Categorical braid group actions
and cactus groups

In this article we work with a simply-laced quantum group $U_{q}(\mathfrak{g})$ of finite type. Recall that we have an associated Cartan datum and a root datum, which consists of:

• •

  A finite set $I$,

• •

  a symmetric bilinear form $(\cdot,\cdot)$ on $\mathbb{Z}I$ satisfying $(i,i)=2$ and $(i,j)\in\{0,-1\}$ for all $i\neq j\in I$,

• •

  a free $\mathbb{Z}$-module $X$, called the weight lattice, and

• •

  a choice of simple roots $\{\alpha_{i}\}_{i\in I}\subset X$ and simple coroots $\{h_{i}\}_{i\in I}\subset X^{\vee}=\operatorname{Hom}(X,\mathbb{Z})$ satisfying $\langle h_{i},\alpha_{j}\rangle=(i,j)$, where $\langle\cdot,\cdot\rangle:X^{\vee}\times X\to\mathbb{Z}$ is the natural pairing.

The quantum group $U_{q}(\mathfrak{g})$ is the unital, associative, $\mathbb{C}(q)$ algebra generated by $E_{i},F_{i},K_{h},(i\in I,h\in X^{\vee})$ subject to relations:

1. (1)

   $K_{0}=1$ and $K_{h}K_{h^{\prime}}=K_{h+h^{\prime}}$ for any $h,h^{\prime}\in X^{\vee}$,

2. (2)

   $K_{h}E_{i}=q^{\langle h,\alpha_{i}\rangle}E_{i}K_{h}$ for any $i\in I,h\in X^{\vee}$,

3. (3)

   $K_{h}F_{i}=q^{-\langle h,\alpha_{i}\rangle}F_{i}K_{h}$ for any $i\in I,h\in X^{\vee}$,

4. (4)

   $E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\frac{K_{i}-K_{i}^{-1}}{q-q^{-1}}$, where we set $K_{i}=K_{h_{i}}$, and

5. (5)

   for all $i\neq j$,

   $$\sum_{a+b=-\langle h_{i},\alpha_{j}\rangle+1}(-1)^{a}E_{i}^{(a)}E_{j}E_{i}^{(b)}=0\quad\text{ and }\quad\sum_{a+b=-\langle h_{i},\alpha_{j}\rangle+1}(-1)^{a}F_{i}^{(a)}F_{j}F_{i}^{(b)}=0,$$

   where $E_{i}^{(a)}=E_{i}^{a}/[a]!,F_{i}^{(a)}=F_{i}^{a}/[a]!$, and $[a]!=\prod_{i=1}^{a}\frac{q^{i}-q^{-i}}{q-q^{-1}}$.

We let $a_{ij}=(i,j)$, so that $(a_{ij})_{i,j\in I}$ is a Cartan matrix. Given $\lambda\in X$ we abbreviate $\lambda_{i}=\langle h_{i},\lambda\rangle$, and let

be the set of dominant weights.

Let $R\subset X$ be the root lattice, defined as the $\mathbb{Z}$-span of the simple roots, and let $R_{+}\subset R$ be the $\mathbb{N}$-span of the simple roots. We define the usual preorder $\succ$ on $X$ by $\lambda\succeq\mu$ if $\lambda-\mu\in R_{+}$. For $\mu\in R$ let $ht(\mu)$ denote the height of $\mu$, i.e. $ht(\sum_{i}a_{i}\alpha_{i})=\sum_{i}a_{i}$.

When convenient, we also view $I$ as the Dynkin diagram of $\mathfrak{g}$, and make reference to subdiagrams or diagram automorphisms of $I$.

Given a $U_{q}(\mathfrak{g})$-module $V$ and $\mu\in X$ we let $V_{\mu}$ denote the $\mu$ weight space of $V$. For $\lambda\in X_{+}$ we let $L(\lambda)$ be the irreducible representation of $U_{q}(\mathfrak{g})$ of highest weight $\lambda$. Let $\mathsf{Iso}_{\lambda}(V)$ denote the $\lambda$-isotypic component of $V$. We say that $V$ is isotypic if there exists $\lambda\in X_{+}$ such that $V=\mathsf{Iso}_{\lambda}(V)$.

