now fix $k>0$ and assume that the claim holds for all y and $k'<k$ .

Assume $u=0$ . If $i=j$ , then M is some multiple of the identity morphism by (2.2). If $i>j$ , then $M=e_{j,i}^{(k)}$ , and if $i<j$ , then $M=f_{i,j}^{(k)}$ . This proves the claim when $u=0$ . We now assume that $u>0$ and that the claim holds for all $u'<u$ .

As $u>0$ , we may write $y=y'e_{[r,s]}^{(t)}$ or $y=y'f_{[r,s]}^{(t)}$ for some $r,s,t$ . We will assume the former, as the latter case is similar. If $r \neq i$ , then $e_{[r,s]}^{(t)}$ moves freely below past the split on the ith strand in M. Then the induction assumption on u may be used to complete the proof of the claim.

So we now assume that $y=y'e_{[i,s]}^{(t)}$ , so that M may be written:

(3.14)

Note that for clarity here we are omitting strands between the ith and sth strands. Using Corollary 3.3.2, any morphisms on strands between the ith and sth strands may be pulled all the way to the right side of the morphism M by introducing crossings. For this reason, any morphisms between the ith and sth strands will not affect our calculations and can be safely ignored.

Using (2.3), M may be rewritten as a linear combination of diagrams of the form

(3.15)

Using (2.1) and (2.2), any diagram as in (3.15) can be written as a linear combination of diagrams of the form

(3.16)

An application of (2.2) allows us to write any diagram as in (3.16) as a linear combination of diagrams of the form

(3.17)

where $k'+k"=k$ . If $k' = 0$ or $k"=0$ , then the claim follows by the induction assumption on u. If $k',k">0$ , then applying the induction assumption on k to the $k'$ strand, and subsequently to the $k"$ strand proves the claim, and completes the proof.

Given $\boldsymbol {a} = (a_{1}, \dotsc , a_{r}) \in {\mathbb {Z}}^{r}$ , we write $|\boldsymbol {a} | = \sum _{i=1}^{r} |a_{i}|$ .

Proof Let ${\textbf {a}},{\textbf {b}} \in {\mathbb {Z}}^m_{\geq 0}$ , and let $\xi :=\xi ^{(0,0,C,0)} \in \mathscr {B}$ be an element in ${\mathrm {Hom}}_{{\mathfrak {gl}\text {-}\mathbf {Web}}}({\textbf {a}},{\textbf {b}})$ as in Proposition 3.5.2. We show by inducting on $n=|{\textbf {a}}|+|{\textbf {b}}|$ that $\xi $ belongs to the set $S_{\{e,f\}}(m)$ from Lemma 3.6.1. Since by Proposition 3.5.2 such elements $\xi $ span ${\mathrm {Hom}}_{{\mathfrak {gl}\text {-}\mathbf {Web}}}({\textbf {a}},{\textbf {b}})$ , this will prove the lemma.

If $n=0$ , then $\xi $ is the identity morphism. Thus, we may assume that $n>0$ , and that the claim holds for all $n'<n$ . If $C=0$ , then $\xi $ is the identity morphism, so assume $C_{ij}>0$ for some $i,j$ . Then, using (2.1), we may write

for some basis element $\xi ' \in {\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}({\textbf {a}}',{\textbf {b}}')$ , where $|{\textbf {a}}'|<|{\textbf {a}}|$ and $|{\textbf {b}}'|<|{\textbf {b}}|$ . Applying the induction assumption on n, we have that $\xi ' \in S_{\{e,f\}}(m)$ . Then, applying Lemma 3.6.1, it follows that $\xi \in S_{\{e,f\}}(m)$ , as desired.

Theorem 3.6.3 Morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ are generated under composition by the set $\{b_{[1],a} \mid a \in {\mathbb {Z}}_{\geq 0}\}$ when $m =1$ , and by

$$ \begin{align*} Y(m):= \left\{ \left. e_{[i,i+1],{\textbf{a}}}^{(t)}, f_{[i,i+1],{\textbf{a}}}^{(t)}, b_{[1,2],{\textbf{a}}}, c_{[1,2],{\textbf{a}}} \; \right| t \in {\mathbb{Z}}_{\geq 0}, 1 \leq i < m, {\textbf{a}} \in {\mathbb{Z}}_{\geq 0}^m \right\}, \end{align*} $$

when $m\geq 2$ .

Proof The statement for $m=1$ follows directly from Proposition 3.5.2, so assume $m \geq 2$ . First, let

$$ \begin{align*} Y'(m):= \left\{ \left. e_{[r,s],{\textbf{a}}}^{(t)}, f_{[r,s],{\textbf{a}}}^{(t)}, b_{[r,s],{\textbf{a}}}, c_{[r,s],{\textbf{a}}}, b_{[u],{\textbf{a}}} \; \right| t \in {\mathbb{Z}}_{\geq 0}, 1 \leq r < s \leq m, u \in [1,m], {\textbf{a}} \in {\mathbb{Z}}_{\geq 0}^m \right\}\!. \end{align*} $$

We first prove a preliminary claim that morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ are generated under composition by $Y'(m)$ . Write $S_m$ (resp. $S^{\prime }_m$ ) for the subcategory of ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ generated by morphisms in $Y(m)$ (resp. $Y'(m)$ ). Let ${\textbf {a}},{\textbf {b}} \in {\mathbb {Z}}^m_{\geq 0}$ . By Proposition 3.5.2, ${\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}({\textbf {a}},{\textbf {b}})$ is spanned by elements of the form $\xi :=\xi ^{(A,B,C,D)} \in \mathscr {B}$ . We show by induction on $n=|{\textbf {a}}|+|{\textbf {b}}|$ that $\xi \in S^{\prime }_m$ .

If $n=0$ , then $\xi $ is the identity morphism. Thus, we may assume $n>0$ , and that the claim holds for all $n'<n$ . Note the following:

• • If $A_{ij}>0$ for any $i<j$ , then by (2.1), $\xi = \xi 'b_{[i,j],{\textbf {a}}}$ for some basis element ${\xi ' \in {\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}({\textbf {a}}', {\textbf {b}})}$ , where $|{\textbf {a}}'|<|{\textbf {a}}|$ .

• • If $B_{ij}>0$ for any $i<j$ , then by (2.1), $\xi = c_{[i,j],{\textbf {b}}'}\xi '$ for some basis element ${\xi ' \in {\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}({\textbf {a}}, {\textbf {b}}')}$ , where $|{\textbf {b}}'|<|{\textbf {b}}|$ .

• • If $D_{i}>0$ for any i, then by (2.1), $\xi = \xi 'b_{[i],{\textbf {a}}}$ for some basis element ${\xi ' \in {\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}({\textbf {a}}', {\textbf {b}})}$ , where $|{\textbf {a}}'|<|{\textbf {a}}|$ .

In any of these cases, applying the induction assumption on n to $\xi '$ completes the proof. Therefore, we may assume $A=B=D=0$ . Then the preliminary claim follows by Lemma 3.6.2, so morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ are generated under composition by $Y'(m)$ .

Since $S_m$ contains $e_{[i,i+1],{\textbf {a}}}^{(t)}$ and $f_{[i,i+1],{\textbf {a}}}^{(t)}$ for all $1 \leq i < m, {\textbf {a}} \in {\mathbb {Z}}_{\geq 0}^m$ , it follows by (2.3) that $S_m$ contains all crossing morphisms which transpose neighboring strands. Then we have that $e_{[r,s],{\textbf {a}}}^{(t)}$ , $f_{[r,s],{\textbf {a}}}^{(t)}$ , $b_{[r,s],{\textbf {a}}}$ , $c_{[r,s],{\textbf {a}}}$ belong to $S_m$ for all $1 \leq r < s \leq m$ , ${\textbf {a}} \in {\mathbb {Z}}^m_{\geq 0}$ , as one may generate these elements by pre- and post-composing the morphisms $e_{[1,2],{\textbf {a}}}^{(t)}, f_{[1,2],{\textbf {a}}}^{(t)}, b_{[1,2],{\textbf {a}}}, c_{[1,2],{\textbf {a}}}$ with sequences of crossing morphisms. Finally, we have that $b_{[u],{\textbf {a}}} = (1/2) [ b_{[u,u+1]}, e_{[u,u+1]}^{(1)}]1_{{\textbf {a}}}$ for $1 \leq u < m$ and $b_{[m],{\textbf {a}}} = (1/2) [ b_{[m-1,m]}, f_{[m-1,m]}^{(1)}]1_{{\textbf {a}}}$ by (3.4) and (3.5), so it follows that $S_m' \subseteq S_m$ , completing the proof.

Corollary 3.6.4 If ${\mathbb {k}}$ is a field of characteristic zero, then morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ are generated under composition by the set $\{b_{[1],a} \mid a \in {\mathbb {Z}}_{\geq 0}\}$ when $m =1$ , and by

$$ \begin{align*} Y_0(m):= \left\{ \left. e_{[i,i+1],{\textbf{a}}}^{(1)}, f_{[i,i+1],{\textbf{a}}}^{(1)}, b_{[1,2],{\textbf{a}}}, c_{[1,2],{\textbf{a}}}\; \right| t \in {\mathbb{Z}}_{\geq 0}, 1 \leq i < m, {\textbf{a}} \in {\mathbb{Z}}_{\geq 0}^m \right\}, \end{align*} $$

when $m \geq 2$ .

3.7 Relations for morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$

We now establish a number of relations which hold among the generators in $Y(m)$ . While not utilized in this paper, these relations will be key in establishing the Howe duality result in [Reference Davidson, Kujawa and Muth13].

Lemma 3.7.1 The following relations hold in ${\mathfrak {p}\text {-}\mathbf {Web}}_m$ , for all valid $1 \leq i, j \leq m$ , and ${\textbf {a}} \in {\mathbb {Z}}_{\geq 0}^m$ .

