2.1 Quantum groups

Our general conventions for quantum groups are the same as in [Reference LusztigLus10], except that we write q in place of Lusztig’s v. Let $\mathbf {f}$ be the polynomial algebra over $\mathbb {Q}(q)$ generated by one element $\theta $ . We write $\theta ^{(n)}$ for the divided power $\theta ^n / [n]!$ . This is a fixed point for the bar involution $\psi :\mathbf {f} \rightarrow \mathbf {f}$ , which is the antilinear involution defined from $\psi (\theta ) := \theta $ . Let $(\cdot ,\cdot ):\mathbf {f} \times \mathbf {f} \rightarrow \mathbb {Q}(q)$ be the nondegenerate symmetric bilinear form from [Reference LusztigLus10, Sec. 1.2.5]. It satisfies

(2.1) $$ \begin{align} \big( \theta^{(m)},\theta^{(n)}\big) = \frac{\delta_{m,n}}{(1-q^{-2}) (1-q^{-4}) \cdots (1-q^{-2n})} \end{align} $$

for $m, n\geq 0$ . Let $R:\mathbf {f}\rightarrow \mathbf {f}$ be the linear map defined by

(2.2) $$ \begin{align} R(1) &= 0, & R\big(\theta^{(n)}\big) &= \frac{q^{n-1} \theta^{(n-1)}}{1-q^{-2}} \end{align} $$

for $n \geq 1$ . This map arises naturally as the adjoint of left multiplication by $\theta $ : we have that

(2.3) $$ \begin{align} ( \theta x, y ) =( x, R(y)) \end{align} $$

for all $x,y \in \mathbf {f}$ . Equivalently, $R(x) = r(x) / (1-q^{-2})$ where r is the map defined in either the first or the second paragraph of [Reference LusztigLus10, Sec. 1.2.13] (the two maps coincide in rank one).

The quantum group ${\mathrm U}={\mathrm U}_q(\mathfrak {sl}_2)$ is the $\mathbb {Q}(q)$ -algebra with generators $e, f, k, k^{-1}$ satisfying the relations

$$ \begin{align*} k e k^{-1} &= q^2 e,& k f k^{-1} &= q^{-2} f,& [e,f] &= \frac{k-k^{-1}}{q-q^{-1}}. \end{align*} $$

Here, we have switched to using lower case for $e,f$ compared to [Reference LusztigLus10] so that we can use the upper case letters $E, F$ for corresponding functors in categorification. The subalgebras of ${\mathrm U}$ generated by f and by e are denoted ${\mathrm U}^-$ and ${\mathrm U}^+$ , respectively. Both are isomorphic to $\mathbf {f}$ via the maps $\mathbf {f} \rightarrow {\mathrm U}^+, x \mapsto x^+$ and $\mathbf {f} \rightarrow {\mathrm U}^-, x \mapsto x^-$ defined so that $\theta ^+ := e$ and $\theta ^- := f$ . The divided powers $e^{(n)} := e^n / [n]!$ and $f^{(n)} := f^n / [n]!$ are the images of $\theta ^{(n)}$ under these maps. There are various useful symmetries:

• • Let $\psi :{\mathrm U} \rightarrow {\mathrm U}$ be the usual bar involution on ${\mathrm U}$ , that is, the antilinear algebra involution that fixes e and f and takes k to $k^{-1}$ .

• • Let $\rho :{\mathrm U} \rightarrow {\mathrm U}$ be the linear algebra antiinvolution such that $\rho (k) = k$ , $\rho (e) = q kf$ , $\rho (f) = q^{-1} e k^{-1}$ .

By [Reference LusztigLus10, Prop. 3.1.6(b)] (or an easy induction exercise using (2.2)), we have that

(2.4) $$ \begin{align} ex^- -x^- e = q^{-1} k R(x)^- - q^{-1} R(x)^- k^{-1} \end{align} $$

for any $x \in \mathbf {f}$ .

We denote the irreducible ${\mathrm U}$ -module of highest weight $\lambda \in \mathbb {N}$ by $V(\lambda )$ . This is generated by a vector $\eta _\lambda $ such that $e \eta _\lambda = 0$ and $k \eta _\lambda = q^\lambda \eta _\lambda $ . There is an antilinear involution $\psi _\lambda :V(\lambda )\rightarrow V(\lambda )$ such that $\psi _\lambda (\eta _\lambda ) = \eta _\lambda $ and $\psi _\lambda (uv) = \psi (u) \psi _\lambda (v)$ for $u \in {\mathrm U}, v \in V(\lambda )$ . Also let $(\cdot ,\cdot )_\lambda :V(\lambda )\times V(\lambda )\rightarrow \mathbb {Q}(q)$ be the unique nondegenerate symmetric bilinear form on $V(\lambda )$ such that

(2.5) $$ \begin{align} (\eta_\lambda,\eta_\lambda)_\lambda &= 1, & (uv_1 ,v_2)_\lambda &= ( v_1, \rho(u) v_2)_\lambda \end{align} $$

for $u \in {\mathrm U}, v_1,v_2 \in V(\lambda )$ . The form $(\cdot ,\cdot )$ on $\mathbf {f}$ can be recovered from these forms on the modules $V(\lambda )$ since we have that

(2.6) $$ \begin{align} (x,y)= \lim_{\lambda \rightarrow \infty} \big(x^- \eta_\lambda, y^- \eta_\lambda \big)_\lambda \end{align} $$

for all $x,y \in \mathbf {f}$ by a special case of [Reference LusztigLus10, Prop. 19.3.7]. The vectors $f^{(n)} \eta _\lambda \:(0 \leq n \leq \lambda )$ give the canonical basis for $V(\lambda )$ . In fact, they give a basis for an integral form ${{}_{\mathbb {Z}}}V(\lambda )$ over ${\mathbb {Z}[q,q^{-1}]}$ . The antiinvolution $\psi _\lambda $ restricts to an antilinear involution of ${{}_{\mathbb {Z}}}V(\lambda )$ , and the values of the form $(\cdot ,\cdot )_\lambda $ on elements of ${{}_{\mathbb {Z}}}V(\lambda )$ lie in $\mathbb {Z}[q,q^{-1}]$ .

For the purposes of categorification, one usually replaces ${\mathrm U}$ by its modified form $\dot {\mathrm U}$ , which is a locally unital algebra $\dot {\mathrm U} = \bigoplus _{\lambda ,\mu \in \mathbb {Z}} 1_\mu \dot {\mathrm U} 1_\lambda $ with a distinguished system $1_\lambda (\lambda \in \mathbb {Z})$ of mutually orthogonal idempotents replacing the diagonal generators $k, k^{-1}$ . The relationship between ${\mathrm U}$ and $\dot {\mathrm U}$ can be expressed either by saying that $\dot {\mathrm U}$ is a $({\mathrm U},{\mathrm U})$ -bimodule, or that ${\mathrm U}$ embeds into the completion of $\dot {\mathrm U}$ consisting of matrices $(a_{\mu ,\lambda })_{\lambda ,\mu \in \mathbb {Z}} \in \prod _{\lambda ,\mu \in \mathbb {Z}} 1_\mu \dot {\mathrm U} 1_\lambda $ such that there are only finitely many nonzero entries in each row and column. The element $k \in {\mathrm U}$ corresponds to the diagonal matrix with $q^\lambda 1_\lambda $ as its $\lambda $ th diagonal entry, while $e,f \in {\mathrm U}$ are identified with the matrices whose only nonzero entries are $1_{\lambda +2} e 1_\lambda \:(\lambda \in \mathbb {Z})$ and $1_{\lambda } f 1_{\lambda +2}\:(\lambda \in \mathbb {Z})$ , respectively.

2.2 The iquantum group and its standard/costandard bases

The iquantum group ${\mathrm U}^\imath (\mathfrak {sl}_2)$ is the subalgebra ${\mathrm U}^{\imath }$ of ${\mathrm U}$ generated by

(2.7) $$ \begin{align} b := f +\rho(f) = f+q^{-1} e k^{-1}. \end{align} $$

As an algebra, ${\mathrm U}^{\imath }$ is uninteresting since it is the free $\mathbb {Q}(q)$ -algebra on b. However, it is an interesting coideal subalgebra of ${\mathrm U}$ for an appropriate choice of comultiplication.

The symmetry $\rho $ of ${\mathrm U}$ restricts to a linear antiinvolution $\rho :{\mathrm U}^{\imath }\rightarrow {\mathrm U}^{\imath }$ with $\rho (b) = b$ . Also, the bar involution $\psi ^\imath :{\mathrm U}^{\imath }\rightarrow {\mathrm U}^{\imath }$ is the unique antilinear involution such that $\psi ^\imath (b) = b$ . We stress a key point: $\psi ^\imath $ is not the restriction of the bar involution $\psi $ on ${\mathrm U}$ ; indeed, the latter does not leave ${\mathrm U}^{\imath }$ invariant. For $\lambda \in \mathbb {N}$ , there is a unique antilinear involution $\psi _\lambda ^\imath :V(\lambda )\rightarrow V(\lambda )$ such that

(2.8) $$ \begin{align} \psi_\lambda^\imath(\eta_\lambda) &= \eta_\lambda,& \psi_\lambda^\imath(uv) &= \psi^\imath(u) \psi_\lambda^\imath(v) \end{align} $$

for all $u \in {\mathrm U}^{\imath }, v \in V(\lambda )$ ; see [Reference Bao and WangBW18b, Cor. 3.11] and [Reference Bao and WangBW18a, Prop. 5.1]. Also, by [Reference Bao and WangBW18a, Lem. 6.25], there is a symmetric bilinear form $(\cdot ,\cdot )^\imath : {\mathrm U}^{\imath }\times {\mathrm U}^{\imath } \rightarrow \mathbb {Q}(q)$ such that

(2.9) $$ \begin{align} ( u_1, u_2)^\imath = \lim_{\lambda\rightarrow \infty} \big(u_1 \eta_\lambda, u_2 \eta_\lambda \big)_\lambda \end{align} $$

for all $u_1,u_2 \in {\mathrm U}^{\imath }$ . From (2.5), we get that

(2.10) $$ \begin{align} (b u_1, u_2)^\imath &= (u_1, b u_2)^\imath \end{align} $$

for any $u_1,u_2 \in {\mathrm U}^{\imath }$ . In [Reference Bao and WangBW18a, Th. 6.27], it is shown that $(\cdot ,\cdot )^\imath $ is nondegenerate. This also follows from the following theorem together with the nondegeneracy of the form $(\cdot ,\cdot )^-$ on ${\mathrm U}^-$ .

Theorem 2.1. There is a unique isomorphism of $\mathbb {Q}(q)$ -vector spaces $\jmath :{\mathrm U}^{\imath } \stackrel {\sim }{\rightarrow } \mathbf {f}$ such that

(2.11) $$ \begin{align} \lim_{\lambda \rightarrow \infty} \big(u \eta_\lambda, x^-\eta_\lambda\big)_\lambda =(\jmath(u),x) \end{align} $$

for all $u \in {\mathrm U}^{\imath }$ and $x \in \mathbf {f}$ . Moreover, the following hold for $u,u_1,u_2 \in {\mathrm U}^{\imath }$ :

1. 1. $\jmath (bu) = \theta \jmath (u) + R(\jmath (u))$ .

2. 2. $(u_1,u_2)^\imath = \big (\jmath (u_1), \jmath (u_2)\big )$ .