The representation $L(\lambda)$ has a canonical basis, which we denote by $\mathbf{B}(\lambda)$ [Lusbook]. We let $v_{\lambda}$ (respectively $v_{\lambda}^{low}$) denote the unique highest weight (respectively lowest weight) element of $\mathbf{B}(\lambda)$.

Let $B=B_{I}$ denote the braid group of type $I$, which is generated by $\theta_{i}\;(i\in I)$ subject to the braid relations:

Let $W=W_{I}$ be the Weyl group of type $I$, which has generators $s_{i}\;(i\in I)$ subject to the braid relations, and in addition the quadratic relation $s_{i}^{2}=1$. Let $w_{0}\in W$ be the longest element. Recall that $W$ acts on $X$ via $s_{i}\cdot\lambda=\lambda-\langle h_{i},\lambda\rangle\alpha_{i}$. We define $\tau:I\to I$ by the equality $\alpha_{\tau(i)}=-w_{0}(\alpha_{i})$ for any $i\in I$.

To $J\subset I$ a subdiagram, we associate $W_{J}\subset W$ the parabolic subgroup, $w_{0}^{J}\in W_{J}$ its longest element, and $\tau_{J}:I\to I$ the bijection given by

For any $w\in W$ we can consider its positive lift $\theta_{w}\in B$, where $\theta_{w}=\theta_{i_{1}}\cdots\theta_{i_{\ell}}$ and $w=s_{i_{1}}\cdots s_{i_{\ell}}$ is any reduced decomposition.

Let $V$ be an integrable representation of $U_{q}(\mathfrak{g})$. A fundamental structure of $V$, discovered by Lusztig, is that it admits (several) braid group symmetries, sometimes referred to as the “quantum Weyl group actions”. To recall this, let $\mathsf{1}_{\mu}$ denote the projection onto the $\mu$ weight space. For each $i\in I$ we define $\mathsf{t}_{i}:V\to V$ by:

Note that the indexing set of this sum is infinite, but the sum itself is finite on $V$. In the notation of [Lusbook], $\mathsf{t}_{i}=\mathsf{t}^{\prime\prime}_{i,-1}$. (Note that the formula given in [Lusbook] is more complicated. This simpler form was initially observed in [CR] in the non-quantum setting, and generalised in [CKM] to the quantum setting.). The assignment $\theta_{i}\mapsto\mathsf{t}_{i}$ defines an action of $B$ on $V$, and so we can unambiguously write $\mathsf{t}_{w}$ for any $w\in W$.

Fix a field $\mathbbm{k}$ of any characteristic. In this paper we will be concerned mostly with abelian $\mathbbm{k}$-linear categories $\mathcal{C}$. We will assume throughout that each block of $\mathcal{C}$ is a finite abelian category [EGNO, Definition 1.8.6].

Recall that a category $\mathcal{A}$ is graded if it is equipped with an auto-equivalence $\langle 1\rangle:\mathcal{A}\to\mathcal{A}$ called the “shift functor”. We let $\langle\ell\rangle$ be the auto-equivalence obtained by applying the shift functor $\ell$ times. We denote by $\mathsf{Irr}(\mathcal{A})$ the set of equivalence classes of simple objects of $\mathcal{A}$ up to shift.

A functor $F:\mathcal{A}\to\mathcal{A}^{\prime}$ between graded categories is graded if it commutes with the shift functors. We denote by $[\mathcal{A}]_{\mathbb{Z}}$ the Grothendieck group of an abelian category $\mathcal{A}$. If $\mathcal{A}$ is graded we denote by $[\mathcal{A}]_{\mathbb{Z}[q,q^{-1}]}$ the quotient of $[\mathcal{A}]_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}[q,q^{-1}]$ by the additional relation $q[M]=[M\langle-1\rangle]$. This quotient is naturally a $\mathbb{Z}[q,q^{-1}]$-module. Set

For a $\mathbbm{k}$-algebra $A$, we let $A\mathrm{-mod}$ be the category of finitely generated $A$-modules, and if $A$ is $\mathbb{Z}$-graded, we let $A\mathrm{-mod}_{\mathbb{Z}}$ be the category of finitely generated $\mathbb{Z}$-graded modules. These are naturally $\mathbbm{k}$-linear abelian categories.