(3.18) $$ \begin{align}&[e_{[i,i+1]}^{(1)}, f_{[j,j+1]}^{(1)}]1_{\mathbf{a}} = \delta_{i,j}(a_i - a_{i+1})1_{\mathbf{a}};\end{align}$$

(3.19) $$ \begin{align} &[e_{[i,i+1]}^{(1)}, e_{[j,j+1]}^{(1)}]1_{\textbf{a}} = 0 &\text{if } j \neq i \pm 1; \end{align} $$

(3.20) $$ \begin{align} &[e_{[i,i+1]}^{(1)},[e_{[i,i+1]}^{(1)},e_{[j,j+1]}^{(1)}]1_{\textbf{a}} = 0 &\text{if }j = i \pm 1; \end{align} $$

(3.21) $$ \begin{align} &[f_{[i,i+1]}^{(1)}, f_{[j,j+1]}^{(1)}]1_{\textbf{a}} = 0 &\text{if } j \neq i \pm 1; \end{align} $$

(3.22) $$ \begin{align} &[f_{[i,i+1]}^{(1)},[f_{[i,i+1]}^{(1)},f_{[j,j+1]}^{(1)}]1_{\textbf{a}} = 0 &\text{if }j = i \pm 1; \end{align} $$

(3.23) $$ \begin{align} &[b_{[i,i+1]},b_{[j,j+1]}]1_{\textbf{a}} =0; \end{align} $$

(3.24) $$ \begin{align} &[c_{[i,i+1]},c_{[j,j+1]}]1_{\textbf{a}} =0; \end{align} $$

(3.25) $$ \begin{align} &[b_{[i,i+1]},c_{[j,j+1]}]1_{\textbf{a}} = \begin{cases} (a_i - a_{i+1})1_{\textbf{a}}, & \text{if } j=i;\\ [e_{[i-1,i]},e_{[i,i+1]}]1_{\textbf{a}}, & \text{if } j=i-1;\\ [f_{[i,i+1]},f_{[i+1,i+2]}]1_{\textbf{a}}, &\text{if } j=i+1;\\ 0, &\text{otherwise}; \end{cases} \end{align} $$

(3.26) $$ \begin{align} &[b_{[i,i+1]},e_{[j,j+1]}^{(1)}]1_{\textbf{a}} = 0 & \text{if }j \neq i,i+1; \end{align} $$

(3.27) $$ \begin{align} &[b_{[i,i+1]},f_{[j,j+1]}^{(1)}]1_{\textbf{a}} = 0 & \text{if }j \neq i,i-1; \end{align} $$

(3.28) $$ \begin{align} &[b_{[i,i+1]},e_{[i,i+1]}^{(1)}]1_{\textbf{a}} = 2b_{[i+1]} = [b_{[i+1,i+2]},f_{[i+1,i+2]}^{(1)}]1_{\textbf{a}}; \end{align} $$

(3.29) $$ \begin{align} &[b_{[i,i+1]},e_{[i+1,i+2]}^{(1)}]1_{\textbf{a}} = b_{[i,i+2]} = [b_{[i+1,i+2]},f_{[i,i+1]}^{(1)}]1_{\textbf{a}}; \end{align} $$

(3.30) $$ \begin{align} &[e_{[j,j+1]}^{(1)},c_{[i,i+1]}]1_{\textbf{a}} = 0 & \text{if }j \neq i-1; \end{align} $$

(3.31) $$ \begin{align} &[f_{[j,j+1]}^{(1)},c_{[i,i+1]}]1_{\textbf{a}} = 0 & \text{if }j \neq i+1; \end{align} $$

(3.32) $$ \begin{align} &[e_{[i,i+1]}^{(1)},c_{[i+1,i+2]}]1_{\textbf{a}} = c_{[i,i+2]} = [f_{[i+1,i+2]}^{(1)},c_{[i,i+1]}]1_{\textbf{a}}; \end{align} $$

(3.33) $$ \begin{align} &[[b_{[i,i+1]},e_{[i,i+1]}^{(1)}],e_{[j,j+1]}^{(1)}]1_{\textbf{a}} = \begin{cases} 2b_{[i+1,i+2]} 1_{\textbf{a}}, & \text{if }j=i+1;\\ 0,&\text{otherwise}; \end{cases} \end{align} $$

(3.34) $$ \begin{align} &[[b_{[1,2]},e_{[1,2]}^{(1)}], f_{[1,2]}^{(1)}] 1_{\textbf{a}}= 2b_{[1,2]}1_{\textbf{a}}; \end{align} $$

(3.35) $$ \begin{align} &[[b_{[1,2]},f_{[1,2]}^{(1)}], f_{[j,j+1]}^{(1)}] 1_{\textbf{a}}= 0; \end{align} $$

(3.36) $$ \begin{align} &[[c_{[i+1,i]},e_{[i,i+1]}^{(1)}],e_{[j,j+1]}^{(1)}] 1_{\textbf{a}} = \begin{cases} c_{[i,i+1]}1_{\textbf{a}}, & \text{if }j=i+1;\\ 0, & \text{if }j \neq i+1,i-1. \end{cases} \end{align} $$

4.1 Definition of ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$

We now introduce an oriented version of ${\mathfrak {p}\text {-}\mathbf {Web}}$ . In this section, we continue to assume ${\mathbb {k}}$ is an integral domain in which $2$ is invertible. It will again be a diagrammatic supercategory given by generators and relations as outlined in Section 2.1. Because many of the constructions and arguments in this section are similar to those given in the previous section, so we will sometimes be brief in explanations.

Definition 4.1.1 The category ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ is the ${\mathbb {k}}$ -linear strict monoidal supercategory defined as follows. Objects are words (including the empty word) from the set

$$ \begin{align*} \left\{ \uparrow_k, \downarrow_k \mid k \in {\mathbb{Z}}_{\geq 1}\right\}. \end{align*} $$

The morphisms are generated as a ${\mathbb {k}}$ -linear monoidal supercategory by the diagrams:

for all $a,b \in {\mathbb {Z}}_{\geq 0}$ . We call these morphisms upward split, upward merge, leftward cap, leftward cup, tag-in, tag-out, and rightward crossing, respectively. The parity is given by declaring the tag-in and tag-out morphisms to be odd and the rest of the generating morphisms to be even.

To describe the defining relations, it will be convenient to first set a diagrammatic shorthand for certain additional morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ . We define morphisms:

which we call upward cap, upward cup, and upward antenna, respectively. We also define, for all $a,b \in {\mathbb {Z}}_{\geq 0}$ , rightward cap and rightward cup morphisms:

For all $a,b \in {\mathbb {Z}}_{\geq 0}$ , define the upward crossing

as in (2.3), with all strands oriented upward. We then define the leftward crossing by composing this with the leftward cap and cup morphisms:

With this notation established, we can now give the defining relations for ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ . The defining relations of ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ are:

Up-arrow relations. Relations (2.1) and (3.7) hold in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ , where we interpret the diagrams as having all strands oriented upward.

Leftward straightening. For all $a \in {\mathbb {Z}}_{\geq 0}$ we have

(4.1)

Left/right crossing inversion. For all $a,b \in {\mathbb {Z}}_{\geq 0}$ , we have

(4.2)

Bubble annihilation. For all $a \in {\mathbb {Z}}_{>0}$ we have

(4.3)

We remark that including the rightward crossing generator along with the left/right crossing inversion relation is equivalent to imposing the relation that the leftward crossing is invertible. While this latter approach is sometimes used in the literature, we chose to make the inverse morphism explicitly part of the definition.

Going forward, it will be convenient to sometimes write $\uparrow _{0}$ or $\downarrow _{0}$ for the empty word (i.e., the monoidal unit object).

4.2 Additional morphisms

We define downward splits and downward merges by

We define downward crossings like so:

4.3 Implied relations for ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$

The following theorem establishes a number of additional relations which follow from the defining relations of ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ . In particular, they show that diagrams that are the same as oriented graphs (which may have edges with tag-in and tag-out diagrams) are equal, up to a sign. In particular, up to a sign, tag-in and tag-out morphisms move freely through crossings and along strands.

Theorem 4.3.1 The following equalities hold in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ for all $a,b,c \in \mathbb {Z}_{\geq 0}$ and all admissible strand orientations:

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

Proof Relations (4.4)–(4.7) hold when all strands are oriented upward, as shown in Theorem 2.3.3. Using this fact, together with (4.1)–(4.3), it is a routine exercise then to prove that the equalities (4.4)–(4.12) hold for all admissible orientations. The equalities (4.13)–(4.17) can be seen to hold thanks to (4.4)–(4.12), Lemma 4.5.1, and Theorem 2.3.3, after noting that

(4.18)

which completes the proof.

For a nonnegative integer a, it will be convenient to adopt the notation $|_{a}:= \uparrow _{a}$ and $|_{-a}:= \downarrow _{a}$ . More generally, for $r \in {\mathbb {Z}}_{\geq 0}$ and $\boldsymbol {a} =(a_1, \ldots , a_{r}) \in {\mathbb {Z}}^{r}$ , we will write

$$ \begin{align*} |_{\boldsymbol{a}}:= |_{a_{1}} \cdots |_{a_{r}} \end{align*} $$

as a shorthand for the latter as an object of ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ . For example, $|_{(6,2,-9)} = \uparrow _{6}\uparrow _{2}\downarrow _{9}$ .

The up, down, left, and right crossings can be used to define crossing morphisms,

$$ \begin{align*} \beta_{|_a |_b}: |_a |_b {\rightarrow} |_b |_a, \end{align*} $$

for arbitrary $a,b \in {\mathbb {Z}}$ . Just as with ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$ , we can use these morphisms to make oriented versions of (2.7) and to verify these make ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ into a symmetric braided monoidal supercategory.

The relations (4.1), (4.10) show that $\uparrow _{a}$ and $\downarrow _{a}$ are left and right duals to each other with the cups and caps as the evaluation and coevaluation morphisms. More generally, using “rainbows” constructed from leftward and rightward caps and cups, we can also construct evaluation and coevaluation morphisms for general objects in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ . For example, the evaluation and coevaluation morphisms for $\downarrow _{a}\uparrow _{b}\downarrow _{c}$ are

Using these, we see that ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ is in fact a rigid category. Altogether, we have the following result.