Proof. Uniqueness of a linear map $\jmath $ satisfying (2.11) follows easily from the nondegeneracy of the form $(\cdot ,\cdot )$ . To prove existence, we can assume that u is a power of b and proceed by induction on degree. Let $\jmath (1) := 1$ , which clearly satisfies (2.11) for all $x \in \mathbf {f}$ . Now assume for some $u \in {\mathrm U}^{\imath }$ that $\jmath (u)$ satisfying (2.11) for all x has been constructed inductively, and consider $\jmath (bu)$ . Using (2.5) and the identity (2.4) multiplied on the left by $q k^{-1}$ , we have that

$$ \begin{align*} \lim_{\lambda \rightarrow \infty} \big( b u \eta_\lambda, x^- \eta_\lambda \big)_\lambda &\!\!\stackrel{(2.5)}{=}\! \lim_{\lambda \rightarrow \infty} \big( u \eta_\lambda, b x^- \eta_\lambda \big)_\lambda = \lim_{\lambda \rightarrow \infty} \big( u \eta_\lambda, f x^- \eta_\lambda + q k^{-1} e x^- \eta_\lambda \big)_\lambda\\ &\!\!\!\stackrel{(2.4)}{=}\! \lim_{\lambda \rightarrow \infty} \big( u \eta_\lambda, f x^- \eta_\lambda + R(x)^- \eta_\lambda - k^{-1} R(x)^- k^{-1} \eta_\lambda \big)_\lambda\\ &= \lim_{\lambda \rightarrow \infty} \big( u \eta_\lambda, (\theta x)^- \eta_\lambda + R(x)^- \eta_\lambda \big)_\lambda\\ &= (\jmath(u), \theta x + R(x)) \stackrel{(2.3)}{=} (\theta \jmath(u)+R(\jmath(u)),x). \end{align*} $$

So $\jmath (bu) := \theta \jmath (u) + R(\jmath (u))$ satisfies (2.11). This proves the existence of a linear map $\jmath $ satisfying (2.11), and at the same time we have established (1). To see that $\jmath $ is a linear isomorphism, it follows easily from (1) that $\jmath (b^n)$ is a monic polynomial of degree n in $\theta $ . Since ${\mathrm U}^{\imath }$ and $\mathbf {f}$ are free on b and on $\theta $ , respectively, it is now clear that $\jmath $ is an isomorphism.

It remains to prove (2). By the definition (2.9) and (2.11), we need to show that

$$ \begin{align*}\lim_{\lambda \rightarrow \infty} \big(u_1\eta_\lambda, \jmath(u_2)^-\eta_\lambda\big)_\lambda= \lim_{\lambda \rightarrow \infty} \big(u_1 \eta_\lambda, u_2 \eta_\lambda\big)_\lambda \end{align*} $$

for all $u_1,u_2 \in {\mathrm U}^{\imath }$ . Note that the limit on the left hand side exists by what we have proved so far. We assume that $u_2$ is a power of b and proceed by induction on its degree. The base case $u_2 = 1$ is clear. Now assume the result has been proved for all $u_1$ and some $u_2$ , and consider $b u_2$ . Using (1), we have that

$$ \begin{align*} \lim_{\lambda \rightarrow\infty} \big(u_1\eta_\lambda, \jmath(b u_2)^- \eta_\lambda \big)_\lambda&=\; \lim_{\lambda \rightarrow\infty} \big(u_1\eta_\lambda, f \jmath(u_2)^- \eta_\lambda + R(\jmath(u_2))^- \eta_\lambda\big)_\lambda\\ &=\; \lim_{\lambda \rightarrow\infty} \big(u_1\eta_\lambda, f \jmath(u_2)^- \eta_\lambda + R(\jmath(u_2))^- \eta_\lambda -k^{-1} R(\jmath(u_2))^- k^{-1}\eta_\lambda\big)_\lambda\\ &\!\!\stackrel{(2.4)}{=}\! \lim_{\lambda \rightarrow\infty} \big(u_1\eta_\lambda, f \jmath(u_2)^- \eta_\lambda + q k^{-1} e \jmath(u_2)^- \eta_\lambda \big)_\lambda= \lim_{\lambda \rightarrow\infty} \big(u_1\eta_\lambda, b \jmath(u_2)^- \eta_\lambda \big)_\lambda\\ &\!\!\stackrel{(2.5)}{=}\! \lim_{\lambda \rightarrow\infty} \big(b u_1\eta_\lambda, \jmath(u_2)^- \eta_\lambda \big)_\lambda = \lim_{\lambda \rightarrow\infty} \big(b u_1\eta_\lambda, u_2 \eta_\lambda \big)_\lambda \stackrel{(2.5)}{=} \lim_{\lambda \rightarrow\infty} \big(u_1\eta_\lambda, b u_2 \eta_\lambda \big)_\lambda. \end{align*} $$

Applying Theorem 2.1, we let be the unique element such that . The elements give a basis for ${\mathrm U}^{\imath }$ , which we call the costandard basis. From Theorem 2.1(2) and (2.1), we get that

(2.12)

for $m,n\geq 0$ . Thus, the costandard basis is an orthogonal basis. The following recurrence relation is easily deduced using Theorem 2.1(1) and (2.2):

(2.13)

for $n \geq 0$ , interpreting as $0$ . Applying the bar involution $\psi ^\imath $ to the costandard basis gives another basis for ${\mathrm U}^{\imath }$ which we call the standard basis. It satisfies the recurrence

(2.14) $$ \begin{align} \delta_0 &= 1,& b \delta_n &= [n+1]\delta_{n+1} + \frac{q^{1-n}}{1-q^{2}} \delta_{n-1}. \end{align} $$

Despite only differing by an application of the bar involution, we generally prefer to work with $\delta _n$ , although has the advantage of being an orthogonal basis.

2.3 Combinatorics of chord diagrams

Next, we investigate the rational functions $w_{m,n}(q) \in \mathbb {Q}(q)$ defined from the expansion

(2.15)

One reason to be interested in these is that

(2.16)

for any $m,n \geq 0$ .

Lemma 2.3. For $0 \leq n \leq m$ , we have that

$$ \begin{align*} w_{0,0}(q) &= 1, & w_{m,n}(q) &= [n] w_{m-1,n-1}(q) + \frac{q^n w_{m-1,n+1}(q)}{1-q^{-2}}, \end{align*} $$

interpreting $w_{m,n}(q)$ as 0 if $n < 0$ or $n> m$ .

Proof. Applying $\jmath $ to gives that $\jmath (b^m) = \sum _{n=0}^m w_{m,n}(q) \theta ^{(n)}$ . Thus, $w_{m,n}(q)$ is the $\theta ^{(n)}$ -coefficient of $\jmath (b^m)$ . Suppose that $m \geq 1$ . By Theorem 2.1(1), we have that $\jmath (b^{m}) = \theta \jmath (b^{m-1}) + R(\jmath (b^{m-1}))$ . Then we observe using (2.2) that the right hand side equals

$$ \begin{align*}\sum_{n=1}^{m} [n] w_{m-1,n-1}(q) \theta^{(n)} + \sum_{n=0}^{m-2} \frac{q^n w_{m-1,n+1}(q)}{1 - q^{-2}} \theta^{(n)}. \end{align*} $$

From this, we see that the coefficient $w_{m,n}(q)$ of $\theta ^{(n)}$ in $\jmath (b^m)$ satisfies the recurrence relation in the statement of the lemma.

We are going to give an elementary combinatorial interpretation of $w_{m,n}(q)$ in terms of certain chord diagrams with n chords tethered to a fixed basepoint and $f=(m-n)/2$ free chords. The notion of a chord diagram is quite standard and we will not give a formal definition, but note in our setup that a pair of chords cannot intersect twice, and a chord cannot have both ends tethered. The following is an example of a chord diagram with $n=3$ tethered chords, $f=4$ free chords, and $c=11$ crossings:

(2.17)

The three tethered chords are the ones attached to the basepoint. We have also numbered the free endpoints of the tethered chords in order going clockwise around the circle. Here is one more example with $n=4, f=3$ , and $c=5$ :

(2.18)

In a chord diagram with f free and n tethered chords, the maximum possible number of crossings is $nf+\frac {1}{2} f(f-1)$ . Counting chord diagrams up to planar isotopy fixing the basepoint, let $N(f,n,c)$ be the number of chord diagrams with f free chords, n tethered chords, and c crossings, and

(2.19) $$ \begin{align} T_{f,n}(q) := \sum_{c=0}^{nf+\frac{1}{2}f(f-1)} N(f,n,c) q^c \in \mathbb{N}[q] \end{align} $$

be the resulting generating function. We obviously have that $T_{0,n}(q) = 1$ , and $T_{1,n}(q) = (n+1)+nq + (n-1)q^2+\cdots + q^n$ . Other examples: $T_{2,0}(q) = 2+q$ and $T_{3,0}(q) = 5+6q+3q^2+q^3$ . Note also that $T_{f,n}(1) = \binom {2f+n}{n} (2f-1)!!$ (here, $n!!$ denotes the double factorial defined recursively by $n!!=n\cdot (n-2)!!$ and $0!!= (-1)!!=1$ ). Let $\{n\}$ be the classical q-integer $1+q+q^2+\cdots +q^{n-1}$ .

Lemma 2.4. The generating function $T_{f,n}(q)$ satisfies the recurrence relation

(2.20) $$ \begin{align} T_{0,0}(q) &= 1,& T_{f,n}(q) &= T_{f,n-1}(q) +\{n+1\} T_{f-1,n+1}(q), \end{align} $$

interpreting $T_{f,n}(q)$ as 0 if n or f is negative.

Proof. It is clear that $T_{0,0}(q) = 1$ . Now suppose that $f+n> 0$ . Let $\mathrm {C}(f,n)$ be the set of chord diagrams with f free and n tethered chords. We are going to construct a set partition

$$ \begin{align*}\mathrm{C}(f,n) = \overline{\mathrm{C}}(f,n)\sqcup \coprod_{i=0}^n \mathrm{C}_i(f,n). \end{align*} $$

Take a chord diagram $D \in \mathrm {C}(f,n)$ . Consider the chord x in D that has the nearest free endpoint to the basepoint measured in a clockwise direction around the circumference of the circle. There are two cases:

• • If x is a tethered chord then we put D into the set $\overline {\mathrm {C}}(f,n)$ and let $\theta (D) \in \mathrm {C}(f,n-1)$ be the chord diagram obtained from D by removing x. Note that $\theta (D)$ has the same number of crossings as D. An example of this situation is given by (2.18); for this $\theta (D)$ is

  

• • Otherwise, x is a free chord. Its furthest endpoint from the basepoint lies between the free endpoints of the ith and $(i+1)$ th tethered chords for some $0 \leq i \leq n$ . We put D into the set $\mathrm {C}_i(f,n)$ and let $\theta _i(D) \in \mathrm {C}(f-1,n+1)$ be the chord diagram obtained from D by replacing x by a tethered chord y with the same furthest endpoint as x. Note that $\theta _i(D)$ has i fewer crossings than D since y crosses i fewer tethered chords compared to x. An example is given by (2.17); for this, we have that $i=2$ and $\theta _2(D)$ is

  

We have now defined the partition of $\mathrm {C}(f,n)$ . It is also clear that $\theta :\overline {\mathrm {C}}(f,n)\stackrel {\sim }{\rightarrow } \mathrm {C}(f,n-1)$ and all $\theta _i:\mathrm {C}_i(f,n) \stackrel {\sim }{\rightarrow } \mathrm {C}(f-1,n+1)$ are bijections. The lemma follows by computing the generating function $T_{f,n}(q)$ using this partition to see that $T_{f,n}(q) = T_{f,n-1}(q) + \sum _{i=0}^n q^i T_{f-1,n+1}(q)$ .

Theorem 2.5. For $0 \leq n \leq m$ with $n \equiv m{\:\left (\mathrm {mod}\ 2\right )}$ , we have that

$$ \begin{align*}w_{m,n}(q) = \begin{cases} [n]!\frac{T_{f,n}(q^2)}{(1-q^{-2})^f} &\text{if }m = n+2f\text{ for some }f \in \mathbb{N}\\ 0&\text{otherwise.} \end{cases} \end{align*} $$

Proof. It is clear from Lemma 2.3 that $w_{m,n}(q) = 0$ if $n \not \equiv m{\:\left (\mathrm {mod}\ 2\right )}$ . Also using Lemma 2.3 it follows that the rational function $\widetilde {T}_{f,n}(q)$ defined from

$$ \begin{align*}\widetilde{T}_{f,n}(q^2) := (1-q^{-2})^f w_{n+2f,n}(q) / [n]! \end{align*} $$

satisfies the recurrence relation in Lemma 2.4. Hence, $\widetilde {T}_{f,n}(q^2) = T_{f,n}(q^2)$ and the result follows.