In this section we introduce our main objects of study: representations of the 2-quantum group on abelian categories, which we refer to as “categorical representations” of $U_{q}(\mathfrak{g})$. This definition of the categorified quantum groups and their categorical actions is originally due to Rouquier and Khovanov-Lauda [Rou2KM, KLI, KLII].

In the literature, there are a number of slightly different-looking examples of 2-representations, depending not just on whether or not one is working with the Khovanov-Lauda or Rouquier 2-categories, but also depending on whether or not one is interested in categorifications of representations of $U_{q}(\mathfrak{g})$ or of $U(\mathfrak{g})$. At the categorical level, the difference between $U_{q}(\mathfrak{g})$ or of $U(\mathfrak{g})$ arises from grading considerations; in categorifications of representations of $U_{q}(\mathfrak{g})$, one works with categories enriched in graded vector spaces, whereas in categorifications of representations of $U(\mathfrak{g})$ no such grading is needed.

Fortunately, the different notions of categorical representations - and in particular relationships between the Rouquier and Khovanov-Lauda frameworks in both graded and ungraded settings, have been brought in line by work of Cautis-Lauda, who proved that in the graded setting, integrable 2-representations of the Rouquier 2-category induce 2-representations of the Khovanov-Lauda 2-category [CaLa], and by work of Brundan [Brundandef], who proved that the underlying ungraded 2-categories of Rouquier and Khovanov-Lauda are equivalent.

The important point for us is that all our theorems, including our main results (Theorem 6.4 and Theorem 6.8), remain true in any 2-representation of the Khovanov-Lauda-Rouqier 2-category, in either the graded or ungraded setting. Since the compatibility of the internal grading with the braid group is of some independent combinatorial interest, we elect to keep track of the gradings in the rest of the paper, and leave it to the reader to verify that the arguments go through while ignoring the gradings and working in the setup of, e.g. Brundan [Brundandef]. To that end, we have chosen to follow the notation and conventions of Cautis-Lauda below.

A categorical representation of $U_{q}(\mathfrak{g})$ consists of the following data:

• •

  A family of graded abelian $\mathbbm{k}$-linear categories $\mathcal{C}_{\mu}$ indexed by $\mu\in X$. We refer to each $\mathcal{C}_{\mu}$ as a weight category.

• •

  Exact graded functors $\mathsf{E}_{i}\mathbbm{1}_{\mu}:\mathcal{C}_{\mu}\to\mathcal{C}_{\mu+\alpha_{i}}$ and $\mathsf{F}_{i}\mathbbm{1}_{\mu}:\mathcal{C}_{\mu}\to\mathcal{C}_{\mu-\alpha_{i}}$, for $i\in I$ and $\mu\in X$. We refer to $\mathsf{E}_{i},\mathsf{F}_{i}$ as Chevalley functors.

• •

  A collection of natural transformations between these functors. We won’t be using directly these natural transformations in this work, so refer the reader to [CaLa, Definition 1.1] for their definition.

This data is subject to the conditions spelled out in items (1)-(5) of [CaLa, Definition 1.1]. We only record those that are relevant for us:

1. (1)

   The functors $\mathsf{E}_{i}\mathbbm{1}_{\mu}$ and $\mathsf{F}_{i}\mathbbm{1}_{\mu}$ are biadjoint up to a specified degree shift (see (3.1) below).

2. (2)

   The powers of $\mathsf{E}_{i}$ carry an action of the KLR algebra associated to $Q$, where $Q$ denotes a choice of units $(t_{ij})_{i\neq j\in I}$ in $\mathbbm{k}^{\times}$. These units satisfy some restrictions which are not relevant for us.