4.4 Isomorphic morphism spaces in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$

We next remind the reader of well-known arguments (e.g., [Reference Etingof, Gelaki, Nikshych and Ostrik16, Proposition 2.10.8]) which use the existence of the braiding morphisms in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ to define isomorphisms between various morphism spaces. Entirely analogous results obviously hold by the same arguments for ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$ .

The symmetric group $\mathfrak {S}_r$ acts on ${\mathbb {Z}}^r$ by place permutation:

$$ \begin{align*} \sigma \cdot {\textbf{a}} = \sigma \cdot (a_1, \ldots, a_r) = (a_{\sigma^{-1}(1)}, \ldots, a_{\sigma^{-1}(r)}). \end{align*} $$

The braiding morphism defines an associated invertible morphism

$$ \begin{align*} \beta_{\sigma,|_{\textbf{a}}} \in {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(|_{\textbf{a}}, |_{\sigma \cdot {\textbf{a}}}), \end{align*} $$

where $\beta ^{-1}_{\sigma , |_{\textbf {a}}} = \beta _{\sigma ^{-1}, |_{\sigma \cdot {\textbf {a}}}} $ . More generally, for $r_1,r_2 \in {\mathbb {Z}}_{\geq 0}$ , $\boldsymbol {a} \in {\mathbb {Z}}^{r_1}$ , $\boldsymbol {b} \in {\mathbb {Z}}^{r_2}$ , $\sigma \in \mathfrak {S}_{r_1}$ , and $\omega \in \mathfrak {S}_{r_2}$ , we have an isomorphism of morphism spaces:

(4.19) $$ \begin{align} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(|_{\boldsymbol{a}}, |_{\boldsymbol{b}}) \xrightarrow{\sim} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(|_{ \sigma \cdot \boldsymbol{a}}, |_{\omega \cdot \boldsymbol{b}}), \end{align} $$

given by

$$ \begin{align*} f \mapsto \beta_{\omega, |_{\boldsymbol{b}}} \circ f \circ \beta_{\sigma^{-1}, |_{\sigma \cdot \boldsymbol{a}}}. \end{align*} $$

Let $\omega _0 \in \mathfrak {S}_r$ be the longest element, so that $\omega _{0}\cdot (a_1, \ldots ,a_r) = (a_r, \ldots , a_1)$ . Then, for $\boldsymbol {a} \in {\mathbb {Z}}_{\geq 0}^{r_1}$ , $\boldsymbol {b} \in {\mathbb {Z}}_{\geq 0}^{r_2}$ , $\boldsymbol {c} \in {\mathbb {Z}}_{\geq 0}^{r_3}$ , and $\boldsymbol {d} \in {\mathbb {Z}}_{\geq 0}^{r_4}$ , we have an isomorphism of Hom spaces:

(4.20) $$ \begin{align} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(\downarrow_{\boldsymbol{a}} \uparrow_{\boldsymbol{b}}, \uparrow_{\boldsymbol{c}} \downarrow_{\boldsymbol{d}}) \xrightarrow{\sim} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_\uparrow}(\uparrow_{\boldsymbol{b}} \uparrow_{\omega_0\boldsymbol{d}} ,\uparrow_{\omega_0 \boldsymbol{a}} \uparrow_{\boldsymbol{c}}), \end{align} $$

given by

with inverse given by an entirely similar map, thanks to (4.1).

Let ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow }$ (resp. ${\mathfrak {p}\text {-}\mathbf {Web}}_{\downarrow }$ ) be the full subcategory of ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ consisting of objects of the form $\uparrow _{\boldsymbol {a}}:=\uparrow _{a_1} \cdots \uparrow _{a_r}$ (resp. $\downarrow _{\boldsymbol {a}}:= \downarrow _{a_1} \cdots \downarrow _{a_r}$ ) for all $r \in {\mathbb {Z}}_{\geq 0}$ and $\boldsymbol {a} =(a_1, \ldots , a_r) \in {\mathbb {Z}}_{\geq 0}^r$ . Combining (4.19) and (4.20) yields the following lemma.

Lemma 4.4.1 Let $\boldsymbol {a} \in {\mathbb {Z}}^{r_1}$ and $\boldsymbol {b} \in {\mathbb {Z}}^{r_2}$ . Then there exists $\boldsymbol {c} \in {\mathbb {Z}}_{\geq 0}^{r_3}$ , $\boldsymbol {d} \in {\mathbb {Z}}_{\geq 0}^{r_4}$ such that $|\boldsymbol {c}| + |\boldsymbol {d}| = |\boldsymbol {a}| + |\boldsymbol {b}|$ and there is a parity preserving isomorphism of morphism spaces

$$ \begin{align*} \Phi:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(|_{\boldsymbol{a}}, |_{\boldsymbol{b}}) \xrightarrow{\sim} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow}}(\uparrow_{\boldsymbol{c}}, \uparrow_{\boldsymbol{d}}) \end{align*} $$

given by

$$ \begin{align*} \Phi:f \mapsto \varphi_2 \circ f \circ \varphi_1, \end{align*} $$

for some invertible morphisms

$$ \begin{align*} \varphi_1 \in {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(\uparrow_{\boldsymbol{c}}, |_{\boldsymbol{a}}) \qquad \text{and} \qquad \varphi_2 \in {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(|_{\boldsymbol{b}}, \uparrow_{\boldsymbol{d}}). \end{align*} $$

Given a supercategory $\mathcal{B}$ , let $\mathcal{B}^{\text {sop}}$ be the category with the same objects and morphisms as $\mathcal{B}$ but with composition given by $\alpha \bullet \beta := (-1)^{{\mathfrak {p}}{\alpha }{\mathfrak {p}}{\beta }}\beta \circ \alpha $ for all homogeneous morphisms $\alpha $ and $\beta $ in $\mathcal{B}$ . Let $\text {refl}:{\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow } {\rightarrow } {\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ be the involutive contravariant superfunctor (in the sense of [Reference Brundan and Ellis5, Reference Lepine21]) given by $D \mapsto (-1)^{k(k-1)/2}D'$ on diagrams, where $D'$ is the reflection of D along a horizontal axis, and k is the number of tag-in and tag-out generators in D. It is easily checked that this is well defined using Theorem 4.3.1. The following lemma is immediate.

4.5 Connecting ${\mathfrak {p}\text {-}\mathbf {Web}}$ to ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$

Lemma 4.5.1 There is a well-defined functor of monoidal supercategories

$$ \begin{align*} \iota_{\uparrow}: {\mathfrak{p}\text{-}\mathbf{Web}} {\rightarrow} {\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow} \end{align*} $$

given on objects by $\iota _\uparrow (k) = \uparrow _k$ and on morphisms by

Proof We begin by proving a claim.

Claim: Let $\boldsymbol {a} \in {\mathbb {Z}}_{\geq 0}^{r}$ , $\boldsymbol {b} \in {\mathbb {Z}}_{\geq 0}^s$ , with $a_1 = b_1=k$ . If $f \in {\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}_\uparrow }(\uparrow _{\boldsymbol {a}}, \uparrow _{\boldsymbol {b}})$ is in the image of $\iota _\uparrow $ , then the morphism

is also in the image of $\iota _\uparrow $ .

We prove the Claim by induction on k, with the base case $k=0$ being trivial. Let $k>0$ and assume that the claim holds for all $m<k$ . By Corollary 6.3.2, we may assume that f is of the form $\iota _\uparrow (\xi )$ for some $\xi \in \mathscr {B}$ . After an isotopy of the strands in $\iota _\uparrow (\xi )$ , we may write

(4.21)

for some morphisms $f_1,f_2, f_3$ in the image of $\iota _\uparrow $ , and some $k' \leq k$ . If $k'=k$ , then the diagram has a bubble, and is thus zero by (4.3). If $k'=0$ , then the loop can be untwisted, using Theorem 4.3.1. So we assume now that $0<k'<k$ . Using (4.11) and (4.12), we may rewrite (4.21) as

where g is a morphism in the image of $\iota _\uparrow $ . Now, applying the inductive assumption for $k'$ , and then for $k-k'$ gives the result, proving the Claim.

Now, we prove the lemma. Let f be a diagram in ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow }$ . Using (4.18), we may assume that f is composed only of upward splits, upward merges, leftward/rightward/upward cups, leftward/rightward/upward caps, and crossings of all orientations. Let c be the number of leftward/rightward cups in f. We prove by induction on c that f is in the image of $\iota _\uparrow $ .

If $c=0$ , then since the domain and the codomain are composed only of up-arrows, it must be that there are no downward strands in f, so f is composed only of upward splits, upward merges, upward cups, upward caps, and upward crossings, and hence f is in the image of $\iota _\uparrow $ .

Now, for the induction step, assume $c>0$ . Select any leftward/rightward cup in f. Again, since the domain and the codomain are composed only of up-arrows, it must be that the downward strand leaving from the cup must lead into a leftward/rightward cap in f. Then, using Theorem 4.3.1, we may pull the downward strand to the left side of the diagram, giving a diagram of the form

where g is a diagram in ${\mathfrak {p}\text {-}\mathbf {Web}}_\uparrow $ with $c-1$ leftward/rightward cups. By the induction assumption, g is in the image of $\iota _\uparrow $ . Thus, by the Claim, f itself is in the image of $\iota _\uparrow $ , completing the proof.

We will show in Theorem 6.8.3 that $\iota _{\uparrow }$ is also faithful.