Corollary 2.6. The bilinear form $(\cdot ,\cdot )^\imath $ on ${\mathrm U}^{\imath }$ satisfies

$$ \begin{align*}(b^m, b^n)^\imath = \begin{cases} \textstyle\frac{T_{f,0}(q^2)}{(1-q^{-2})^f} &\text{if }m+n=2f\text{ for some }f\in \mathbb{N}\\ 0&\text{otherwise.} \end{cases} \end{align*} $$

For example, Corollary 2.6 implies the following:

(2.21) $$ \begin{align} (b,b)^\imath = (1, b^2)^\imath &=\frac{1}{1-q^{-2}}, & (b^2, b^2)^\imath = (b, b^3)^\imath = (1, b^4)^\imath &= \frac{2+q^2}{(1-q^{-2})^2}. \end{align} $$

The generating function $T_{f,0}(q)$ for ordinary chord diagrams has been studied classically; for example, see [Reference RiordanRio75]. Our more general tethered chord diagrams will show up again in a slightly different guise later in the article; see Example 5.2.

2.4 The icanonical basis

So far we have not used the parameter $t \in \{0,1\}$ , but all subsequent results depend on it. To avoid notational confusion, it is helpful to appeal to the construction from [Reference Bao and WangBW18b, Chap. 4] and [Reference Bao and WangBW18a, Sec. 3.7], which shows that ${\mathrm U}^{\imath }$ has a modified form $\dot {\mathrm U}^{\imath } = \dot {\mathrm U}^{\imath } 1_{\bar 0} \oplus \dot {\mathrm U}^{\imath } 1_{\bar 1}$ . We will denote the summands here simply by ${\mathrm U}_0^{\imath }$ and ${\mathrm U}_1^{\imath }$ since they are actually unital algebras. In fact, the map ${\mathrm U}^{\imath } \rightarrow \dot {\mathrm U}_t^{\imath }, u \mapsto u 1_t$ is an algebra isomorphism. We use this to transport all of the results about ${\mathrm U}^{\imath }$ established so far to ${\mathrm U}_t^{\imath }$ , and work only with the latter from now on. In particular, ${\mathrm U}_t^{\imath }$ is freely generated by $b = b 1_t$ , it has the symmetries $\rho $ and $\psi ^\imath $ fixing b as before, it possesses a bilinear form $(\cdot ,\cdot )^\imath $ as in (2.9), there is a linear isomorphism $\jmath :{\mathrm U}_t^{\imath }\stackrel {\sim }{\rightarrow }\mathbf {f}$ as in Theorem 2.1, and we have the standard basis $\delta _n\:(n \geq 0)$ for ${\mathrm U}_t^{\imath }$ satisfying (2.14). However, one should have in mind that ${\mathrm U}_t^{\imath }$ is a subalgebra not of the original quantum group ${\mathrm U}$ but rather of the summand of the completion of $\dot {\mathrm U}$ consisting of matrices $(a_{\mu ,\lambda })_{\mu ,\lambda \in \mathbb {Z}} \in \prod _{\lambda ,\mu \in \mathbb {Z}} 1_\mu \dot {\mathrm U} 1_\lambda $ such that $a_{\mu ,\lambda } = 0$ if $\lambda ,\mu \not \equiv t {\:\left (\mathrm {mod}\ 2\right )}$ . This means that ${\mathrm U}_t^{\imath }$ should only be allowed to act on ${\mathrm U}$ -modules whose weights satisfy $\lambda \equiv t {\:\left (\mathrm {mod}\ 2\right )}$ . For example, the definition (2.9) of the form $(\cdot ,\cdot )^\imath $ on ${\mathrm U}_t^{\imath }$ should really be written now as

(2.22) $$ \begin{align} ( u_1, u_2)^\imath = \lim_{\substack{\lambda\rightarrow \infty\\ \lambda\equiv t{\:\left(\mathrm{mod}\ 2\right)}}} \big(u_1 \eta_\lambda, u_2 \eta_\lambda \big)_\lambda \end{align} $$

for all $u_1,u_2 \in {\mathrm U}_t^{\imath }$ .

By the integrality properties from [Reference Bao and WangBW18b, Th. 4.18] and [Reference Bao and WangBW18a, Th. 5.3], the symmetry $\psi _\lambda ^\imath $ restricts to an antilinear involution on ${{}_{\mathbb {Z}}}V(\lambda )$ . Applying [Reference Bao and WangBW18b, Th. 4.20] and [Reference Bao and WangBW18a, Th. 5.7], we define the icanonical basis for $V(\lambda )$ to be the unique ${\mathbb {Z}[q,q^{-1}]}$ -basis $b^{(n)} \eta _\lambda \:(0 \leq n \leq \lambda )$ for ${{}_{\mathbb {Z}}}V(\lambda )$ such that each $b^{(n)} \eta _\lambda $ is $\psi _\lambda ^\imath $ -invariant and

$$ \begin{align*}b^{(n)} \eta_\lambda - f^{(n)}\eta_\lambda \in\sum_{m=0}^\lambda q^{-1}\mathbb{Z}[q^{-1}] f^{(m)} \eta_\lambda. \end{align*} $$

As the notation suggests, for $\lambda \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ , the vector $b^{(n)} \eta _\lambda $ is obtained by applying an element $b^{(n)} \in {\mathrm U}_t^{\imath }$ to $\eta _\lambda $ . In fact, there is a unique element $b^{(n)} \in {\mathrm U}_t^{\imath }\:(n \geq 0)$ such that $b^{(n)} \eta _\lambda $ is the icanonical basis element of $L(\lambda )$ for all $0\leq n \leq \lambda $ with $\lambda \equiv t {\:\left (\mathrm {mod}\ 2\right )}$ ; see [Reference Bao and WangBW18b, Chap. 4] and [Reference Berman and WangBW18c, Th. 2.10, Th. 3.6]. The elements $b^{(n)}\:(n \geq 0)$ thus defined give a remarkable basis for ${\mathrm U}_t^{\imath }$ again called the icanonical basis.

Closed formulae for the icanonical basis elements were worked out in [Reference Berman and WangBW18c] (see also [Reference Bao and WangBW18b]): for $n\geq 0$ , we have that

(2.23) $$ \begin{align} b^{(n)} := \begin{cases} \frac{1}{[n]!} \prod\limits_{\substack{k=0\\k\equiv t{\:\left(\mathrm{mod}\ 2\right)}}}^{n-1} \!\!\!\left(b^2 - [k]^2\right) &\text{if }n\text{ is even}\\ \frac{b}{[n]!} \prod\limits_{\substack{k=1\\k\equiv t{\:\left(\mathrm{mod}\ 2\right)}}}^{n-1} \!\!\!\left(b^2 - [k]^2\right) &\text{if }n\text{ is odd.} \end{cases} \end{align} $$

This is also known as the idivided power. It is straightforward to check from (2.23) that the icanonical basis satisfies the recurrence relation

(2.24) $$ \begin{align} b^{(0)} &= 1,& b b^{(n)} &= [n+1]b^{(n+1)} + \delta_{n\equiv t} [n]b^{(n-1)}, \end{align} $$

for any $n \geq 0$ .

Theorem 2.7. For $n \geq 0$ , we have that

(2.25) $$ \begin{align} b^{(n)} &= \sum_{m=0}^{\lfloor \frac{n}{2}\rfloor} \frac{q^{m(2m+1-2\delta_{n\equiv t})}}{(1-q^{4}) (1-q^{8})\cdots (1-q^{4m})} \delta_{n-2m}, \end{align} $$

(2.26) $$ \begin{align} \delta_n &= \sum_{m=0}^{\lfloor \frac{n}{2}\rfloor}(-1)^m \frac{q^{m(2\delta_{n\not\equiv t}+1)}}{(1-q^{4}) (1-q^{8})\cdots (1-q^{4m})} b^{(n-2m)}. \end{align} $$

Corollary 2.8. The icanonical basis of ${\mathrm U}_t^{\imath }$ is almost orthonormal in the sense that

for $m,n \geq 0$ .

Remark 2.9. Using (2.25), one can derive the following explicit formula for the pairings between icanonical basis elements:

$$ \begin{align*} (b^{(n)},b^{(m)})^\imath &= \sum_{\substack{0 \leq i \leq \min(m,n)\\ i\equiv n \equiv m{\:\left(\mathrm{mod}\ 2\right)}}} \frac{q^{-\frac{1}{2}(n-i) (n-i+1-2\delta_{n\equiv t})-\frac{1}{2}(m-i) (m-i+1-2\delta_{m\equiv t})}}{ \prod_{j=1}^i (1-q^{-2j}) \prod_{k=1}^{\frac{n-i}{2}}(1-q^{-4k}) \prod_{l=1}^{\frac{m-i}{2}}(1-q^{-4l})} \end{align*} $$

for any $m,n \geq 0$ . This is 0 if $m \not \equiv n{\:\left (\mathrm {mod}\ 2\right )}$ .

The icanonical basis in fact gives a basis for an integral form ${{}_{\mathbb {Z}}}{\mathrm U}_t^{\imath }$ of ${\mathrm U}_t^{\imath }$ over ${\mathbb {Z}[q,q^{-1}]}$ . Equivalently, we have that

$$ \begin{align*}{{}_{\mathbb{Z}}}{\mathrm U}_t^{\imath} = \left\{u \in {\mathrm U}_t^{\imath}\:\big|\:u \left({{}_{\mathbb{Z}}}V(\lambda)\right) \subseteq {{}_{\mathbb{Z}}}V(\lambda)\text{ for all }\lambda \in \mathbb{N}\text{ with }\lambda\equiv t{\:\left(\mathrm{mod}\ 2\right)}\right\}, \end{align*} $$

from which one sees that ${{}_{\mathbb {Z}}}{\mathrm U}_t^{\imath }$ is a ${\mathbb {Z}[q,q^{-1}]}$ -subalgebra of ${\mathrm U}_t^{\imath }$ . Since both $\rho $ and $\psi ^\imath $ fix each of the icanonical basis elements $b^{(n)}$ , they restrict to symmetries on ${{}_{\mathbb {Z}}}{\mathrm U}_t^{\imath }$ . Also, the form on ${\mathrm U}_t^{\imath }$ restricts to $(\cdot ,\cdot )^\imath :{{}_{\mathbb {Z}}}{\mathrm U}_t^{\imath }\times {{}_{\mathbb {Z}}}{\mathrm U}_t^{\imath }\rightarrow \mathbb {Z}[q,q^{-1}]$ . From (2.14), it is apparent that $\delta _n \in {\mathrm U}_t^{\imath }$ does not lie in the integral form.

2.5 The character ring

The character ring is the ring for a formal variable $\chi $ . It is natural to consider it from a representation-theoretic perspective (see subsection 5.4). We view as a left ${\mathrm U}_t^{\imath }$ -module so that

(2.27) $$ \begin{align} b \chi^n &= \begin{cases} \chi^{n-1}&\text{if }n> 0\\ 0&\text{if }n=0, \end{cases} \end{align} $$

and extending in the natural way to power series. We identify the character ring with the full linear dual $\left ({\mathrm U}_t^{\imath }\right )^*$ so that $\sum _{n \geq 0} f_n(q) \chi ^n$ is the function mapping $b^n$ to $f_n(q)$ . Thus, the topological basis for given by the monomials $\chi ^n\:(n \geq 0)$ is dual to the basis $b^n\:(n \geq 0)$ of ${\mathrm U}_t^{\imath }$ . Then we let $\ell _n, \bar \delta _n$ , and be the unique elements of the character ring such that

(2.28)

for $m,n \geq 0$ . The topological bases $\ell _n\:(n \geq 0)$ , $\bar \delta _n\:(n \geq 0)$ and for give the dual canonical basis, proper standard basis, and proper costandard basis, respectively.