3. (3)

   We have the following isomorphisms:

   	$\displaystyle\mathsf{F}_{i}\mathsf{E}_{i}\mathbbm{1}_{\mu}$	$\displaystyle\cong\mathsf{E}_{i}\mathsf{F}_{i}\mathbbm{1}_{\mu}\oplus_{[-\langle h_{i},\mu\rangle]}\mathbbm{1}_{\mu},\text{ if }\langle h_{i},\mu\rangle\leq 0,$
   	$\displaystyle\mathsf{E}_{i}\mathsf{F}_{i}\mathbbm{1}_{\mu}$	$\displaystyle\cong\mathsf{F}_{i}\mathsf{E}_{i}\mathbbm{1}_{\mu}\oplus_{[\langle h_{i},\mu\rangle]}\mathbbm{1}_{\mu},\text{ if }\langle h_{i},\mu\rangle\geq 0,$
   	$\displaystyle\mathsf{E}_{i}\mathsf{F}_{j}\mathbbm{1}_{\mu}$	$\displaystyle\cong\mathsf{F}_{j}\mathsf{E}_{i}\mathbbm{1}_{\mu}.$

   This notation is explained as follows: for a Laurent polynomial $f=\sum f_{a}q^{a}$, $\oplus_{f}A$ is a direct sum over $a\in\mathbb{Z}$ of $f_{a}$ copies of $A\langle a\rangle$, and $[n]:=q^{n-1}+q^{n-3}+\cdots+q^{1-n}$.

Usually we just say that $\mathcal{C}=\bigoplus_{\mu}\mathcal{C}_{\mu}$ is a categorical representation of $U_{q}(\mathfrak{g})$ (the remaining data is implicit). Given an integrable $U_{q}(\mathfrak{g})$-module $V$, we say that $\mathcal{C}$ is a categorification of $V$ if $\mathcal{C}$ is a categorical representation of $U_{q}(\mathfrak{g})$ such that $[\mathcal{C}]_{\mathbb{C}(q)}\cong V$ as $U_{q}(\mathfrak{g})$-modules.

An additive categorification is a categorical representation on a graded additive $\mathbbm{k}$-linear category $\mathcal{V}$ satisfying the same conditions as above, except the Chevalley functors are of course only required to be additive. We let $\mathcal{V}^{i}$ be the idempotent completion of $\mathcal{V}$.

Given an abelian category $\mathcal{C}$, we consider the additive category $\mathcal{C}\mathrm{-proj}$ defined as the full subcategory of projective objects in $\mathcal{C}$. Note that if $\mathcal{C}$ is a categorical representation of $U_{q}(\mathfrak{g})$, then $\mathcal{C}\mathrm{-proj}$ naturally inherits the structure of an additive categorification.

As a consequence of this definition, and in particular condition (2), there exist divided power functors $\mathsf{E}_{i}^{(r)}\mathbbm{1}_{\lambda}\subset\mathsf{E}_{i}^{r}\mathbbm{1}_{\lambda},\mathsf{F}_{i}^{(r)}\mathbbm{1}_{\lambda}\subset\mathsf{F}_{i}^{r}\mathbbm{1}_{\lambda}$ which categorify the usual divided powers on the level of the quantum group. Again, we refer the reader to [CaLa] and references therein for further details. We note that their adjoints are related as follows:

We recall the definition of a crystal.

Any $U_{q}(\mathfrak{g})$-representation $V$ has a corresponding $\mathfrak{g}$-crystal $\mathbf{B}=\mathbf{B}_{V}$. The underlying set of $\mathbf{B}$ is in natural bijection with a particular basis of $V$ (the “global crystal basis” or “canonical basis”). The maps $\widetilde{e}_{i},\widetilde{f}_{i}$ are related to the raising and lowering Chevalley operators; vaguely speaking they encode information about the leading terms of the Chevalley operators acting on this basis. In particular, for an integral dominant weight $\lambda$, the canonical basis $\mathbf{B}(\lambda)$ of $L(\lambda)$ carries a natural crystal structure [GL92].

The crystal of $V$ naturally arises via categorical representation theory. Namely, as we describe in the next proposition, if $\mathcal{C}$ is a categorification of $V$, then $\mathsf{Irr}(\mathcal{C})$ carries a crystal structure isomorphic to $\mathbf{B}_{V}$. This follows from [CR, Proposition 5.20] and [LV], and is explained in detail in [BDcryst].