5.1 The Lie superalgebra of type P

In this section, let ${\mathbb {k}}$ be a field of characteristic different from two. Let $I=I_{n|n}$ be the index set $\left\{1, \dotsc ,n, -1, \dotsc , -n \right\}$ with fixed order $1 < \dotsb < n < -1 < \dotsb < -n$ . Let $|{~\cdot ~}|: I {\rightarrow } {\mathbb {Z}}_{2}$ be the function defined by $|{i}| = \bar {0}$ if $i>0$ and $|{i}|=\bar {1}$ if $i <0$ . Let $V=V_{n}$ be the vector space with distinguished basis $\left\{v_{i} \mid i \in I \right\}$ . We define a ${\mathbb {Z}}_{2}$ -grading on V by declaring $|{v_{i}}|=|{i}|$ for all $i \in I$ . Let ${\mathfrak {gl}} (V) = {\mathfrak {gl}}(n|n)$ denote the superspace of all linear endomorphisms of V. Then ${\mathfrak {gl}} (V)$ is a Lie superalgebra via the graded version of the commutator bracket:

$$\begin{align*}[f,g] = f \circ g - (-1)^{|{f}|\; |{g}|}g \circ f \end{align*}$$

for all homogeneous $f,g \in {\mathfrak {gl}} (V)$ . As done here, we frequently only give a formula for homogeneous elements and leave it understood that the general case is obtained via linearity.

Define an odd supersymmetric nondegenerate bilinear form on V by declaring

(5.1) $$ \begin{align} (v_{i},v_{j}) = (v_{j}, v_{i}) = \delta_{i,-j} \end{align} $$

for $i,j \in I$ . Here, odd means that the associated linear map $V \otimes V {\rightarrow } {\mathbb {k}}$ is an odd map of superspaces, while supersymmetric means that $(v,w) = (-1)^{|{v}| |{w}|} (w,v)$ for all homogeneous $v,w \in V$ .

Define a Lie superalgebra ${\mathfrak {g}} = {\mathfrak {p}} (n) \subseteq {\mathfrak {gl}} (V)$ consisting of all linear maps which preserve the bilinear form given in (5.1). That is, for all homogeneous $v,w \in V$ ,

$$\begin{align*}{\mathfrak{p}}(n) = \left\{f \in {\mathfrak{gl}} (V) \, \left| \, (f(v),w) + (-1)^{|{f}||{v}|}(v,f(w))=0 \right. \right\}. \end{align*}$$

The supercommutator restricts to define a Lie superalgebra structure on ${\mathfrak {p}} (n)$ .

With respect to our choice of basis, we can describe ${\mathfrak {p}}(n)$ as the $2n \times 2n$ matrices defined over ${\mathbb {k}}$ of the form

(5.2) $$ \begin{align} {\mathfrak{p}} (n) = \left\{\left(\begin{matrix} A & B \\ C & -A^{t} \end{matrix} \right) \right\}, \end{align} $$

where $A,B,C$ are $n \times n$ matrices with entries from ${\mathbb {k}}$ with B symmetric, C skew-symmetric, and where $A^{t}$ denotes the transpose of A. In terms of (5.2), the ${\mathbb {Z}}_{2}$ -grading is given by declaring ${\mathfrak {g}}_{\bar {0}}$ as the subspace of all such matrices where $B=C=0$ and ${\mathfrak {g}}_{\bar {1} }$ as the subspace of all such matrices where $A=0$ .

A (left) ${\mathfrak {p}} (n)$ -supermodule is a ${\mathbb {Z}}_{2}$ -graded ${\mathbb {k}}$ -vector space with a left ${\mathbb {k}}$ -linear action of ${\mathfrak {p}} (n)$ which respects the ${\mathbb {Z}}_{2}$ -grading and which satisfies graded versions of the usual axioms required of a module for a Lie algebra. For example, the natural supermodule is $V_{n}$ with ${\mathfrak {p}}(n)$ -supermodule structure given by matrix multiplication. Since we will only consider supermodules we usually leave the prefix “super” implicit going forward.

We allow for all (not just parity preserving) ${\mathfrak {p}}(n)$ -module homomorphisms. Consequently, the set of all ${\mathfrak {p}}(n)$ -homomorphisms between two modules is naturally a ${\mathbb {Z}}_{2}$ -graded vector space. Explicitly, $f: M {\rightarrow } N$ is a homogeneous ${\mathfrak {p}} (n)$ -module homomorphism if f is a linear map which satisfies $f(M_{s}) \subseteq N_{s+|{f}|}$ for $s\in {\mathbb {Z}}_{2}$ and $f(x.m) = (-1)^{|{x}||{f}|}x.f(m)$ for all homogeneous $x \in {\mathfrak {p}} (n)$ and $m\in M$ .

Since the enveloping superalgebra $U({\mathfrak {p}} (n))$ is a Hopf superalgebra, the category of ${\mathfrak {p}} (n)$ -modules is a monoidal supercategory in the sense of [Reference Brundan and Ellis5]. In what follows, we study particular monoidal sub-supercategories of this category. For every $k \geq 0$ , let $S^{k}(V_{n})$ denote the kth symmetric power of the natural module $V_{n}$ (by convention, $S^{0}(V_{n}) = {\mathbb {k}}$ , the trivial module). Let ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}} $ denote the full monoidal sub-supercategory of ${\mathfrak {p}}(n)$ -modules generated by $\left\{S^{k}(V_{n}) \mid k \geq 0 \right\}$ , and let ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S, S^{*}}}}$ denote the full monoidal sub-supercategory of ${\mathfrak {p}}(n)$ -modules generated by $\left\{S^{k}(V_{n}), S^{k}(V_{n})^{*} \mid k \geq 0 \right\}$ . That is, ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}}$ is the full subcategory of ${\mathfrak {p}} (n)$ -modules consisting of objects of the form

$$ \begin{align*} S^{a_1}(V_n) \otimes \cdots \otimes S^{a_k}(V_n), \end{align*} $$

ranging over all $k \in {\mathbb {Z}}_{\geq 0}$ and $\boldsymbol {a} = (a_1, \ldots , a_k) \in {\mathbb {Z}}_{\geq 0}^k$ . The objects of ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S, S^{*}}}}$ are similar except some symmetric powers are replaced by their duals.

5.2 Basic ${\mathfrak {p}}(n)$ -module maps

To connect our diagrammatic categories to the representation theory of ${\mathfrak {p}} (n)$ , we introduce certain explicit ${\mathfrak {p}}(n)$ -module homomorphisms in the categories ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}}$ and ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S, S^{*}}}}$ .

We can view the symmetric bisuperalgebra

(5.3) $$ \begin{align} S(V_n) = \bigoplus_{k \in {\mathbb{Z}}_{\geq 0}}S^k(V_n) \end{align} $$

as the enveloping superalgebra for the abelian Lie superalgebra $V_{n}$ . This endows $S(V_{n})$ with the structure of a ${\mathbb {Z}}$ -graded Hopf superalgebra. In particular, it admits an associative product $m: S(V_n) \otimes S(V_n) {\rightarrow } S(V_n)$ and coassociative coproduct $\Delta : S(V_n) {\rightarrow } S(V_n) \otimes S(V_n)$ . The product is the usual concatenation product and the coproduct is given on generators $v \in V_{n}$ by $\Delta (v) = v \otimes 1 + 1 \otimes v$ . We stress that the multiplication in $S(V_n)$ is supercommutative, meaning that $v w = (-1)^{|{v}||{w}|} w v$ for all homogeneous $v,w \in V_n$ . Corresponding to the direct sum decomposition (5.3), there are also projections $p_k:S(V_n) {\rightarrow } S^k(V_n)$ and inclusions $\iota _k: S^k(V_n) {\rightarrow } S(V_n)$ for all $k \in {\mathbb {Z}}_{\geq 0}$ . Each of the maps $\Delta , m, p_k, \iota _k$ is a $U({\mathfrak {p}}(n))$ -module homomorphism, and we use them to construct several module homomorphisms which will be used in the sequel.

We define the split $U({\mathfrak {p}}(n))$ -module morphism

$$ \begin{align*} \text{spl}^{a,b}_{a+b}: S^{a+b}(V_n) {\rightarrow} S^a(V_n) \otimes S^b(V_n) \end{align*} $$

via $\text {spl}^{a,b}_{a+b}:= (p_a \otimes p_b) \circ \Delta \circ \iota _{a+b}$ . Explicitly, we have

$$ \begin{align*} \text{spl}^{a,b}_{a+b}(x_1 \cdots x_{a+b})= \sum_{\substack{T = \{t_1< \cdots< t_a\} \\ U = \{u_1< \cdots<u_b\} \\ T \cup U = \{1,\ldots, a+b\} }} (-1)^{\varepsilon(T,U)}x_{t_1} \cdots x_{t_a} \otimes x_{u_1} \cdots x_{u_b}, \end{align*} $$

for all homogeneous $x_1, \ldots , x_{a+b} \in V_n$ , where $\varepsilon (T,U) \in {\mathbb {Z}}_2$ is defined by

$$ \begin{align*} \varepsilon(T,U)=\#\left\{ (t,u) \in T \times U \mid t>u, \,\bar{x}_t = \bar{x}_u = \bar 1\right\}. \end{align*} $$

Similarly, define the merge $U({\mathfrak {p}}(n))$ -module morphism

$$ \begin{align*} \operatorname{mer}_{a,b}^{a+b}: S^a(V_n) \otimes S^b(V_n) {\rightarrow} S^{a+b}(V_n) \end{align*} $$

via $\text {mer}_{a,b}^{a+b}:= p_{a+b} \circ m \circ (\iota _a \otimes \iota _b)$ , or, explicitly,

$$ \begin{align*} \operatorname{mer}_{a,b}^{a+b}(x_1 \cdots x_a \otimes y_1 \dotsc y_b) = x_1 \cdots x_a y_1 \cdots y_b, \end{align*} $$

for all $x_1, \ldots , x_a, y_1, \ldots , y_b \in V_n$ . Both the split and merge maps are even (i.e., parity preserving).

As the odd bilinear form used to define ${\mathfrak {p}} (n)$ is supersymmetric, it factors through to define the odd antenna $U({\mathfrak {p}}(n))$ -module homomorphism $\text {ant}{}:S^2(V_n) {\rightarrow } {\mathbb {k}}$ given by

$$ \begin{align*} \operatorname{ant} (x_1x_2) = (x_1,x_2) \end{align*} $$

for all $x_1, x_2 \in V_n$ .