There is a bar involution on the character ring, which is the antilinear map

(2.29)

This is compatible with the bar involution on ${\mathrm U}_t^{\imath }$ in the sense that $\bar \psi ^\imath (ug) = \psi ^\imath (u) \bar \psi ^\imath (g)$ for $u \in {\mathrm U}_t^{\imath }$ and . Also, the bar involution on is related to the bar involution on ${\mathrm U}_t^{\imath }$ by the formula

(2.30) $$ \begin{align} \big(\bar\psi^\imath(g)\big)(u) = \overline{g(\psi^\imath(u))}. \end{align} $$

This follows easily from (2.29) as $\psi ^\imath (b^n) = b^n$ for each $n \geq 0$ . Using (2.30) and the definitions, it follows that the dual canonical basis elements $\ell _n$ are fixed by $\bar \psi ^\imath $ , and .

From Theorem 2.7, we get that

(2.31) $$ \begin{align} \bar\delta_n &= \sum_{m=0}^\infty \frac{q^{m(2m+1-2\delta_{n\equiv t})}}{(1-q^{4}) (1-q^{8})\cdots (1-q^{4m})} \ell_{n+2m}, \end{align} $$

(2.32) $$ \begin{align} \ell_n &= \sum_{m=0}^\infty(-1)^m \frac{q^{m(2\delta_{n\not\equiv t}+1)}}{(1-q^{4}) (1-q^{8})\cdots (1-q^{4m})} \bar\delta_{n+2m}. \end{align} $$

for $n \geq 0$ . Also the following recurrence relations may be deduced from (2.14) and (2.24):

(2.33) $$ \begin{align} b \bar \delta_{n} &= [n]\bar\delta_{n-1} + \frac{q^{-n}}{1-q^{2}}\bar\delta_{n+1}, \end{align} $$

(2.34) $$ \begin{align} b \ell_n &= [n] \ell_{n-1} + \delta_{n\not\equiv t} [n+1] \ell_{n+1} \end{align} $$

for any $n \geq 0$ .

We proceed to derive explicit formulae for $\bar \delta _n$ and $\ell _n$ as formal series in $\chi $ .

Lemma 2.10. For $n \geq 0$ , we have that

$$ \begin{align*}\bar\delta_n = [n]! \sum_{f \geq 0} \frac{T_{f,n}(q^{-2})}{(1-q^{2})^f} \chi^{n+2f}. \end{align*} $$

Theorem 2.12. We have that

(2.35) $$ \begin{align} \ell_n = [n]! \chi^n \prod_{\substack{1 \leq k\leq n+1 \\k\equiv t {\:\left(\mathrm{mod}\ 2\right)}}} \frac{1}{1-[k]^2 \chi^2} = [n]! \sum_{m \geq 0} \left(\sum_{\alpha \in \mathcal{P}_t(m\times n)} [\alpha_1+1]^2\cdots [\alpha_m+1]^2\right)\, \chi^{n+2m} \end{align} $$

where $\mathcal {P}_t(m\times n)$ is the set of $\alpha \in \mathbb {N}^m$ with $0 \leq \alpha _1 \leq \cdots \leq \alpha _m \leq n$ and $\alpha _i\not \equiv t {\:\left (\mathrm {mod}\ 2\right )}$ for each i.

Proof. The second equality follows by expanding the product. To prove the first equality, we proceed by induction on n. The induction base follows from Lemma 2.11. For the induction step, take $n> 0$ . The constant term of $\ell _n$ is 0 since $\ell _n(1) = \ell _n(b^{(0)}) = 0$ so we have that $b \ell _n = \ell _n / \chi $ by (2.27). Suppose first that $n\equiv t{\:\left (\mathrm {mod}\ 2\right )}$ . Then (2.34) shows that

(2.36) $$ \begin{align} \ell_n = [n]\ \chi \ell_{n-1} \end{align} $$

and we easily get done by induction in this case. When $n\not \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ , (2.34) gives that

$$ \begin{align*}\ell_n = [n]\ \chi \ell_{n-1} + [n+1]\ \chi \ell_{n+1} = [n]\ \chi \ell_{n-1} + [n+1]^2 \chi^2 \ell_n. \end{align*} $$

Hence,

(2.37) $$ \begin{align} \ell_n = \frac{[n]\ \chi}{1-[n+1]^2 \chi^2} \ell_{n-1}, \end{align} $$

and again the result follows by induction.

Corollary 2.13. For $n \geq 0$ , we have that

$$ \begin{align*}b^n= \sum_{m=0}^{\lfloor \frac{n}{2}\rfloor}[n-2m]! \left(\sum_{\alpha \in \mathcal{P}_t(m\times (n-2m))} [\alpha_1+1]^2\cdots [\alpha_m+1]^2\right) b^{(n-2m)}. \end{align*} $$

3.1 Definition and basic properties

We use the usual string calculus for morphisms in strict monoidal categories; our general convention is that $f \circ g$ denotes composition of f drawn on top of g (“vertical composition”) and $f \star g$ denotes the tensor product of f drawn to the left of g (“horizontal composition”). We always draw string diagrams so that the underlying strings are smooth curves. Recall the following definition from [Reference Brundan, Wang and WebsterBWW24, Def. 2.1].

Definition 3.1. The nil-Brauer category $\mathbf {NB}_t$ is the strict graded monoidal category with one generating object B (whose identity endomorphism will be represented diagrammatically by the unlabeled string  ) and four generating morphisms

(3.1)

subject to the following relations:

(3.2)

(3.3)

(3.4)

(3.5)

The following relations are easily derived from the defining relations in [Reference Brundan, Wang and WebsterBWW24, (2.6)–(2.8)]:

(3.6)

(3.7)

(3.8)

In view of the last relation from (3.4) and the first relation from (3.6), we can unambiguously denote the morphisms in these two equations by the “pitchforks” and , respectively. Together with the last relation of (3.3), it then follows that a string diagram with no dots can be deformed under planar isotopy without changing the morphism that it represents. This is not true in the presence of dots due to the sign in the last relations of (3.5) and (3.8) – there is a sign change whenever a dot slides across the critical point of a cup or cap.

The relations discussed so far imply that there are strict graded monoidal functors

(3.9) $$ \begin{align} {\scriptstyle\mathrm{R}}:\mathbf{NB}_t &\rightarrow \mathbf{NB}_t^{{\mathrm{rev}}}, &B &\mapsto B, & s &\mapsto (-1)^{\bullet(s)} s^{\leftrightarrow},\end{align} $$

(3.10) $$ \begin{align} {\scriptstyle\mathrm{T}}:\mathbf{NB}_t &\rightarrow \mathbf{NB}_t^{{\mathrm{op}}}, &B &\mapsto B, & s &\mapsto s^{\updownarrow}. \end{align} $$

Here, for a string diagram s we use $s^{\updownarrow }$ and $s^{\leftrightarrow }$ to denote its reflection in a horizontal or vertical axis, and $\bullet (s)$ denotes the total number of dots in the diagram. The category $\mathbf {NB}_t$ is pivotal with duality functor ${\scriptstyle \mathrm {D}} := {\scriptstyle \mathrm {R}} \circ {\scriptstyle \mathrm {T}} = {\scriptstyle \mathrm {T}} \circ {\scriptstyle \mathrm {R}}$ , which rotates a string diagram s through $180^\circ $ then scales by $(-1)^{\bullet (s)}$ .

3.2 Generating functions for dots and bubbles

Next we recall the generating function formalism from [Reference Brundan, Wang and WebsterBWW24, Sec. 2]. We denote the rth power of under vertical composition simply by labeling the dot with r. More generally, given a polynomial $f(x) = \sum _{r\geq 0} c_r x^r \in \Bbbk [x]$ and a dot in some string diagram s, we denote

$$ \begin{align*}\sum_{r \geq 0} c_r \times (\text{the morphism obtained from }s\text{ by labeling the dot by }r) \end{align*} $$

by attaching what we call a pin to the dot, labeling the node at the head of the pin by $f(x)$ :

(3.11)

In the drawing of a pin, the arm and the head of the pin can be moved freely around larger diagrams so long as the dot at the pointy end stays put – these are not part of the string calculus. More generally, $f(x)$ here could be a polynomial with coefficients in the algebra $\Bbbk (\!( u^{-1} )\!)$ of formal Laurent series in an indeterminate $u^{-1}$ ; then the string s decorated with a pin labeled $f(x)$ defines a generating function of morphisms.

We will use the following shorthands for the generating functions of [Reference Brundan, Wang and WebsterBWW24, (2.14)–(2.15)]:

(3.12)

(3.13)

The notation here is motivated by the following standard trick: for any $f(x) \in \Bbbk [x]$ , we have that

(3.14)

where $[-]_{u^r}$ denotes the $u^r$ -coefficient of the formal Laurent series inside the brackets. These identities follow by using linearity to reduce to the case that $f(x) = x^n$ for $n \geq 0$ , then explicitly computing coefficients on both sides. As we do with ordinary dots, we denote the nth power of one of these “dot generating functions” by labeling them also by n. This makes sense for any $n \in \mathbb {Z}$ since we have by the definitions that

The endomorphisms (3.12) and (3.13) obviously commute with each other and all other pins. Note also that ${\scriptstyle \mathrm {T}}$ and ${\scriptstyle \mathrm {R}}$ satisfy

(3.15)

Another useful trick is to apply the substitution $u \mapsto -u$ ; this interchanges and .

It is clear from the last relation in (3.4) that and similarly for cups; hence, we have that

(3.16)

Further useful relations involving these generating functions are

(3.17)

These are also noted in [Reference Brundan, Wang and WebsterBWW24, (2.19)–(2.20)]. Equating the coefficients of $u^{-n-1}$ , we obtain

(3.18)

(3.19)

Now consider the “dotted bubble generating function”

(3.20)

This is often useful, but even more important will be the renormalization

(3.21)

Its $u^{-r}$ -coefficients $\mathbb {O}_r$ are given explicitly by

(3.22)

for $r \geq 1$ . Note also by (3.15) and (3.16) that $\mathbb {O}(u)$ is invariant under ${\scriptstyle \mathrm {R}}$ and ${\scriptstyle \mathrm {T}}$ .

Theorem 3.3 [Reference Brundan, Wang and WebsterBWW24, Th. 2.5].

The following relations hold in $\mathbf {NB}_t$ :

(3.23)

(3.24)

(3.25)

(3.26)

Corollary 3.4. The following relations hold in $\mathbf {NB}_t$ :

(3.27)

(3.28)

Corollary 3.5. For $n \geq 0$ , we have that

3.3 The basis theorem

Let $\Lambda $ be the graded algebra of symmetric functions over $\Bbbk $ . Adopting standard notation, this is freely generated either by the elementary symmetric functions $e_r\:(r> 0)$ or by the complete symmetric functions $h_r\:(r> 0)$ ; our convention for the grading puts these in degree $2r$ . The two families of generators are related by the identity

(3.29) $$ \begin{align} e(-u) h(u) = 1 \end{align} $$

where

(3.30) $$ \begin{align} e(u) &= \sum_{r \geq 0} u^{-r} e_r,& h(u) &= \sum_{r \geq 0} u^{-r} h_r \end{align} $$

are the corresponding generating functions, and $e_0 = h_0 = 1$ by convention. It is also convenient to interpret $e_r$ and $h_r$ as $0$ when $r < 0$ .