A categorification of a simple representation (respectively an isotypic representation) is called a simple categorification (respectively an isotypic categorification). There is a distinguished categorification of $L(\lambda)$ called the minimal categorification and denoted $\mathcal{L}(\lambda)$ [CR, Rou2KM, KLI, Webmerged, KK]. It is characterized by the fact that $\mathcal{L}(\lambda)_{\lambda}\cong\mathcal{L}(\lambda)_{w_{0}(\lambda)}\cong\mathbbm{k}\mathrm{-mod}_{\mathbb{Z}}$. We let $\mathbbm{k}_{low}\in\mathcal{L}(\lambda)_{w_{0}(\lambda)},\mathbbm{k}_{high}\in\mathcal{L}(\lambda)_{\lambda}$ be the generators.

The Jordan-Hölder Theorem for categorical representations will play an important role in our work. This was originally developed by Rouquier for additive categorifications [Rou2KM], and in this section we transfer these results to the abelian setting. To set this up, recall that given finite abelian $\mathbbm{k}$-linear categories $\mathcal{A},\mathcal{B}$ the Deligne tensor product $\mathcal{A}\otimes_{\mathbbm{k}}\mathcal{B}$ is universal for the functor assigning to every such abelian category $\mathcal{C}$ the category of bilinear bifunctors $\mathcal{A}\times\mathcal{B}\to\mathcal{C}$ right exact in both variables [EGNO, Definition 1.11.1]. The tensor product is again a finite abelian $\mathbbm{k}$-linear category, and there is a bifunctor $\mathcal{A}\times\mathcal{B}\to\mathcal{A}\otimes_{\mathbbm{k}}\mathcal{B},(X,Y)\mapsto X\otimes Y$. This construction enjoys the following properties:

1. (1)

   The tensor product is unique up to unique equivalence,

2. (2)

   for finite $\mathbbm{k}$-algebras $A,B$ we have that $(A\mathrm{-mod})\otimes_{\mathbbm{k}}(B\mathrm{-mod})\cong(A\otimes_{\mathbbm{k}}B)\mathrm{-mod}$, and

3. (3)

   $\operatorname{Hom}_{\mathcal{A}\otimes_{\mathbbm{k}}\mathcal{B}}(X_{1}\otimes Y_{1},X_{2}\otimes Y_{2})\cong\operatorname{Hom}_{\mathcal{A}}(X_{1},Y_{1})\otimes\operatorname{Hom}_{\mathcal{B}}(X_{2},Y_{2})$.

Let $\mathcal{C}$ be a categorical representation, and let $\mathcal{A}$ be a finite $\mathbbm{k}$-linear abelian category. We can endow $\mathcal{C}\otimes_{\mathbbm{k}}\mathcal{A}$ with a structure of a categorical representation, by setting $(\mathcal{C}\otimes_{\mathbbm{k}}\mathcal{A})_{\mu}=\mathcal{C}_{\mu}\otimes_{\mathbbm{k}}\mathcal{A}$, defining Chevalley functors $\mathsf{E}_{i}\mathbbm{1}_{\mu}\otimes\mathbbm{1}_{\mathcal{A}}$, etc. If $\mathcal{C}$ is a simple categorification, then clearly $\mathcal{C}\otimes_{\mathbbm{k}}\mathcal{A}$ is an isotypic categorification. Conversely, we have:

Let $\mathcal{C}$ be a categorical representation of $U_{q}(\mathfrak{g}),\mu\in X$ and $i\in I$. We define a complex of functors $\Theta_{i}\mathbbm{1}_{\mu}$, supported in nonpositive cohomological degrees, where for $r\geq 0$ the $-r$ component is

The differential $d^{r}:(\Theta_{i}\mathbbm{1}_{\mu})^{-r}\to(\Theta_{i}\mathbbm{1}_{\mu})^{-r+1}$ is defined using the counits of the bi-adjunctions relating $\mathsf{E}_{i}$ and $\mathsf{F}_{i}$ (see [Cauclasp, Section 4] for details). This produces a functor $\Theta_{i}\mathbbm{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to D^{b}(\mathcal{C}_{s_{i}(\mu)})$, which following Chuang and Rouquier we call the Rickard complex.