For any $k \in {\mathbb {Z}}_{\geq 0}$ , we have the evaluation $U({\mathfrak {p}}(n))$ -module homomorphism

$$ \begin{gather*} \operatorname{eval}_k: S^k(V_n)^{*} \otimes S^k(V_n) {\rightarrow} {\mathbb{k}}\\ f \otimes x \mapsto f(x). \end{gather*} $$

Dualizing the evaluation map yields the coevaluation $U({\mathfrak {p}}(n))$ -module homomorphism

$$\begin{align*}\text{coeval}_k: {\mathbb{k}} {\rightarrow} S^k(V_n) \otimes S^k(V_n)^{*}. \end{align*}$$

In particular, we have

$$ \begin{gather*} \operatorname{coeval}_1: {\mathbb{k}} {\rightarrow} V_n \otimes V_n^{*}, \\ 1 \mapsto \sum_{i \in I_m} v_i \otimes v_i^{*}, \end{gather*} $$

where $\{v_i^{*} \mid i \in I\}$ is the dual basis for $V_n^{*}$ defined by $v_{i}^{*}(v_{j})=\delta _{i,j}$ .

The odd nondegenerate bilinear form $(\cdot , \cdot )$ induces an odd $U({\mathfrak {p}}(n))$ -module isomorphism

$$ \begin{gather*} D:V_n {\rightarrow} V_{n}^{*}, \\ v_i \mapsto (v_{i}, -)= v_{-i}^{*}. \end{gather*} $$

Using this isomorphism, we define the odd cap and cup $U({\mathfrak {p}}(n))$ -module homomorphisms by

$$ \begin{align*} \cap:= & \operatorname{eval}_1 \circ (D \otimes \operatorname{id}) : V_n^{\otimes 2} {\rightarrow} {\mathbb{k}},\\ \cup:= & (\operatorname{id} \otimes D^{-1})\circ \operatorname{coeval}_1 : {\mathbb{k}} {\rightarrow} V_n^{\otimes 2}. \end{align*} $$

On our basis for $V_{n}$ , these maps are given by

$$ \begin{align*} \cap(v_i \otimes v_j) = \delta_{i,-j} \qquad \qquad \text{and} \qquad \qquad \cup(1)= \sum_{i \in I_m} (-1)^{\bar i} v_i \otimes v_{-i}. \end{align*} $$

Finally, for any two ${\mathfrak {p}} (n)$ -modules M and N, we have the even “tensor swap” homomorphism

$$ \begin{gather*} \tau_{M,N}: M \otimes N {\rightarrow} N \otimes M, \\ m \otimes n \mapsto (-1)^{{\mathfrak{p}}{m} {\mathfrak{p}} {n}}n \otimes m, \end{gather*} $$

for all homogeneous $m\in M$ and $n\in N$ . Note that

(5.4) $$ \begin{align} \tau_{V_n,V_n} = (\operatorname{spl}_{2}^{1,1} \circ \operatorname{mer}_{1,1}^2) - (\operatorname{id}_1 \otimes \operatorname{id}_1). \end{align} $$

Compare with (2.3) when $a=b=1$ .

Note that $\operatorname {spl}^{a,b}_{a+b}$ , $\operatorname {mer}^{a+b}_{a,b}$ , ${\mathrm {ev}}_{k}$ , ${\mathrm {coev}}_{k}$ , and the tensor swap are in fact ${\mathfrak {gl}} (V)$ -equivariant. On the other hand, $\text {ant}$ , the odd cup, and the odd cap are only ${\mathfrak {p}} (n)$ -equivariant.

6.1 The functor $G: {\mathfrak {p}\text {-}\mathbf {Web}} {\rightarrow } {\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}$

Unless otherwise stated, in this section, ${\mathbb {k}}$ is a field of characteristic not two. Recall ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}}$ denotes the monoidal supercategory of ${\mathfrak {p}} (n)$ -modules generated by symmetric powers of the natural module $V_n$ .

Theorem 6.1.1 There is a well-defined functor

$$ \begin{align*} G: {\mathfrak{p}\text{-}\mathbf{Web}} {\rightarrow} {\mathfrak{p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}} \end{align*} $$

given on objects by $G(k) = S^k(V_n)$ and on morphisms by

6.2 The crossing morphism in ${{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}}$

For short, let

$$\begin{align*}\tau_{a,b}: S^{a}(V_{n}) \otimes S^{b}(V_{n}) {\rightarrow} S^{b}(V_{n}) \otimes S^{a}(V_{n}) \end{align*}$$

be the tensor swap map introduced in Section 5.2.

Lemma 6.2.1 For all $a,b \in {\mathbb {Z}}_{\geq 0}$ , we have

6.3 Basis theorems for ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$

We now prove that the sets introduced in Section 3.5 form ${\mathbb {k}}$ -bases for the morphism spaces of ${\mathfrak {gl}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}$ .

Theorem 6.3.1 Assume $|\boldsymbol {a}| + |\boldsymbol {b}| \leq 2n$ . Then

$$ \begin{align*} \left\{ \left. G(\xi^{(A,B,C,D)})\; \right| (A,B,C,D) \in \chi(\boldsymbol{a}, \boldsymbol{b}) \right\} \end{align*} $$

is a family of linearly independent morphisms in ${\mathrm { Hom}}_{{\mathfrak {p}}(n)}(S^{\boldsymbol {a}}(V_n),S^{\boldsymbol {b}}(V_n))$ .

Proof Let $(A,B,C,D) \in \chi (\boldsymbol {a},\boldsymbol {b})$ . For all $i=1, \ldots , t$ and $j=1,\ldots ,u$ , define

$$ \begin{align*} \texttt{r}_i = \sum_{\ell =1}^u C_{i\ell}, \qquad \qquad |C| = \sum_{k =1}^t \texttt{r}_k, \qquad \text{ and } \qquad P_{ij} = \sum_{k=1}^{i-1} \texttt{r}_k + \sum_{\ell =1 }^{j-1} C_{i \ell}. \end{align*} $$

For $X \in \left\{A, B \right\}$ , let

$$ \begin{align*} \left( (i_1^{X}, j_1^{X}), (i_2^{X}, j_2^{X}), \ldots, (i_\alpha^{X}, j_\alpha^{X}) \right) \end{align*} $$

be an irredundant list of all pairs of indices $(i,j)$ such that $i<j$ and $X_{ij} = 1$ . Let

$$ \begin{align*} (i_1^D, i_2^D, \ldots, i_\delta^D) \end{align*} $$

be an irredundant list of all indices i such that $D_i = 1$ . It follows from the fact that $|\boldsymbol {a}| + |\boldsymbol {b}| \leq 2n$ and the definition of the set $\chi (\boldsymbol {a},\boldsymbol {b})$ that $|C|+\alpha + \beta + \delta \leq n$ .

We define the following elements of $S(V_n)^{\otimes t}$ :

$$ \begin{align*} v^{(A,B,C,D),1}&:= v_1 \cdots v_{\texttt{r}_1} \otimes v_{\texttt{r}_1 +1} \cdots v_{\texttt{r}_1 + \texttt{r}_2} \otimes \cdots \otimes v_{\texttt{r}_1 + \cdots + \texttt{r}_{t-1}+1} \cdots v_{|C|} \\ v^{(A,B,C,D),2}&:= \prod_{k=1}^\alpha 1 \otimes \cdots \otimes 1 \otimes v_{|C|+ k} \otimes 1 \otimes \cdots \otimes 1 \otimes v_{-|C|-k} \otimes 1 \otimes \cdots \otimes 1, \\ v^{(A,B,C,D),3}&:= \prod_{k=1}^\delta 1 \otimes \cdots \otimes 1 \otimes v_{|C| + \alpha + k} v_{-|C| - \alpha - k} \otimes 1 \otimes \cdots \otimes 1, \end{align*} $$

where the vectors $v_{|C|+ k}$ and $ v_{-|C|-k}$ appear in the $i_k^A$ -th and $j_k^A$ -th slots, respectively, and the term $v_{|C| + \alpha + k} v_{-|C| - \alpha - k}$ appears in the $i_k^D$ -th slot.

We also define the following elements of $S(V_n)^{\otimes u}$ :

$$ \begin{align*} w^{(A,B,C,D),1} &:= \prod_{i=1}^t \prod_{j=1}^u 1 \otimes \cdots 1 \otimes v_{P_{ij}+1} \cdots v_{P_{ij}+C_{ij}} \otimes 1 \otimes \cdots \otimes 1, \\ w^{(A,B,C,D),2} &:= \prod_{k=1}^\beta 1 \otimes \cdots \otimes 1 \otimes v_{|C|+ k} \otimes 1 \otimes \cdots \otimes 1 \otimes v_{-|C|-k} \otimes 1 \otimes \cdots \otimes 1, \end{align*} $$

where the term $v_{P_{ij}+1} \cdots v_{P_{ij}+C_{ij}}$ appears in the jth slot, and the vectors $v_{|C|+ k}$ and $ v_{-|C|-k}$ appear in the $i_k^B$ -th and $j_k^B$ -th slots, respectively.