Following [Reference MacdonaldMac15, Ch. III, Sec. 8], we define a power series and elements $q_r\;(r \geq 0)$ of $\Lambda $ so that

(3.31) $$ \begin{align} q(u) = \sum_{r\geq 0} u^{-r} q_r := e(u)h(u). \end{align} $$

By (3.29), we have that

(3.32) $$ \begin{align} q(u) q(-u) = 1. \end{align} $$

Equivalently, $q_0 = 1$ and

(3.33) $$ \begin{align} q_{2r} &= (-1)^{r-1}{\textstyle\frac{1}{2}}q_r^2 + \sum_{s=1}^{r-1}(-1)^{s-1}q_s q_{2r-s} \end{align} $$

for $r \geq 1$ ; cf. [Reference MacdonaldMac15, (III.8.2’)]. As with $e_r$ and $h_r$ , we adopt the convention that $q_r = 0$ for $r < 0$ .

The graded subalgebra of $\Lambda $ generated by all $q_r\:(r \geq 0)$ is denoted $\Gamma $ . As explained in [Reference MacdonaldMac15], $\Gamma $ is freely generated by $q_1,q_3,q_5,\dots $ (and it has a distinguished basis given by the Schur Q-functions $Q_\lambda $ indexed by all strict partitions). It follows that $\Gamma $ is generated by the elements $q_r\:(r \geq 0)$ subject only to the relations (3.32). Hence, (3.25) is all that is needed to establish the existence of a graded algebra homomorphism

(3.34)

By [Reference Brundan, Wang and WebsterBWW24, Cor. 5.4], this is actually an isomorphism.

Now we recall the basis theorem for morphism spaces in $\mathbf {NB}_t$ , which is the main result of [Reference Brundan, Wang and WebsterBWW24]. For $m, n \geq 0$ , any morphism $f:B^{\star n} \rightarrow B^{\star m}$ is represented by a linear combination of $m \times n$ string diagrams, that is, string diagrams with m boundary points at the top and n boundary points at the bottom that are obtained by composing the generating morphisms from (3.1). It follows that $ {\mathrm {Hom}}_{\mathbf {NB}_t}(B^{\star n}, B^{\star m})$ is 0 unless $m \equiv n {\:\left (\mathrm {mod}\ 2\right )}$ . The individual strings in an $m \times n$ string diagram s are of four basic types: cups (with two boundary points on the top edge), caps (with two boundary points on the bottom edge), propagating strings (with one boundary point at the top and one at the bottom), and internal bubbles (no boundary points). We define an equivalence relation $\sim $ on the set of $m \times n$ string diagrams by declaring that $s \sim s'$ if their strings define the same matching on the set of $m+n$ boundary points. We say that s is reduced if the following properties hold:

• • There are no internal bubbles.

• • Propagating strings have no critical points ( $\!\! =$ points of slope 0).

• • Cups and caps each have exactly one critical point.

• • There are no double crossings ( $\!\! =$ two different strings which cross each other at least twice).

These assumptions imply in particular that there are no self-intersections ( $=$ crossings of a string with itself). Fix a set $\overline {\mathrm {D}}(m\times n)$ of representatives for the $\sim $ -equivalence classes of undotted reduced $m \times n$ string diagrams. The total number of such diagrams is $(m+n-1)!!$ if $m \equiv n {\:\left (\mathrm {mod}\ 2\right )}$ , and there are none otherwise. For each of these $\sim $ -equivalence class representatives, we also choose distinguished points in the interior of each of its strings that are away from points of intersection. Then let $\mathrm {D}(m\times n)$ be the set of all morphisms $f:B^{\star n} \rightarrow B^{\star m}$ which can be obtained by taking an element of $\overline {\mathrm {D}}(m\times n)$ then adding dots labeled by nonnegative multiplicities at each of the distinguished points on the strings.

Now we can make the first significant connection between $\mathbf {NB}_t$ and the iquantum group. Recall the bilinear form $(\cdot ,\cdot )^\imath :{\mathrm U}_t^{\imath } \times {\mathrm U}_t^{\imath } \rightarrow \mathbb {Q}(q)$ from (2.22).

Proof. We compare the explicit combinatorial formula for $(b^m, b^n)^\imath $ from Corollary 2.6 with the graded rank of $ {\mathrm {Hom}}_{\mathbf {NB}_t}(B^{\star n}, B^{\star m})$ as a free graded $\Gamma $ -module computed via Theorem 3.6. If $m \not \equiv n {\:\left (\mathrm {mod}\ 2\right )}$ then $(b^m,b^n)^\imath = 0$ and $\mathrm {D}(m\times n)$ is empty, and the result is clear. Now assume that $m \equiv n {\:\left (\mathrm {mod}\ 2\right )}$ and let $f := (m+n)/2$ . There is an obvious bijection between equivalence classes of $m \times n$ string diagrams and chord diagrams with f free chords and no tethered chords. This just arises by identifying the $(m+n)$ boundary points of strings in an $m \times n$ string diagram with the $(m+n)$ endpoints of chords in a chord diagram in some fixed way that preserves the clockwise ordering, then replacing strings by chords so that the underlying matching of these points is preserved. In a string diagram, each crossing is of degree $-2$ , so it contributes $q^{-2}$ to the graded rank. The dots placed at the f distinguished points produce the factor $1 / (1-q^{2})^f$ , this being $\dim _{q} \Bbbk [x_1,\dots ,x_f]$ with $x_i$ in degree 2. Recalling the definition of the generating function $T_{f,0}(q)$ from (2.19), we deduce that

$$ \begin{align*}\mathrm{rank}_q {\mathrm{Hom}}_{\mathbf{NB}_t}(B^{\star n}, B^{\star m}) = \sum_{s \in \mathrm{D}(m\times n)} q^{\deg(s)} = T_{f,0}(q^{-2}) / (1-q^{2})^f, \end{align*} $$

which is $\overline {(b^m,b^n)^\imath }$ according to Corollary 2.6.

3.4 Central elements

Recall that the center $Z(\mathbf {A})$ of a category $\mathbf {A}$ means the algebra of endomorphisms of its identity endofunctor. Thus, elements of $Z(\mathbf {NB}_t)$ consist of tuples $(z_n)_{n \geq 0}$ for elements $z_n \in {\mathrm {End}}_{\mathbf {NB}_t}(B^{\star n})$ such that $z_m \circ f = f \circ z_n$ for all $m,n \geq 0$ and $f \in {\mathrm {Hom}}_{\mathbf {NB}_t}(B^{\star n}, B^{\star m})$ . In this subsection, we are going to use the dotted bubbles to construct many – conjecturally, all – elements of $Z(\mathbf {NB}_t)$ .

Since and $2$ is invertible in $\Bbbk $ , it makes sense to take the square roots $\sqrt {\mathbb {O}(\pm u)}$ ; we choose the ones that are positive in the sense that they again lie in . We have that $\sqrt {\mathbb {O}(-u)} = \left (\sqrt {\mathbb {O}(u)}\right )^{-1}$ by (3.25). Taking the square roots of both sides of (3.26), both of which lie in , we obtain

(3.35)

Let $e_{r,n}, h_{r,n}, q_{r,n} \in \Bbbk [x_1,\dots ,x_n]^{S_n}$ be the symmetric polynomials in n variables obtained by specializing the symmetric functions $e_r, h_r, q_r$ from (3.30) and (3.31). We have that

(3.36) $$ \begin{align} q_{r,n} = \sum_{s=0}^r e_{s,n} h_{r-s,n}. \end{align} $$

Moreover,

(3.37)

In the statement of the next theorem, for a polynomial $f \in \Bbbk [x_1,\dots ,x_n]$ , we use the notation $f 1_n = 1_n f$ to denote the endomorphism of $B^{\star n}$ defined by interpreting $x_i$ as , that is, the dot on the ith string.

Proof. We need to show that $q_{r,m} 1_m \circ f = f \circ q_{r,n} 1_n$ for any $f \in {\mathrm {Hom}}_{\mathbf {NB}_t}\left (B^{\star n}, B^{\star m}\right )$ . By (3.35), we have that

(3.38)

The result follows from this since the expression on the right hand side clearly has the desired property by the interchange law.

3.5 Cyclotomic quotients

In this subsection, the grading plays no role – we view $\mathbf {NB}_t$ simply as a strict $\Bbbk $ -linear monoidal category. Let $\mathbf {V}$ be a strict $\mathbf {NB}_t$ -module category. This means that $\mathbf {V}$ is a $\Bbbk $ -linear category, and we are given a strict $\Bbbk $ -linear monoidal functor $\mu $ from $\mathbf {NB}_t$ to the strict $\Bbbk $ -linear monoidal category $\mathbf {End}_\Bbbk (\mathbf {V})$ whose objects are $\Bbbk $ -linear endofunctors of $\mathbf {V}$ and whose morphisms are natural transformations. We denote the endofunctor $\mu (B):\mathbf {V}\rightarrow \mathbf {V}$ simply by B. The string calculus for monoidal categories extends to the module category $\mathbf {V}$ in the usual way. We will represent the identity endomorphism of an object V of $\mathbf {V}$ by the labeled string on the right hand boundary of the diagram. For a string diagram s representing a morphism in $ {\mathrm {Hom}}_{\mathbf{NB}_{t}}(B^{\star n},B^{\star m})$ , we represent the morphism $\mu (s)_V:B^n V \rightarrow B^m V$ diagrammatically by .

Let $V \in \mathbf {V}$ be any object. We write $Z_V$ for the center of its endomorphism algebra. Any dotted bubble evaluates to an element of $Z_V$ . In particular, the generating function $\mathbb {O}(u)$ for dotted bubbles from (3.21) evaluates to a power series . Let

(3.39) $$ \begin{align} \theta_V: Z_V[u] \rightarrow {\mathrm{End}}_{\mathbf{V}}(BV) \end{align} $$

be the $\Bbbk $ -algebra homomorphism mapping $z \in Z_V$ to $B z : BV \rightarrow BV$ and u to the endomorphism of $BV$ defined by the dot. We represent $\theta _V(f(u))$ diagrammatically by . The coefficients of $f(x)$ are elements of $Z_V$ which could be represented by coupons on the right-hand string, but we will not attempt to do that here. Let $I_V := \ker \theta _V \unlhd Z_V[u]$ .

Theorem 3.10. Let V be an object of an $\mathbf {NB}_t$ -module category $\mathbf {V}$ as above, and $f(u) \in I_V$ . The formal Laurent series $f^*(u) := (-1)^t f(-u) \mathbb {O}_V(-u) \in Z_V(\!( u^{-1})\!)$ is a polynomial in $I_V$ with the same constant term as $f(u)$ , and $f^{**}(u) = f(u)$ . Moreover, $\tfrac {f(u)+f^*(u)}{2}$ and $\tfrac {f(u)-f^*(u)}{2u}$ are $*$ -invariant elements of $I_V$ . Assuming in addition that $f(u) \in I_V$ is monic,

$$ \begin{align*}g(u) := \begin{cases} \tfrac{f(u)+f^*(u)}{2}&\text{if }\deg f(u) \equiv t {\:\left(\mathrm{mod}\ 2\right)}\\ \tfrac{f(u)-f^*(u)}{2u}&\text{if }\deg f(u) \not\equiv t {\:\left(\mathrm{mod}\ 2\right)} \end{cases} \end{align*} $$

is a monic polynomial in $I_V$ with $\deg g(u) = \deg f(u)$ or $\deg g(u) = \deg f(u)-1$ in the two cases, such that $\mathbb {O}_V(u) = (-1)^t g(-u) / g(u)$ .

Proof. For $r \geq 0$ , we have that

Thus, for $r \geq 0$ , we have that $\left [f^*(u)\right ]_{u^{-r}} =\left [f(-u)\right ]_{u^{-r}} = \delta _{r,0} f(0)$ . This shows that $f^*(u)$ is a polynomial in $Z_V[u]$ with the same constant term as $f(u)$ . Also $f^{**}(u) = f(u) \mathbb {O}_V(u)\mathbb {O}_V(-u)$ , which equals $f(u)$ by (3.25).