It’s straightforward to verify that the Rickard complex $\Theta_{i}\mathbbm{1}_{\mu}$ categorifies Lusztig’s braid group operators $\mathsf{t}_{i}\mathsf{1}_{\mu}$ ([Cauclasp, Section 2]). On the level of categories we have the following two theorems of Chuang-Rouquier and Cautis-Kamnitzer, which are the fundamental results about Rickard complexes. Note that the latter theorem was conjectured in [Rou2KM, Conjecture 5.19].

This action is “weak” since we don’t make any claim on the canonicity of the functorial isomorphisms. Nevertheless, for $w\in W$ we define $\Theta_{w}\mathbbm{1}_{\mu}:=\Theta_{i_{1}}\circ\cdots\circ\Theta_{i_{\ell}}\mathbbm{1}_{\mu}$, where $w=s_{i_{1}}\cdots s_{i_{\ell}}$ is a reduced expression. Thus $\Theta_{w}\mathbbm{1}_{\mu}$ is defined up to isomorphism, but not canonical isomorphism. Luckily, everything we do in this paper only requires $\Theta_{w}\mathbbm{1}_{\mu}$ to be defined up to isomorphism.

As a consequence of their proof of Theorem 3.11, Chuang and Rouquier show that the inverse of $\Theta_{i}\mathbbm{1}_{\mu}$ is its right adjoint. We denote this functor by $\Theta_{i}^{\prime}\mathbbm{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to D^{b}(\mathcal{C}_{s_{i}(\mu)})$, so that $\Theta_{i}\Theta_{i}^{\prime}\mathbbm{1}_{\mu}\cong\Theta^{\prime}_{i}\Theta_{i}\mathbbm{1}_{\mu}\cong\mathbbm{1}_{\mu}$. As a complex of functors, $\Theta_{i}^{\prime}\mathbbm{1}_{\mu}$ is supported in nonnegative cohomological degrees, where for $s\geq 0$ the $s$ component is

Let $\mathcal{T}$ be a triangulated category with shift functor $[1]:\mathcal{T}\rightarrow\mathcal{T}$. In the cases of most interest to us, $\mathcal{T}$ is a subcategory of a derived category, in which case $[1]$ is the homological shift functor. Suppose $\mathcal{T}$ has a t-structure $t=(\mathcal{T}^{\leq 0},\mathcal{T}^{\geq 0})$, with heart $\mathcal{T}^{\heartsuit}=\mathcal{T}^{\leq 0}\cap\mathcal{T}^{\geq 0}$ [BBD]. Recall that a triangulated functor $F:\mathcal{T}\to\mathcal{S}$ between triangulated categories with $t$-structure is $t$-exact if $F(\mathcal{T}^{\leq 0})\subseteq\mathcal{S}^{\leq 0}$ and $F(\mathcal{T}^{\geq 0})\subseteq\mathcal{S}^{\geq 0}$. We let $F[p]:\mathcal{T}\to\mathcal{S}$ denote the pre-composition of $F$ with the $p$-shift $[p]$.

Now let $\mathcal{S}\subset\mathcal{T}$ be a thick triangulated subcategory, and consider the quotient functor $Q:\mathcal{T}\to\mathcal{T}/\mathcal{S}$. Following [CRperv], we say that $t$ is compatible with $\mathcal{S}$ if $t_{\mathcal{T}/\mathcal{S}}=(Q(\mathcal{T}^{\leq 0}),Q(\mathcal{T}^{\geq 0}))$ is a t-structure on $\mathcal{T}/\mathcal{S}$. By [CRperv, Lemmas 3.3 & 3.9], if $t$ is compatible with $\mathcal{S}$ then $(\mathcal{T}/\mathcal{S})^{\heartsuit}=\mathcal{T}^{\heartsuit}/\mathcal{T}^{\heartsuit}\cap\mathcal{S}$, and $t_{\mathcal{S}}=(\mathcal{S}\cap\mathcal{T}^{\leq 0},\mathcal{S}\cap\mathcal{T}^{\geq 0})$ is a t-structure on $\mathcal{S}$ such that $\mathcal{S}^{\heartsuit}=\mathcal{T}^{\heartsuit}\cap\mathcal{S}$.