Considering $S(V_n)^{\otimes t}$ and $S(V_n)^{\otimes u}$ as associative algebras, we define

$$ \begin{align*} v^{(A,B,C,D)}&:=v^{(A,B,C,D),1 }\cdot v^{(A,B,C,D),2}\cdot v^{(A,B,C,D),3} \in S^{\boldsymbol{a}}(V_n),\\ w^{(A,B,C,D)}&:=w^{(A,B,C,D),1 }\cdot w^{(A,B,C,D),2} \in S^{\boldsymbol{b}}(V_n). \end{align*} $$

Since $S^{\boldsymbol {b}}(V_n)$ has a ${\mathbb {k}}$ -basis of tensor products of monomials in $\{v_i \mid i \in I_n\}$ , we may define a linear projection map

$$ \begin{align*} p_{(A,B,C,D)}: S^{\boldsymbol{b}}(V_n) {\rightarrow} {\mathbb{k}}\{ w^{(A,B,C,D)}\}. \end{align*} $$

We define a partial order $\succeq $ on $\chi (\boldsymbol {a},\boldsymbol {b})$ by setting

$$ \begin{align*} (A',B',C',D') \succeq (A,B,C,D) \end{align*} $$

if and only if

$$ \begin{align*} A^{\prime}_{ij} \leq A_{ij}, \qquad B^{\prime}_{ij} \leq B_{ij}, \qquad C^{\prime}_{ij} \geq C_{ij}, \qquad D^{\prime}_{i} \leq D_{i} \qquad \text{ for all }i,j. \end{align*} $$

It is straightforward to check, with the aid of Lemma 6.2.1, that

(6.1) $$ \begin{align} p_{(A,B,C,D)} \circ G(\xi^{(A',B',C',D')})(v^{(A,B,C,D)}) = \begin{cases} \pm w^{(A,B,C,D)}, & \text{if } (A',B',C',D') = (A,B,C,D),\\ 0, & \text{if } (A',B',C',D') \not \succeq (A,B,C,D). \end{cases} \end{align} $$

Now, assume that there exist nontrivial scalars $c_{(A',B',C',D')} \in {\mathbb {k}}$ such that

$$ \begin{align*} \sum_{(A',B',C',D') \in \chi(\boldsymbol{a},\boldsymbol{b})} c_{(A',B',C',D')} G(\xi^{(A',B',C',D')}) = 0. \end{align*} $$

Let $(A,B,C,D) \in \chi (\boldsymbol {a},\boldsymbol {b})$ be maximal in the $\succeq $ order such that $c_{(A,B,C,D)} \neq 0$ . Then we have by (6.1) that

$$ \begin{align*} 0&=\sum_{(A',B',C',D') \in \chi(\boldsymbol{a},\boldsymbol{b})} c_{(A',B',C',D')} p_{(A,B,C,D)} \circ G(\xi^{(A',B',C',D')})(v^{(A,B,C,D)})\\& = \pm c_{(A,B,C,D)}w^{(A,B,C,D)}, \end{align*} $$

a contradiction.

Corollary 6.3.2 The set

$$ \begin{align*} \mathscr{B}:=\left\{ \left. \xi^{(A,B,C,D)} \right| (A,B,C,D) \in \chi(\boldsymbol{a}, \boldsymbol{b}) \right\} \end{align*} $$

is a ${\mathbb {k}}$ -basis for ${\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(\boldsymbol {a},\boldsymbol {b})$ .

The previous corollary along with the results of Section 4.4 and Theorem 6.8.3 provide a basis theorem for ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ , as well. The following result is also immediate.

Corollary 6.3.3 The set

$$ \begin{align*} \mathscr{B}:=\left\{\left. \xi^{(0,0,C,0)} \right| (0,0,C,0) \in \chi(\boldsymbol{a}, \boldsymbol{b}) \right\} \end{align*} $$

is a ${\mathbb {k}}$ -basis for ${\mathrm { Hom}}_{{\mathfrak {gl}\text {-}\mathbf {Web}}}(\boldsymbol {a},\boldsymbol {b})$ .

These morphism spaces have been studied in, e.g., [Reference Rose and Tubbenhauer24, Reference Tubbenhauer, Vaz and Wedrich28]. Also, this basis should be compared with the basis of “reduced chicken foot diagrams” given in [Reference Brundan, Entova-Aizenbud, Etingof and Ostrik6].

6.4 ${\mathfrak {p}\text {-}\mathbf {Web}}$ and the marked Brauer category

We now explain how the marked Brauer category introduced in [Reference Kujawa and Tharp20] can be viewed as a subcategory of ${\mathfrak {p}\text {-}\mathbf {Web}}$ . It should be noted that in [Reference Kujawa and Tharp20] diagrams were read top-to-bottom, contrary to the convention here. However, using the functor $\text {refl}$ described in Section 4.4, one can easily translate between the two conventions. In this section, we assume that ${\mathbb {k}}$ is a field of characteristic different from two.

Definition 6.4.1 The marked Brauer category $\mathcal{B}$ is the ${\mathbb {k}}$ -linear strict monoidal supercategory generated by a single object $\bullet $ . For $k \in {\mathbb {Z}}_{\geq 0}$ , we will use the notation $[k]$ to designate the object $\bullet ^{\otimes k}$ . The category $\mathcal{B}$ has generating morphisms:

We call these morphisms twist, cap, and cup, respectively. The ${\mathbb {Z}}_2$ -grading is given by declaring twists to have parity $\bar 0$ , and caps and cups to have parity $\bar 1$ . The defining relations of $\mathcal{B}$ are:

(6.2)

(6.3)

(6.4)

(6.5)

(6.6)

Let ${\mathfrak {p}}(n)\text{-}\mathrm{Mod}_{V}$ denote the monoidal supercategory of ${\mathfrak {p}}(n)$ -modules generated by the natural module $V_n$ . That is, ${\mathfrak {p}}(n)\text{-}\mathrm{mod}_{V}$ is the full subcategory of ${\mathfrak {p}}(n)\text{-}\mathrm{mod}$ consisting of objects of the form $\left\{ V_{n}^{\otimes k} \mid k \in {\mathbb {Z}}_{\geq 0}\right\}$ .

Theorem 6.4.2 [Reference Kujawa and Tharp20, Theorem 5.2.1]

There is a well-defined functor of monoidal supercategories

$$ \begin{align*} F: \mathcal{B} {\rightarrow} {\mathfrak{p}}(n)\text{-}\mathrm{Mod}_{V} \end{align*} $$

given by $F(\bullet ) = V_{n}$ and on morphisms by

Theorem 6.4.3 If ${\mathbb {k}}$ is a field of characteristic zero, then the functor F is full. That is, for all $a,b \in {\mathbb {Z}}_{\geq 0}$ , the induced map of superspaces,

$$ \begin{align*} F:{\mathrm{Hom}}_{\mathcal{B}}([a],[b]) \xrightarrow{\sim} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}(V_n^{\otimes a}, V_n^{\otimes b}), \end{align*} $$

is surjective.

Let ${\mathfrak {p}\text {-}\mathbf {Web}}_1$ be the full monoidal sub-supercategory of ${\mathfrak {p}\text {-}\mathbf {Web}}$ whose objects are tuples consisting of only ones, including the empty tuple.

Theorem 6.4.4 There is an isomorphism of monoidal supercategories

$$ \begin{align*} F':\mathcal{B} {\rightarrow} {\mathfrak{p}\text{-}\mathbf{Web}}_1 \end{align*} $$

given on objects by $\bullet \mapsto 1$ and on morphisms by

6.5 The functor $G_{\uparrow \downarrow }: {\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow } {\rightarrow } {{\mathfrak {p}}(n)\text{-}\mathrm{mod}_{\mathcal{S, S^{*}}}}$

Theorem 6.5.1 There is a well-defined functor of monoidal supercategories

$$ \begin{align*} G_{\uparrow \downarrow}: {\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow} {\rightarrow} {{\mathfrak{p}}(n) \text{-}\mathrm{mod}_{\mathcal{S, S^{*}}}} \end{align*} $$

given on objects by $G_{\uparrow \downarrow }(\uparrow _a) = S^a(V_n)$ and $G_{\uparrow \downarrow }(\downarrow _a)=S^a(V_n)^{*}$ . The functor is given on morphisms by

Proof Note that upward caps, upward cups, and upward antennas are sent to $\cup $ , $\cap $ , and $\text {ant}$ , respectively, so the defining up-arrow relations of ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ are preserved by $G_{\uparrow \downarrow }$ thanks to Theorem 6.1.1. We now check (4.1)–(4.3).

Let $B_a$ be a homogeneous ${\mathbb {k}}$ -basis for $S^a(V_n)$ , with dual basis $\{x^{*} \mid x \in B_a\}$ for $S^a(V_n)^{*}$ . Then, for $x,y \in B_a$ , we have

$$ \begin{align*} \text{eval}_a(x^{*} \otimes y) = \delta_{x,y}, \qquad \text{and} \qquad \text{coeval}_a(1) = \sum_{x \in B_a} x \otimes x^{*}. \end{align*} $$

So, for all $x \in B_a$ , we have

$$ \begin{align*} (\text{id} \otimes \text{eval}_a) \circ (\text{coeval}_a \otimes \text{id})(x)= \sum_{y \in B_a}(\text{id} \otimes \text{eval}_a)(y \otimes y^{*} \otimes x)= \sum_{y \in B_a} \delta_{y,x}y = x \end{align*} $$

and

$$ \begin{align*} (\text{eval}_a \otimes \text{id})\circ (\text{id} \otimes \text{coeval}_a)(x^{*}) = \sum_{y \in B_a} (\text{eval}_a \otimes \text{id})(x^{*} \otimes y \otimes y^{*}) = \sum_{y \in B_a} \delta_{x,y}y^{*} = x^{*}, \end{align*} $$

proving (4.1).

By Lemma 6.2.1, we have

From this, it is immediate that for all $x \in B_a$ , $y \in B_b$ , the image of the leftward crossing under $G_{\uparrow \downarrow }$ is $\tau _{S^a(V_n)^{*}, S^b(V_n)}$ , and thus we have that relation (4.2) is preserved.

Finally, to check relation (4.3), we note that

$$ \begin{align*} \text{eval}_a \circ \tau_{S^a(V_n)^{*}, S^a(V_n)} \circ \text{coeval}_{a}(1) &= \sum_{x \in B_a} \text{eval}_a \circ \tau_{S^a(V_n)^{*}, S^a(V_n)}(x \otimes x^{*} )\\ &= \sum_{x \in B_a} (-1)^{|x|} \text{eval}_a (x^{*} \otimes x) = \sum_{x \in B_a} (-1)^{|x|} = 0, \end{align*} $$

completing the proof.

6.6 Explosion and contraction

In this section, ${\mathbb {k}}$ can be an integral domain. For short, given $k \geq 1 $ , we write $y_k \in {\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(k,1^k)$ and $z_k \in {\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(1^k,k)$ for the morphisms defined by (3.11).