Since $f^*(u)$ and $f(u)$ are polynomials with the same constant terms, $\frac {(-u)^r f^*(u) - u^r f(u)}{2u}$ is a polynomial in u for any $r \geq 0$ . So it is an element of $Z_V[u]$ . Its image under $\theta _V$ is

Thus, we have proved that $\frac {(-u)^r f^*(u) - u^r f(u)}{2u} \in I_V$ for any $r \geq 0$ . The $r=0$ and $r=1$ cases show that $\tfrac {f(u)+f^*(u)}{2}$ and $\tfrac {f(u)-f^*(u)}{2u}$ both belong to $I_V$ , which implies that $f^*(u) \in I_V$ . Moreover, it is easy to check using $f^{**}(u) = f(u)$ that both of these elements are $*$ -invariant.

Finally, we assume that $f(u)$ is monic and consider the two cases separately:

• • If $\deg f(u) \equiv t {\:\left (\mathrm {mod}\ 2\right )}$ , then both $f(u)$ and $f^*(u)$ are monic, so $g(u) := \frac {f(u) + f^*(u)}{2} \in I_V$ is monic of the same degree as $f(u)$ .

• • If $\deg f(u) \not \equiv t {\:\left (\mathrm {mod}\ 2\right )}$ then both $f(u)$ and $-f^*(u)$ are monic, so $g(u) := \frac {f(u)-f^*(u)}{2u} \in I_V$ is monic of one less degree than $f(u)$ .

The equality $g^*(u) = g(u)$ means that $(-1)^t g(-u) \mathbb {O}_V(-u) = g(u)$ , so $g(u) \mathbb {O}_V(u) = (-1)^t g(-u)$ . Since $g(u)$ is monic, it is invertible in $Z_V(\!( u^{-1})\!)$ , and the final assertion of the theorem follows.

Some of the ideas in the proof of Theorem 3.10 were suggested by Anthropic’s Claude AI. In the first version of this paper, we were only able to prove the slightly weaker result in the following corollary, which is all that is needed later on in the article.

4.1 Extended graphical calculus

We begin by introducing some further diagrammatical shorthands in the spirit of the “thick calculus” of [Reference Khovanov, Lauda, Mackaay and StošićKLMS12]. We denote the tensor product $|^{\star a}$ of a strings by a single thick string labeled by a. A thick cup or cap labeled by a denotes that number of nested ordinary cups or caps (no crossings). Sometimes it is notationally convenient to be able to split thick strings into thinner ones or to merge thinner strings to obtain thicker ones: the diagrams

simply represent the identity morphisms $B^{\star n} \rightarrow B^{\star a} \star B^{\star b}$ and $B^{\star a} \star B^{\star b} \rightarrow B^{\star n}$ for $a+b=n$ . We will often omit a thickness label on a thick string when it can be inferred from others in the diagram.

For $a+b=n$ , the thick crossing

denotes the morphism $B^{\star a} \star B^{\star b} \rightarrow B^{\star b} \star B^{\star a}$ obtained by composing ordinary crossings according to a reduced expression for the longest of the minimal length $S_n / (S_a \times S_b)$ -coset representatives. We use a thick string decorated with a cross to denote the composition of thin crossings corresponding to a reduced expression for the longest element $w_n$ . For example:

When working with these morphisms, we will often make implicit use of various obvious consequences of the braid relations, such as

In view of the pitchfork relations, one can also draw this cross at the critical point of a thick cup or cap without there being any ambiguity as to the meaning:

We use a dot on a string of thickness n labeled by $\alpha \in \mathbb {N}^n$ to denote the tensor product of dots on ordinary strings labeled by the parts of $\alpha $ :

The n-tuples $\rho _n := (n-1,n-2,\dots ,1,0) \in \mathbb {N}^n$ and $\varpi _{r,n} := (1,\dots ,1,0,\dots ,0) \in \mathbb {N}^n$ with r entries equal to $1$ followed by $(n-r)$ entries equal to $0$ will appear often. To simplify notation, we allow the subscript n to be omitted in these when used to label a node on a string of thickness n:

Generalizing the notation (3.11), given a polynomial $f= \sum _{\alpha \in \mathbb {N}^n} c_\alpha x_1^{\alpha _1} \cdots x_n^{\alpha _n} \in \Bbbk [x_1,\dots ,x_n]$ , the pin

denotes the endomorphism $f 1_n = 1_n f$ of $B^{\star n}$ . Often this f will be the elementary symmetric polynomial $e_{r,n} := \sum _{1 \leq i_1 < \cdots < i_r \leq n} x_{i_1} \cdots x_{i_r}$ . Again, if this is pinned to a string of thickness n, we allow the subscript n to be dropped, writing simply

since the number n of variables in the elementary symmetric polynomial can be inferred from the thickness of the string.

Lemma 4.1. For $0 \leq r \leq n$ , we have that

(4.1)

Lemma 4.2. For $0 \leq i \leq n$ , we have that

(4.2)

Proof. We just prove the first identity; the second then follows on applying ${\scriptstyle \mathrm {R}}$ . By Theorem 3.6, the lowest nonzero degree of $ {\mathrm {End}}_{\mathbf {NB}_t}(B^{\star (n+1)})$ is $-n(n+1)$ , and the diagram on the left hand side of the identity is of degree $-n(n-1)-4n+2i$ . If $i < n$ then $-n(n-1)-4n+2i < -n(n+1)$ so the expression is 0.

To prove the result in the remaining case that $i=n$ , we proceed by induction on n. Assume the result is true for n and consider the next case

In this expression, the term before the summation is 0 by the degree argument given already, the first term in the summation is 0 unless $a=0$ by the defining relations (3.2), and similarly the second term in the summation is 0 unless $b=0$ . So

where we used the induction hypothesis for the second equality.

Corollary 4.3. For $0 \leq i \leq n+1$ , we have that

(4.3)

Proof. As usual, we just prove the first equality. By the braid relation then Corollary 3.5 and Lemma 4.2, we get that

Proof. This follows by induction on n. For the induction step, we have that

using Lemma 4.2 for the first equality and the induction hypothesis for the second one.

Corollary 4.5. For $0 \leq r \leq n$ , we have that

(4.4)

The remaining relations to be established in this subsection are more complicated. The guiding principle here is that relations in the nil-Hecke algebra can be ported to the nil-Brauer category providing there are enough additional strings to eliminate the cup/cap term in the dot sliding relation (3.8).

Lemma 4.6. For $0 \leq i \leq n+1$ , we have that

(4.5)

Proof. We first slide both sets of i dots downwards past the crossing using (3.18) and (3.19) to see that

So

Now the lemma follows using also the identities

These are consequences of Lemma 4.2 and Corollary 4.3.

Lemma 4.7. For $i,j \geq 0$ with $i+j \leq 2n+3$ , we have that

(4.6)

Proof. We assume that $i \leq j$ , and proceed by induction on $j-i$ . The base case $j-i=0$ follows by Lemma 4.6. For the induction step, suppose that $i < j$ and $i+j \leq 2n+3$ . By induction, we have that

Then we vertically compose on top with $e_{1,2n+2}=\frac {1}{2}q_{1,2n+2}(x_1,\dots ,x_{2n+2})$ , using the centrality from Theorem 3.8 to commute this down to the middle; it becomes $e_{1,2} = x_1+x_2$ in the middle on the left hand side and $e_{1,0} = 0$ in the middle on the right hand side. We deduce that

If $j-i=1$ , the last two terms are the same as the first two terms, and the result follows on dividing by 2. If $j-i> 1$ we use the induction hypothesis to simplify the last two terms to obtain

The result follows.

Corollary 4.8. For $\alpha \in \mathbb {N}^{n+1}$ and $1 \leq i \leq n$ such that $\alpha _i + \alpha _{i+1} \leq 2n+1$ , we have that

(4.7)

where $\widehat \alpha := (\alpha _1,\dots ,\alpha _{i-1},\alpha _{i+2},\dots ,\alpha _{n+1}) \in \mathbb {N}^{n-1}$ and $s_i \alpha $ is the tuple obtained from $\alpha $ by permuting the ith and $(i+1)$ th entries.

Proof. Let $\beta := (\alpha _1,\dots ,\alpha _{i-1})$ and $\gamma := (\alpha _{i+2},\dots ,\alpha _{n+1})$ . By Lemma 4.7, we have that

Corollary 4.9. For $\alpha \in \mathbb {N}^{n+1}$ and $1 \leq i \leq n$ such that $\alpha _i =\alpha _{i+1} \leq n$ , we have that

(4.8)

where $\widehat \alpha := (\alpha _1,\dots ,\alpha _{i-1},\alpha _{i+2},\dots ,\alpha _{n+1}) \in \mathbb {N}^{n-1}$ .

Lemma 4.10. The following relation holds for any $n \geq 1$ and $0 \leq r \leq n$ :

(4.9)

interpreting the final term as 0 in case $r \leq 1$ .

Proof. We proceed by induction on n. The result is trivial when $n = 1$ . It is also clear when $r=0$ thanks to Corollary 4.4. Now suppose that $n \geq 1$ and $0 \leq r \leq n$ , and consider

Here, we commuted the single dot upward through the thick string. In the summation, the second term is 0 always, and the first term is 0 unless $a=0$ . So the expression simplifies to give

(4.10)

If $r=0$ , we simplify this using Corollary 4.4, then Lemma 4.2, then induction to obtain

as required for the induction step. Now suppose that $r \geq 1$ and consider (4.10) again. Letting $\alpha := \varpi _{r+1,n+1} - \varpi _{1,n+1}+\rho _{n+1} = (n,n,\dots )$ , we use Corollary 4.8 and induction to simplify the first term:

which is the second term we need to prove the induction step. Turning our attention to the second term on the right hand side of (4.10), it remains to show that

assuming $r \geq 1$ . By the induction hypothesis plus the identities $e_{r,n} = e_{r,n-1} + e_{r-1,n-1} x_n$ then $e_{r+1,n} + e_{r,n} x_{n+1}=e_{r+1,n+1}$ , we have that

Corollary 4.11. The following relation holds for any $n \geq 1$ and $0 \leq r \leq n$ :

(4.11)

4.2 Recurrence relation for idempotents

Corollary 4.4 obviously implies that

(4.12)

is a homogeneous idempotent for each $n \geq 0$ . For example:

These are likely already familiar expressions, since the same diagrams are often used to represent distinguished primitive idempotents in the nil-Hecke algebra.

In the remainder of the section, we are going to show that the idempotents $\mathbf {e}_n\:(n \geq 0)$ give a full set of primitive homogeneous idempotents in $\mathbf {NB}_t$ . The first step, accomplished in this subsection, is to decompose $B \star \mathbf {e}_n$ as a sum of mutually orthogonal conjugates of $\mathbf {e}_{n+1}$ and $\mathbf {e}_{n-1}$ . We begin by introducing two more families of endomorphisms of $B^{\star (n+1)}$ : for $0 \leq r \leq n$ let

(4.13)

(4.14)

Recalling the convention that the elementary symmetric function $e_r = 0$ for $r < 0$ and, of course, $e_0=1$ , we have that

(4.15) $$ \begin{align} \mathbf{e}_{0,n} &=\mathbf{e}_{n+1},& \mathbf{f}_{0,n} &= 0. \end{align} $$

By Lemma 4.10 and Corollary 4.11, the definitions (4.13) and (4.14) can be written equivalently as

(4.16)

(4.17)

(4.18)

where we interpret terms involving the undefined symbols $\varpi _{r-1}$ for $r = 0$ and $\varpi _{r-2}$ for $r=0$ or $1$ as 0.

Example 4.12. If $n=0$ then and $\mathbf {f}_{0,0} = 0$ . If $n=1$ then

If $n=2$ then

If $n=3$ then

Proof. For this calculation, it is convenient to drop the $\rho $ from the top of the diagrams, so we set

Notice that

We in fact show that $B \star \mathring {\mathbf {e}}_n = \sum _{r=0}^n (\mathring {\mathbf {e}}_{r,n}+\mathring {\mathbf {f}}_{r,n})$ . The first step is the same as in the proof of [Reference Khovanov, Lauda, Mackaay and StošićKLMS12, Lem. 2.13]:

The last two terms in this expression are equal to $\mathring {\mathbf {e}}_{n,n}+\mathring {\mathbf {f}}_{n,n}$ . It remains to show that the first term is equal to $\sum _{r=0}^{n-1} (\mathring {\mathbf {e}}_{r,n}+\mathring {\mathbf {f}}_{r,n})$ :

The first summation gives the remaining terms $\sum _{r=0}^{n-1}(\mathring {\mathbf {e}}_{r,n}+\mathring {\mathbf {f}}_{r,n})$ that we want, and the second summation is 0 thanks to Corollaries 4.5 and 4.11.