Let $k \in {\mathbb {Z}}_{\geq 0}$ and $\boldsymbol {a} = (a_1, \ldots , a_k) \in {\mathbb {Z}}_{\geq 0}^k$ . We identify $\boldsymbol {a}$ with the object $(a_1, \dotsc , a_{k})$ of ${\mathfrak {p}\text {-}\mathbf {Web}}$ . We will also use the following associated notation:

$$ \begin{align*} |\boldsymbol{a}|:=a_1 + \cdots + a_{k}, \qquad \boldsymbol{a}!:= a_1! \cdots a_{k}! \qquad y_{\textbf{a}} := y_{a_1} \otimes \cdots \otimes y_{a_{k}}, \qquad z_{\textbf{a}} := z_{a_1} \otimes \cdots \otimes z_{a_{k}}. \end{align*} $$

For any objects $\boldsymbol {a}, \boldsymbol {b}$ in ${\mathfrak {p}\text {-}\mathbf {Web}}$ , we have linear maps:

$$ \begin{align*} \mathrm{exp}_{\boldsymbol{a},\boldsymbol{b}}:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}(\boldsymbol{a}, \boldsymbol{b}) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}(1^{|\boldsymbol{a}|}, 1^{|\boldsymbol{b}|}), \qquad f \mapsto y_{\boldsymbol{b}} \circ f \circ z_{\boldsymbol{a}}, \end{align*} $$

and

$$ \begin{align*} \mathrm{con}_{\boldsymbol{a}, \boldsymbol{b}}:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}(1^{|\boldsymbol{a}|}, 1^{|\boldsymbol{b}|}) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}(\boldsymbol{a}, \boldsymbol{b}), \qquad g \mapsto z_{\boldsymbol{b}} \circ g \circ y_{\boldsymbol{a}}. \end{align*} $$

We refer to these maps as explosion and contraction, respectively. See the proof of [Reference Rose and Tubbenhauer24, Theorem 1.10] for a picture showing them in use.

It should be noted that the explosion and contraction morphisms only involve ${\mathfrak {gl}\text {-}\mathbf {Web}}$ morphisms and calculations. They first appear in [Reference Rose and Tubbenhauer24] and are now a standard tool in this area.

6.7 Putting things together, ${\mathfrak {p}\text {-}\mathbf {Web}}$ edition

Let $r,s \in {\mathbb {Z}}_{\geq 0}$ , $\boldsymbol {a} \in {\mathbb {Z}}_{\geq 0}^r$ , $\boldsymbol {b} \in {\mathbb {Z}}_{\geq 0}^s$ . By Theorem 6.1.1, we may define linear maps:

$$ \begin{gather*} \widetilde{\mathrm{exp}}_{\boldsymbol{a},\boldsymbol{b}}:{\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left( S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})\right) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left( V_{n}^{|\boldsymbol{a}|}, V_{n}^{|\boldsymbol{b}|}\right),\\ f \mapsto G\left( y_{\boldsymbol{b}}\right) \circ f \circ G\left(z_{\boldsymbol{a}}\right), \end{gather*} $$

and

$$ \begin{gather*} \widetilde{\mathrm{con}}_{\boldsymbol{a}, \boldsymbol{b}}:{\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left(V_{n}^{|\boldsymbol{a}|}, V_{n}^{|\boldsymbol{b}|}\right) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left(S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})\right),\\ g \mapsto G\left(z_{\boldsymbol{b}}\right) \circ g \circ G\left(y_{\boldsymbol{a}}\right). \end{gather*} $$

Lemma 6.7.1 For all objects $\boldsymbol {a}$ and $\boldsymbol {b}$ in ${\mathfrak {p}\text {-}\mathbf {Web}}$ , the following diagram commutes:

Theorem 6.7.2 If $\boldsymbol {a},\boldsymbol {b}$ are such that $|\boldsymbol {a}| + |\boldsymbol {b}| \leq 2n$ , then the map

$$ \begin{align*} G:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}(\boldsymbol{a},\boldsymbol{b}) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left(S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})\right) \end{align*} $$

is injective. If ${\mathbb {k}}$ is a field of characteristic zero, then the functor G is full.

Proof The injectivity statement follows from Theorem 6.3.1 and Corollary 6.3.2.

Now, assume that ${\mathbb {k}}$ has characteristic zero and consider the diagram in Lemma 6.7.1. Since $F'$ is an isomorphism by Theorem 6.4.4 and F is surjective by Theorem 6.4.3, the map G on the right is surjective. To see surjectivity of G along the left side, let $\varphi \in {\mathrm { Hom}}_{{\mathfrak {p}}(n)}(S^{\boldsymbol {a}}(V_{n}), S^{\boldsymbol {b}}(V_{n}))$ . Then, by surjectivity of the G on the right, there exists $\theta \in {\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}_1^+}(1^{|\boldsymbol {a}|}, 1^{|\boldsymbol {b}|})$ such that

$$ \begin{align*} G\left(\theta \right) = \widetilde{\mathrm{exp}}_{\boldsymbol{a}, \boldsymbol{b}}(\varphi) = G\left(y_{\boldsymbol{b}} \right) \circ \varphi \circ G\left(z_{\boldsymbol{a}} \right). \end{align*} $$

Then we compute

$$ \begin{align*} G\left(\frac{\text{con}_{\boldsymbol{a},\boldsymbol{b}}(\theta)}{\boldsymbol{a}!\boldsymbol{b}!}\right) &= \frac{1}{\boldsymbol{a}!\boldsymbol{b}!} G(z_{\boldsymbol{b}} \circ \theta \circ y_{\boldsymbol{a}}) = \frac{1}{\boldsymbol{a}!\boldsymbol{b}!} Gz_{\boldsymbol{b}} \circ G\theta \circ Gy_{\boldsymbol{a}} = \frac{1}{\boldsymbol{a}!\boldsymbol{b}!} Gz_{\boldsymbol{b}} \circ Gy_{\boldsymbol{b}} \circ \varphi \circ Gz_{\boldsymbol{a}}. \circ Gy_{\boldsymbol{a}}\\[4pt] &=\frac{1}{\boldsymbol{a}!\boldsymbol{b}!} G(z_{\boldsymbol{b}} \circ y_{\boldsymbol{b}}) \circ \varphi \circ G(z_{\boldsymbol{a}} \circ y_{\boldsymbol{a}}) = \frac{1}{\boldsymbol{a}!\boldsymbol{b}!} G\left( \boldsymbol{b}! \cdot \text{id}_{\boldsymbol{b}}\right) \circ \varphi \circ G\left( \boldsymbol{a}!\cdot \text{id}_{\boldsymbol{a}}\right) = \varphi, \end{align*} $$

as desired. That is, G along the left side of the diagram is surjective, completing the proof.

Via the functor G, Theorem 6.7.2 shows that the basis for ${\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}}(\boldsymbol {a},\boldsymbol {b})$ given in Corollary 6.3.2 could be considered a “stable basis,” or a “basis at infinity” for ${\mathrm { Hom}}_{{\mathfrak {p}}(n)}\left(S^{\boldsymbol {a}}(V_{n}), S^{\boldsymbol {b}}(V_{n})\right)$ in characteristic zero, since G defines an isomorphism whenever $n \gg 0$ .

6.8 Putting things together, ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ edition

Let $r,s \in {\mathbb {Z}}_{\geq 0}$ and $\boldsymbol {a} \in {\mathbb {Z}}^r$ , $\boldsymbol {b} \in {\mathbb {Z}}^s$ . Given a nonnegative integer a, it will be convenient to adopt the notation $S^{a}(V_{n}):= S^a(V_{n})$ , $S^{-a}(V_{n}):= S^a(V_{n})^{*}$ , and, more generally,

$$ \begin{align*} S^{\boldsymbol{a}}(V_{n}):= S^{a_1}(V_{n}) \otimes \cdots \otimes S^{a_r}(V_{n}). \end{align*} $$

Let $\boldsymbol {c}, \boldsymbol {d}, \varphi _1, \varphi _2$ be as in Lemma 4.4.1. By Theorem 6.5.1, we have an invertible linear map:

$$ \begin{gather*} \tilde{\Phi}:{\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left( S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})\right) \xrightarrow{\sim} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left(S^{\boldsymbol{c}}(V_{n}), S^{\boldsymbol{d}}(V_{n})\right)\\[4pt] f \mapsto G_{\uparrow \downarrow}\left(\varphi_2 \right)\circ f \circ G_{\uparrow \downarrow}\left(\varphi_1 \right). \end{gather*} $$

Lemma 6.8.1 For any $\boldsymbol {a} \in {\mathbb {Z}}^{r}$ and $\boldsymbol {b} \in {\mathbb {Z}}^{s}$ , let $\boldsymbol {c}$ and $\boldsymbol {d}$ be as above. Then the following diagram commutes:

Theorem 6.8.2 If $\boldsymbol {a} \in {\mathbb {Z}}^{r}$ and $\boldsymbol {b} \in {\mathbb {Z}}^{s}$ are such that $|\boldsymbol {a}| + |\boldsymbol {b}| \leq 2n$ , then the map

$$ \begin{align*} G_{\uparrow \downarrow}:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}\left( |_{\boldsymbol{a}},|_{\boldsymbol{b}} \right) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}\left( S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})\right) \end{align*} $$

is injective. If ${\mathbb {k}}$ is a field of characteristic zero, then the functor $G_{\uparrow \downarrow }$ is full.

Theorem 6.8.3 The functor

$$ \begin{align*} \iota_\uparrow: {\mathfrak{p}\text{-}\mathbf{Web}} {\rightarrow} {\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow} \end{align*} $$

is an isomorphism of categories.

Proof The functor is bijective on objects, and full by Theorem 4.5.2. For any fixed pair of objects $\boldsymbol {c}$ and $\boldsymbol {d}$ in ${\mathfrak {p}\text {-}\mathbf {Web}}$ , we may choose n such that $2n\geq |\boldsymbol {c}| +|\boldsymbol {d}|$ and consider the diagram in Lemma 6.8.1. Then the map G is injective by Theorem 6.7.2, forcing

$$ \begin{align*} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}\left(\boldsymbol{c},\boldsymbol{d} \right) \xrightarrow{\sim} {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_\uparrow}\left( \uparrow_{\boldsymbol{c}}, \uparrow_{\boldsymbol{d}} \right), \end{align*} $$

as required.

6.9 A particular spanning set

In the follow-up paper [Reference Davidson, Kujawa and Muth13], it will be useful to work with a particular spanning set for ${\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }}\left(\emptyset , \uparrow ^{\boldsymbol {a}}\downarrow ^{\boldsymbol {b}} \right)$ . As the relevant diagrammatics are already defined herein, we request the reader’s indulgence in including the necessary result here. In light of the results of Section 6.4, readers familiar with explosion/contraction arguments will note that the spanning set is a contraction of the Brauer diagram basis for ${\mathrm {Hom}}_{\mathcal{B}}\left(\emptyset , |\boldsymbol {a}| + |\boldsymbol {b}| \right)$ given in [Reference Kujawa and Tharp20].

Lemma 6.9.1 Assume that ${\mathbb {k}}$ is a field of characteristic zero, and $\boldsymbol {a}, \boldsymbol {b} \in {\mathbb {Z}}_{\geq 0}^{m}$ . Then ${\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }}\left(\emptyset , \uparrow ^{\boldsymbol {a}}\downarrow ^{\boldsymbol {b}} \right)$ is spanned by diagrams of the form

(6.7)

where all strands except those at the very top of the diagram are thin, the diagram has $0 \leq j \leq |\boldsymbol {a}|/2$ upward-oriented cups, $|\boldsymbol {b}|/2 - |\boldsymbol {a}|/2 + j$ downward-oriented cups, $|\boldsymbol {a}| - 2j$ leftward-oriented cups, $\sigma $ consists only of upward-oriented crossings, and $\tau $ consists only of downward-oriented crossings.

Proof Let $\boldsymbol {b}' = (b_m, \ldots , b_1)$ . We first consider ${\mathrm {Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }}\left(\uparrow ^{\boldsymbol {b}' }, \uparrow ^{\boldsymbol {a}} \right)$ . It follows from Theorem 6.4.4 and Lemma 6.7.1 that this morphism space is spanned by diagrams of the form

where D is a marked Brauer diagram as defined in [Reference Kujawa and Tharp20, Section 2.2]. Using the relations discussed in loc. cit., one can replace D with $\sigma \circ E \circ \tau '$ , where $\sigma $ and $\tau '$ consist of only upward-oriented crossings, and where E is a diagram of the form

By Lemma 4.4.1, there is an isomorphism of superspaces

$$\begin{align*}{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}\left(\uparrow^{\boldsymbol{b}' }, \uparrow^{\boldsymbol{a}} \right) \cong {\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}\left(\emptyset, \uparrow^{\boldsymbol{a}}\downarrow^{\boldsymbol{b}} \right) \end{align*}$$

given by

Applying this map to the elements of our spanning set and using the relations in Theorem 4.3.1 to pull $\tau '$ and the upward-oriented caps to the right side of the diagram, it follows that ${\mathrm { Hom}}_{{\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }}\left(\emptyset , \uparrow ^{\boldsymbol {a}}\downarrow ^{\boldsymbol {b}} \right)$ is spanned by diagrams of the form (6.7).

7.1 Faithfulness

Throughout this section, ${\mathbb {k}}$ is an algebraically closed field of characteristic zero. The assumption that ${\mathbb {k}}$ is algebraically closed allows us to cite the results from [Reference Coulembier10] used below. We expect that it is not necessary. We also remark that Coulembier describes how to construct the morphism $f_{n+1}$ . It would be interesting to describe it explicitly in terms of diagrams.

Let $\mathcal{I}_{n}$ denote the kernel of the functor

$$\begin{align*}F: \mathcal{B} {\rightarrow} {\mathfrak{p}} (n)\text{-modules}. \end{align*}$$

That is, $\mathcal{I}_{n}$ is the tensor ideal given by

$$\begin{align*}\mathcal{I}_{n}\left([a],[b] \right) = \left\{ g \in {\mathrm{Hom}}_{ \mathcal{B}}\left([a],[b] \right) \mid F(g) =0 \right\} \end{align*}$$

for all objects $[a]$ and $[b]$ in $\mathcal{B}$ . The following is a reformulation of [Reference Coulembier10, Theorem 8.3.1] so that it applies to the category $\mathcal{B}$ .

Proof By [Reference Coulembier10, Theorem 8.3.1], there is a morphism $f_{n+1} \in \operatorname {End}_{\mathcal{B}}\left( [\ell ] \right)$ which is in the kernel of the functor F. Since F is a functor of ${\mathbb {k}}$ -linear monoidal categories, the tensor ideal generated by $f_{n+1}$ is also contained in the kernel of F.

On the other hand, let $g \in {\mathrm {Hom}}_{\mathcal{B}}\left( [a], [b] \right)$ be a nonzero element in the kernel of F. By pre- and post-composing with cup and cap diagrams much as in Section 4.4, one can define an isomorphism of superspaces

(7.1) $$ \begin{align} {\mathrm{Hom}}_{\mathcal{B}}\left( [a], [b] \right) \xrightarrow{\cong} {\mathrm{Hom}}_{\mathcal{B}}\left( \left[r \right], \left[r \right] \right), \end{align} $$

where $r=(a+b)/2$ . Let $g' \in {\mathrm {Hom}}_{\mathcal{B}}\left( \left[r \right], \left[r \right] \right)$ be the image of g under this isomorphism. Since F is a monoidal functor, $g'$ is in the kernel of the map $F:\operatorname {End}_{\mathcal{B}}\left( \left[r \right] \right) {\rightarrow } {\mathrm {Hom}}_{{\mathfrak {p}} (n)}\left(V^{\otimes r} \right)$ . By [Reference Coulembier10, Theorem 8.3.1], $r \geq \ell $ and the kernel of this superalgebra homomorphism is generated as an ideal by $f_{n+1} \otimes \operatorname {Id}_{\bullet }^{\otimes \left(r-\ell \right)}$ . Thus, $g' = \sum _{i} a_{i}\left(f_{n+1} \otimes \operatorname {Id}_{\bullet }^{\otimes \left(r-\ell \right)} \right)b_{i}$ for some $a_{i},b_{i} \in \operatorname {End}_{\mathcal{B}}\left( \left[r \right] \right)$ . Applying the inverse of (7.1) to this expression shows that g lies in the tensor ideal generated by $f_{n+1}$ , proving the claim.

We abuse notation and write $f_{n+1}$ for the morphism $F'(f_{n+1})$ and $\iota _{\uparrow }(F'(f_{n+1}))$ in ${\mathfrak {p}\text {-}\mathbf {Web}}$ and ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ , respectively.

We end this section by pointing out that under the assumptions of the present section, the injectivity statements of Theorems 6.7.2 and 6.8.2 can be made sharp.

Proposition 7.1.3 Let $n \geq 1$ and set $\ell = (n+1)(n+2)/2$ . Then, the maps

$$ \begin{align*} G &:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}}(\boldsymbol{a},\boldsymbol{b}) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}(S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})),\\ G_{\uparrow \downarrow} &:{\mathrm{Hom}}_{{\mathfrak{p}\text{-}\mathbf{Web}}_{\uparrow \downarrow}}(|_{\boldsymbol{a}},|_{\boldsymbol{b}}) {\rightarrow} {\mathrm{Hom}}_{{\mathfrak{p}}(n)}(S^{\boldsymbol{a}}(V_{n}), S^{\boldsymbol{b}}(V_{n})) \end{align*} $$

are injective if and only if $ |\boldsymbol {a} | + |\boldsymbol {b} | < (n+1)(n+2)$ .

Proof Doing a diagram chase using Lemmas 6.7.1 and 6.8.1, one can show that the injectivity of these maps is equivalent to the injectivity of the map

$$\begin{align*}F: \operatorname{End}_{\mathcal{B}}\left([r], [r] \right) {\rightarrow} \operatorname{End} _{{\mathfrak{p}} (n)}\left(V^{\otimes r}, V^{\otimes r} \right), \end{align*}$$

where $r = \left(|\boldsymbol {a} | + |\boldsymbol {b} | \right)/2$ . However, by [Reference Coulembier10, Theorem 8.3.1], this map is injective if and only if $r < \ell $ . This proves the claim.

7.2 Category equivalences

Let

$$\begin{align*}\mathcal{B}(n),\;\; {{\mathfrak{p}} (n)\text{-}\mathbf{Web}},\;\; \text{and} \;\; {{\mathfrak{p}} (n)\text{-}\mathbf{Web}}_{\uparrow \downarrow} \end{align*}$$

be the quotient of $\mathcal{B}$ , ${\mathfrak {p}\text {-}\mathbf {Web}}$ , and ${\mathfrak {p}\text {-}\mathbf {Web}}_{\uparrow \downarrow }$ , respectively, by the tensor ideal generated by $f_{n+1}$ , where $f_{n+1}$ is the morphism given in Section 7.1.

As $f_{n+1}$ is in the kernel of the functors F, G, and $G_{\uparrow \downarrow }$ , they induce well-defined functors which we call by the same name:

(7.2) $$ \begin{align} \begin{aligned} F &: \mathcal{B}(n) {\rightarrow} {\mathfrak{p}}(n)\text{-}\mathrm{Mod}_{V}, \\ G&: {{\mathfrak{p}} (n)\text{-}\mathbf{Web}} {\rightarrow} {{\mathfrak{p}}(n)\text{-}\mathrm{mod}_{\mathcal{S}}}, \\ G_{\uparrow \downarrow} &: {{\mathfrak{p}} (n)\text{-}\mathbf{Web}}_{\uparrow \downarrow} {\rightarrow} {{\mathfrak{p}}(n)\text{-}\mathrm{mod}_{\mathcal{S, S^{*}}}}. \end{aligned} \end{align} $$