Now we introduce several more families of morphisms in $\mathbf {NB}_t$ for $0 \leq r\leq n$ and $1 \leq s \leq n$ :

(4.19)

(4.20)

again interpreting the undefined term involving $\varpi _{s-2}$ when $s=1$ as 0. Note that $\mathbf {u}_{0,n} = \mathbf {v}_{0,n} = \mathbf {e}_{n+1}$ thanks to Corollary 4.4, hence, $\mathbf {w}_{0,n} = 0$ . The same corollary also implies easily that $\mathbf {e}_{n+1}\circ \mathbf {u}_{r,n} = \mathbf {u}_{r,n}$ , $\mathbf {v}_{r,n} \circ \mathbf {e}_{n+1} = \mathbf {v}_{r,n}$ , $\mathbf {e}_{n-1} \circ \mathbf {x}_{s,n} = \mathbf {x}_{s,n}$ , and $\mathbf {y}_{s,n} \circ \mathbf {e}_{n-1} = \mathbf {y}_{s,n}$ .

Lemma 4.15. For $0 \leq r,s \leq n$ , we have that

(4.21)

Proof. This is clear for $n=0$ so assume $n \geq 1$ . By the definitions and Corollary 4.4, we have that

where $\alpha = (n-s,n,n-1,\dots ,n-r+1,n-r-1,\dots ,1,0) \in \mathbb {N}^{n+1}$ . If $s=r$ then $\alpha $ is a rearrangement of $\rho _{n+1}$ , so this is equal to $\mathbf {e}_{n+1}$ thanks to Corollary 4.8. If $0 < s \neq r$ then $\alpha $ has two entries equal to $n-s < n$ , so this is 0 by Corollaries 4.8 and 4.9. Finally, if $0=s \neq r$ then $\alpha = (n,n,n-1,\dots ,n-r+1,n-r-1,\dots ,1,0)$ and Corollary 4.9 gives the exceptional formula in this case, referring to (4.17) to see the appropriate form of $\mathbf {f}_{r,n}$ .

Corollary 4.16. For $0 \leq r,s \leq n$ , we have that

(4.22) $$ \begin{align} \mathbf{e}_{r,n} \circ \mathbf{e}_{s,n} = \begin{cases} -\mathbf{f}_{r,n}&\text{if }s=0 < r\text{ and }n\not\equiv t{\:\left(\mathrm{mod}\ 2\right)}\\ \delta_{r,s} \mathbf{e}_{r,n}&\text{otherwise.} \end{cases} \end{align} $$

Proof. By Lemma 4.14, we have that $\mathbf {e}_{r,n} \circ \mathbf {e}_{s,n} = \mathbf {v}_{r,n} \circ \mathbf {u}_{r,n} \circ \mathbf {v}_{s,n} \circ \mathbf {u}_{s,n}$ . Except in the case $s=0< r$ and $n\not \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ , we have that $\mathbf {u}_{r,n} \circ \mathbf {v}_{s,n} = \delta _{r,s} \mathbf {e}_{n+1}$ by Lemma 4.15, and $\mathbf {e}_{n+1} \circ \mathbf {u}_{r,n} = \mathbf {u}_{r,n}$ by Corollary 4.4. The conclusion then follows using that $\mathbf {v}_{r,n} \circ \mathbf {u}_{r,n} = \mathbf {e}_{r,n}$ once again. Suppose from now on that $s=0 < r$ and $n\not \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ . Then, using the form of $\mathbf {f}_{r,n}$ from (4.18), Lemma 4.15 gives instead that

It remains to apply Corollary 4.3 to see that this is equal to $-\mathbf {f}_{r,n}$ ; for this (4.17) is most convenient.

Proof. When $n=t=1$ this follows immediately from the first relation from (3.3). Now suppose that $n \geq 2$ . Since $\mathbf {x}_{r,n}$ is a sum of two terms (the second being 0 in case $r=1$ ), so too is $\mathbf {x}_{r,n}\circ \mathbf {y}_{s,n}$ . We compute the two terms separately. The first term is

where we used Corollary 4.4 for the first equality and Corollary 4.3 for the last one. If $r=1$ (when we already know that the second term is 0) this is $\delta _{s,1} \mathbf {e}_{n-1}$ by Corollary 4.4, and we are done. Assuming from now on that $r \geq 2$ , the second term is

where $\alpha = (n-s,n-2,\dots ,n-r+1,n-r-1,\dots ,1,0) \in \mathbb {N}^{n-1}$ . If $s=1$ this cancels with the first term to give 0, and we are done. Assuming from now on that $s \geq 2$ , the first term is 0, and it just remains to apply Corollaries 4.8 and 4.9 to rewrite the second term, noting that $n\equiv t{\:\left (\mathrm {mod}\ 2\right )}$ so the first term on the right hand side of (4.7) is 0, as is the right hand side of (4.8). We get 0 if $r \neq s$ and, after one more application of Corollary 4.4, we get $\mathbf {e}_{n-1}$ if $r=s$ , as claimed.

Proof. We first consider $\mathbf {x}_{s,n} \circ \mathbf {v}_{r,n}$ . Since $\mathbf {x}_{s,n}$ is a sum of two terms, so too is $\mathbf {x}_{s,n} \circ \mathbf {v}_{r,n}$ . We show that both of these terms are 0. The first term is

This is 0 by Corollary 4.3 since $n-1\not \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ . The second term is 0 automatically if $s=1$ , so we are done in this case. When $s \geq 2$ , the second term equals

which is 0 by the second relation from (3.7).

Now consider $\mathbf {u}_{r,n} \circ \mathbf {y}_{s,n}$ for $0 \leq r \leq n$ and $1 \leq s \leq n$ . For notational convenience, we in fact show that $\mathring {\mathbf {u}}_{r,n} \circ \mathbf {y}_{s,n} = 0$ , where . Applying Corollary 4.4 as usual, we have that

This is of degree $2(r-s)-n(n-1)$ while by Theorem 3.6 the lowest nonzero degree of the graded vector space $ {\mathrm {Hom}}_{\mathbf {NB}_t}(B^{\star (n-1)}, B^{\star (n+1)})$ is $-n(n-1)$ , so it is automatically 0 if $r < s$ . Assume henceforth that $r \geq s$ . When $n=t=1$ , so $r=s=1$ , it is easy to see that we get 0 using Corollary 3.5, so assume also that $n \geq 2$ .

In this paragraph, we treat the case that $r> s$ . We have that $\varpi _{r,n}+\rho _n = (n,n-1,\dots ,n-s,\dots ,n-r+1,n-r-1,\dots ,1,0) \in \mathbb {N}^n$ . Let $\alpha := (n-s,n,n-1,\dots ,\widehat {n-s},\dots ,n-r+1,n-r-1,\dots ,1,0)\in \mathbb {N}^n$ , that is, we have moved the entry $n-s$ to the beginning. Let $\beta := (n-s,\alpha _1,\dots ,\alpha _{n-1})$ . We have that

In checking the second equality here, one also needs to observe that the term arising from the first term on the right hand side of (4.7) (which can definitely appear as $n-1\not \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ ) is 0 due to the second relation from (3.7). Now we have that $\beta _1=\beta _2=n-s$ , so this is 0 by Corollary 4.9; again, when $s=1$ , the term arising from the right hand side of (4.8) vanishes due to (3.7).

Finally, we need to treat the case that $r=s$ (and $n \geq 2$ still). We let $\alpha := \varpi _{r,n}+\rho _n = (n,n-1,\dots ,n-r+1,n-r-1,\dots ,1,0) \in \mathbb {N}^n$ , $\beta := (n-s,\alpha _1,\dots ,\alpha _{n-1})$ , and $\gamma := (n-s,\alpha _2,\dots ,\alpha _n)$ . As $r=s\geq 1$ , the tuple $\gamma $ is a permutation of $\rho _n$ , and $\alpha _1 = n$ . Using Corollary 4.8 several more times like in the previous paragraph, we get that

This is 0 by Corollary 4.3, using that $n-1\not \equiv t{\:\left (\mathrm {mod}\ 2\right )}$ .

Proof. (1) The fact that $\mathbf {e}_{r,n}\:(0 \leq r \leq n)$ are mutually orthogonal idempotents follows from Corollary 4.16. The fact that $\mathbf {f}_{s,n}\:(1 \leq s \leq n)$ are mutually orthogonal idempotents follows from Corollary 4.18. The orthogonality of each $\mathbf {e}_{r,n}\:(0 \leq r \leq n)$ with each $\mathbf {f}_{s,n}\:(1 \leq s \leq n)$ follows from Corollary 4.20. These idempotents sum to $B \star \mathbf {e}_n$ by Lemma 4.13. Also $\mathbf {u}_{r,n} \circ \mathbf {v}_{r,n} = \mathbf {e}_{n+1}$ by Lemma 4.15, and $\mathbf {v}_{r,n} \circ \mathbf {u}_{r,n} = \mathbf {e}_{r,n}$ by Lemma 4.14. Finally, $\mathbf {x}_{s,n} \circ \mathbf {y}_{s,n} = \mathbf {e}_{n-1}$ by Lemma 4.17, and $\mathbf {y}_{s,n} \circ \mathbf {x}_{s,n} = \mathbf {f}_{s,n}$ by Lemma 4.14.

(2) We first show that $\mathbf {e}_{r,n}+\mathbf {f}_{r,n}\:(0 \leq r \leq n)$ are mutually orthogonal idempotents by checking that

$$ \begin{align*}(\mathbf{e}_{r,n}+\mathbf{f}_{r,n}) \circ (\mathbf{e}_{s,n}+\mathbf{f}_{s,n}) = \delta_{r,s} (\mathbf{e}_{r,n}+\mathbf{f}_{r,n}) \end{align*} $$

for $0 \leq r,s \leq n$ . If $r=0$ this follows because $\mathbf {f}_{0,n}=0$ , $\mathbf {e}_{0,n} \circ \mathbf {e}_{s,n} = \delta _{0,s} \mathbf {e}_{0,n}$ and, assuming $s> 0$ , we have that $\mathbf {e}_{0,n} \circ \mathbf {f}_{s,n} = - \mathbf {e}_{0,n} \circ \mathbf {e}_{s,n} \circ \mathbf {e}_{0,n} = 0$ , all by Corollary 4.16. If $r> 0$ and $s=0$ it follows because $\mathbf {e}_{r,n} \circ \mathbf {e}_{0,n} = -\mathbf {f}_{r,n}$ and $\mathbf {f}_{r,n} \circ \mathbf {e}_{0,n} = - \mathbf {e}_{r,n} \circ \mathbf {e}_{0,n} \circ \mathbf {e}_{0,n} = - \mathbf {e}_{r,n} \circ \mathbf {e}_{0,n} = \mathbf {f}_{r,n}$ by Corollary 4.16. Finally, suppose that $1 \leq r,s \leq n$ . Then by Corollary 4.16 we have that

$$ \begin{align*} (\mathbf{e}_{r,n}+\mathbf{f}_{r,n})\circ (\mathbf{e}_{s,n}+\mathbf{f}_{s,n})&= \mathbf{e}_{r,n} \circ \mathbf{e}_{s,n} + \mathbf{e}_{r,n} \circ \mathbf{f}_{s,n} + \mathbf{f}_{r,n} \circ \mathbf{e}_{s,n} + \mathbf{f}_{r,n} \circ \mathbf{f}_{s,n}\\ &= \mathbf{e}_{r,n} \circ \mathbf{e}_{s,n} - \mathbf{e}_{r,n} \circ \mathbf{e}_{s,n} \circ \mathbf{e}_{0,n} - \mathbf{e}_{r,n} \circ \mathbf{e}_{0,n} \circ\mathbf{e}_{s,n} + \mathbf{e}_{r,n} \circ \mathbf{e}_{0,n} \circ \mathbf{e}_{s,n} \circ \mathbf{e}_{0,n}\\ &= \delta_{r,s} \mathbf{e}_{r,n} - \delta_{r,s} \mathbf{e}_{r,n} \circ \mathbf{e}_{0,n} = \delta_{r,s}(\mathbf{e}_{r,n} + \mathbf{f}_{r,n}). \end{align*} $$

We have that $\sum _{r=0}^n (\mathbf {e}_{r,n}+\mathbf {f}_{r,n}) = B \star \mathbf {e}_n$ by Lemma 4.13. Finally, using Lemmas 4.14 and 4.15, Corollary 4.16, and $\mathbf {u}_{0,n} = \mathbf {v}_{0,n} = \mathbf {e}_{0,n}$ , we have that

$$ \begin{align*} \mathbf{w}_{r,n} \circ \mathbf{v}_{r,n} &= \mathbf{u}_{r,n} \circ \mathbf{v}_{r,n} - \mathbf{u}_{r,n} \circ \mathbf{u}_{0,n} \circ \mathbf{v}_{r,n} = \mathbf{e}_{n+1},\\ \mathbf{v}_{r,n} \circ \mathbf{w}_{r,n} &= \mathbf{v}_{r,n} \circ \mathbf{u}_{r,n} - \mathbf{v}_{r,n} \circ \mathbf{u}_{r,n} \circ \mathbf{e}_{0,n} =\mathbf{e}_{r,n} - \mathbf{e}_{r,n} \circ \mathbf{e}_{0,n} = \mathbf{e}_{r,n}+\mathbf{f}_{r,n} \end{align*} $$

for $1 \leq r \leq n$ .

4.3 Locally unital graded algebras and modules

Before explaining the full significance of Theorem 4.21, we need to review some basic terminology. Suppose that $\mathbf {A}$ is any small graded category and let $\mathcal {I}$ be its object set. The path algebra of $\mathbf {A}$ is the graded algebra

$$ \begin{align*}A = \bigoplus_{i,j \in \mathcal{I}} 1_i\, A 1_j \qquad\text{where}\qquad 1_i\, A 1_j := {\mathrm{Hom}}_{\mathbf{A}}(j,i), \end{align*} $$

with multiplication induced by composition in $\mathbf {A}$ . In general, this is locally unital rather than unital, equipped with the distinguished system $1_i\:(i \in \mathcal {I})$ of mutually orthogonal idempotents arising from the identity endomorphisms of the objects of $\mathbf {A}$ .

The center $Z(A)$ is the commutative subalgebra of the unital graded algebra $\prod _{i \in \mathcal {I}} 1_i A 1_i$ consisting of tuples $(z_i)_{i \in \mathcal {I}}$ such that $\theta \, z_j = z_i\, \theta $ for all $i,j \in \mathcal {I}$ and $\theta \in 1_i A 1_j$ . This is a direct translation of the definition of the center of the category $\mathbf {A}$ . Given a (unital) commutative graded algebra R, we say that A is a graded R-algebra if we are given a unital graded algebra homomorphism $\eta :R \rightarrow Z(A)$ . Then each subspace $1_i A 1_j$ is naturally a graded R-module.

By a graded left A-module, we mean a module V as usual which is itself locally unital in the sense that $V = \bigoplus _{i \in \mathcal {I}} 1_i V$ . We sometimes refer to $1_i V$ as the i-weight space of V. There are also the obvious notions of graded right A-modules and, given another locally unital graded algebra B, graded $(A,B)$ -bimodules. For graded left A-modules V and W and $d \in \mathbb {Z}$ , we write $ {\mathrm {Hom}}_{A}(V,W)_d$ for the vector space of all ordinary A-module homomorphisms $f:V \rightarrow W$ such that $f(V_n) \subseteq W_{n+d}$ for each $n \in \mathbb {Z}$ . Then the graded vector space

$$ \begin{align*}{\mathrm{Hom}}_{A}(V,W) := \bigoplus_{d \in \mathbb{Z}} {\mathrm{Hom}}_{A}(V,W)_d \end{align*} $$

is a morphism space in the graded category $A\mathrm {\!-gMod}$ of graded left A-modules. We denote the underlying category consisting of the same objects but just the degree-preserving morphisms by $A\mathrm {\!-gmod}$ . This is the usual Abelian category of graded left A-modules. It is equipped with the upward grading shift functor q defined as in the General conventions, and we have that

(4.23) $$ \begin{align} {\mathrm{Hom}}_A(V,W)_d = \left(q^{-d} {\mathrm{Hom}}_A(V,W)\right)_0= {\mathrm{Hom}}_A(V, q^{-d} W)_0 = {\mathrm{Hom}}_A(q^{d} V, W)_0. \end{align} $$

We use the symbol $\cong $ to denote (degree-preserving) isomorphism in $A \mathrm {\!-gmod}$ . We write $ {\mathrm {End}}_A(V)$ for $ {\mathrm {Hom}}_A(V,V)$ , which is a graded algebra.

Let $A\mathrm {\!-pgmod}$ be the full subcategory of $A\mathrm {\!-gmod}$ consisting of finitely generated projective graded modules. Also let $K_0(A)$ denote the split Grothendieck group of the additive category $A\mathrm {\!-pgmod}$ . This is a $\mathbb {Z}[q,q^{-1}]$ -module with the action of q induced by the grading shift functor. One could also define $K_0(A)$ equivalently as the split Grothendieck group of the graded Karoubi envelope of $\mathbf {A}$ , since the latter category is contravariantly equivalent to $A\mathrm {\!-pgmod}$ by Yoneda’s Lemma. We will not take this point of view here, but note that some care is needed in the identification since contravariant equivalences interchange q with $q^{-1}$ .

Assume in this paragraph that A is locally finite dimensional and bounded below, meaning that for every $i,j \in \mathcal {I}$ , the graded vector space $1_i\, A 1_j$ is locally finite dimensional, that is, each of its graded pieces $1_i\, A_d 1_j$ is finite dimensional, and $1_i\, A_d 1_j = 0$ for $d \ll 0$ . Then $K_0(A)$ can be understood in purely combinatorial terms. To explain what we mean, referring to [Reference BrundanBru25, Sec. 2] for more details, we note to start with that the weight spaces of any irreducible graded left A-module L are finite dimensional, and Schur’s Lemma holds:

(4.24) $$ \begin{align} {\mathrm{End}}_A(L) = \Bbbk. \end{align} $$

We say that a graded left A-module V is locally finite dimensional if $1_i V_d$ is finite dimensional for each $i \in \mathcal {I}$ and $d \in \mathbb {Z}$ , and bounded below if for each $i \in \mathcal {I}$ we have that $1_i V_d = 0$ for $d \ll 0$ . Since the distinguished projective modules $A 1_i\:(i \in \mathcal {I})$ are locally finite dimensional and bounded below, it follows that any finitely generated graded left A-module also has these properties. Any graded left A-module has an injective hull in $A\mathrm {\!-gmod}$ , and any finitely generated graded left A-module has a projective cover in $A\mathrm {\!-gmod}$ , the latter being a summand of a finite direct sum of degree-shifted copies of the distinguished projective modules $A 1_i\:(i \in \mathcal {I})$ . Let $L(b)\:(b \in \mathcal {B})$ be a full set of representatives for the irreducible graded left A-modules (up to isomorphism and grading shift), and define $P(b)$ to be a projective cover of $L(b)$ . The graded multiplicity of $L(b)$ in a locally finite-dimensional graded module V is the formal series

$$ \begin{align*} [V:L(b)]_q &:=\sum_{d \in \mathbb{Z}} \max \bigg( \left|\{r=1,\dots,n\:|\:V_r / V_{r-1} \cong q^d L(b) \}\right| \:\bigg|\: \begin{array}{ll} \text{for all finite graded filtrations}\\ 0 = V_0 \subseteq \cdots \subseteq V_n = V \end{array} \bigg) q^d. \end{align*} $$

Schur’s Lemma implies that

(4.25) $$ \begin{align} [V:L(b)]_q = \dim_q {\mathrm{Hom}}_A(P(b),V). \end{align} $$

Note also that this belongs to $\mathbb {N}(\!( q)\!)$ when V is finitely generated. Finally, any finitely generated projective graded left A-module P satisfies

(4.26) $$ \begin{align} P \cong \bigoplus_{b \in \mathcal{B}} P(b)^{\oplus \overline{\dim_{q} {\mathrm{Hom}}_A(P, L(b))}}. \end{align} $$

Now it follows that $K_0(A)$ is a free $\mathbb {Z}[q,q^{-1}]$ -module with basis $[P(b)]\:(b \in \mathcal {B})$ .

Another basic notion involves induction and restriction. For this, we start with a pair of small graded categories, $\mathbf {A}$ and $\mathbf {B}$ , with object sets denoted $\mathcal {I}$ and $\mathcal {J}$ , respectively. Let A and B be their path algebras. Given a graded functor $F: \mathbf {A} \rightarrow \mathbf {B}$ , there is a graded functor

(4.27) $$ \begin{align} {\mathrm{Res}}_F:B\mathrm{\!-gMod} \rightarrow A\mathrm{\!-gMod} \end{align} $$

called restriction along F . This takes a graded left B-module V to the graded vector space

$$ \begin{align*}1_F V := \bigoplus_{i \in \mathcal{I}} 1_{F i} V \end{align*} $$

with $\theta \in 1_i A 1_j= {\mathrm {Hom}}_{\mathbf {A}}(j,i)$ acting as the linear map $F\theta :1_{Fj} V \rightarrow 1_{Fi} V$ between the summands indexed by j and i, and as 0 on all other summands. This notation is for graded left B-modules, but it is readily adapted to a graded right B-module V, letting

$$ \begin{align*}V 1_F := \bigoplus_{i \in \mathbf{A}} V 1_{Fi} \end{align*} $$

which is a graded right A-module. The functor $ {\mathrm {Res}}_F$ is isomorphic to $\bigoplus _{i \in \mathcal {I}} {\mathrm {Hom}}_B(B 1_{Fi},-)$ . Hence, by adjointness of tensor and hom for locally unital algebras (e.g., see [Reference Brundan and StroppelBS24, Lem. 2.7]), it has a left adjoint

(4.28) $$ \begin{align} {\mathrm{Ind}}_F := B 1_F \otimes_A -&:A\mathrm{\!-gMod} \rightarrow B\mathrm{\!-gMod}, \end{align} $$

where $B 1_F$ is the graded $(B,A)$ -bimodule obtained by restricting the regular $(B,B)$ -bimodule B on the right. We refer to $ {\mathrm {Ind}}_F$ as induction along F . If $\alpha :F \Rightarrow G$ is a graded natural transformation between graded functors $F,G:\mathbf {A} \rightarrow \mathbf {B}$ , we obtain graded bimodule homomorphisms $B 1_G \rightarrow B 1_F$ and $1_F B \rightarrow 1_G B$ defined by the linear maps $1_j B 1_{Gi} \rightarrow 1_{j} B 1_{Fi}, \theta \mapsto \theta \circ \alpha _i$ and $1_{Fi} B 1_j \rightarrow 1_{Gi} B 1_j, \theta \mapsto \alpha _i \circ \theta $ , respectively, for $i \in \mathcal {I}, j \in \mathcal {J}$ . These bimodule homomorphisms define graded natural transformations $ {\mathrm {Ind}}_\alpha : {\mathrm {Ind}}_G \Rightarrow {\mathrm {Ind}}_F$ and $ {\mathrm {Res}}_\alpha : {\mathrm {Res}}_F \Rightarrow {\mathrm {Res}}_G$ .