2.1 Algebras

In this section, we introduce the algebraic construction that will unify the perspectives appearing here. These algebras have appeared in many previous works of the author and collaborators [Reference Kamnitzer, Tingley, Webster, Weekes and YacobiKTW⁺19, Reference WebsterWeba, Reference WebsterWeb15, Reference WebsterWeb16, Reference WebsterWeb17b, Reference WebsterWeb19], as well as the line of research of Khovanov, Lauda, Sussan, and Yonezawa [Reference Khovanov, Lauda, Sussan and YonezawaKLSY18, Reference Khovanov and SussanKS18].

Definition 2.1 A KLRW diagram is a collection of finitely many oriented curves in $\mathbb{R}\times [0,1]$ whose projection to the second factor is a diffeomorphism. Each curve is either:

• • colored red and labeled with the integer m and decorated with finitely many dots;

• • colored black and labeled with $i\in [1,m-1]$ and decorated with finitely many dots. Let $v_i$ be the number of black strands with label i.

The diagram must be locally of the form

with each curve oriented in the negative direction. In particular, no red strands can ever cross. Each curve must meet both $y=0$ and $y=1$ at distinct points from the other curves.

Readers familiar with the conventions of [Reference WebsterWeb16, Reference WebsterWeb17b], etc. might be surprised to see dots on red strands as well as black. This corresponds to the “canonical deformation” discussed in [Reference WebsterWeb19, Section 2.7] or the “redotting” of [Reference Khovanov and SussanKS18].

We’ll only consider KLRW diagrams up to isotopy. Since the orientation on a diagram is clear, we typically won’t draw it.

We call the lines $y=0,1$ the bottom and top of the diagram. Reading across the bottom and top from left to right, we obtain a sequence $\mathbf {i}=(i_1, \dots , i_V)$ of elements of $[1,m]$ labeling both red and black strands, where V is the total number of strands.

Definition 2.2 Given KLRW diagrams a and b, their composition $ab$ is given by stacking a on top of b and attempting to join the bottom of a and top of b. If the sequences from the bottom of a and top of b don’t match, then the composition is not defined and by convention is 0, which is not a KLRW diagram, just a formal symbol.

Fix a field and let ${\tilde {{\tilde {\mathbb{T}}}}}$ be the formal span over of KLRW diagrams (up to isotopy). The composition law induces an algebra structure on ${\tilde {{\tilde {\mathbb{T}}}}}$ .

Let $e(\mathbf {i})$ be the unique crossingless, dotless diagram where the sequence at top and bottom are both $\mathbf {i}$ .

Throughout, we fix integers $m,n,$ and $\chi $ an integral weight $(\chi _1,\dots , \chi _n)\in \mathbb{Z}^n$ such that $\chi _1\leq \cdots \leq \chi _n$ . Fix a dimension vector $\mathbf {v}=(v_1,\dots , v_{m-1})\in \mathbb{Z}_{\geq 0}^{m-1}$ ; by convention, we take $v_{m}=n$ . Consider the set $\Omega =\{(i,j) \mid i\in [1,m], j\in [1,v_i]\}$ . Let $\prec $ be a total order on $\Omega $ such that

(2.1) $$ \begin{align} (i,1)\preceq \cdots \preceq (i,v_i). \end{align} $$

This is equivalent to choosing a word $\mathbf {i} =(i_1,\dots , i_{N}),$ where $N=|\Omega |$ and $i_k=i$ for $v_i$ different indices k.

We will want to weaken this definition a bit and allow $\preceq $ to be a total preorder (that is, a relation which is transitive and reflexive, but not necessarily anti-symmetric). In this case, we have an induced equivalence relation $(i,k)\approx (j,\ell )$ if $(i,k)\preceq (j,\ell )$ and $(i,k)\succeq (j,\ell )$ . We assume that our preorder satisfies the condition that

(2.2) $$ \begin{align} (i,k)\not\approx (j,\ell)\text{ whenever } i\neq j. \end{align} $$

We can still attach a word $\mathbf {i}$ to such a preorder; two equivalent elements give the same letter in the word $\mathbf {i}$ , so it doesn’t matter whether they have a chosen order. We can thus think of a preorder as corresponding to a word in the generators with some subsets where the same letter appears multiple times together grouped together. We can represent this within the word itself by replacing $(i,\dots , i)$ with $i^{(a)}$ . Thus, for our purposes, $(3,2,2,3,1,3)$ and $(3,2^{(2)},3,1,3)$ are different words with different associated preorders. Every such word has a unique totalization satisfying (2.1).

Definition 2.5 Let $\mathbb{\tilde {T}}$ be the quotient of ${\tilde {{\tilde {\mathbb{T}}}}}$ by the following local relations between KLRW diagrams. We draw these below as black, but the same relations apply to red strands (always taken with the label m):

(2.2a)

(2.2b)

(2.2c)

(2.2d)

Let $e_\chi $ be the idempotent given by summing $e(\mathbf {i})$ for all $\chi $ -parabolic total orders satisfying (2.1). For a non-total order satisfying (2.1) and (2.2), we associate the “divided power” idempotent $e'(\mathbf {i})$ which acts on each equivalence class of consecutive strands by a primitive idempotent in the nilHecke algebra (for example, that introduced in [Reference Khovanov, Lauda, Mackaay and StošićKLMS12, (2.18)]). So, for example, we associate to $(3,2^{(2)},3,1,3)$ the idempotent

Let $S_\chi $ be the stabilizer of $\chi $ in $S_n$ ; this naturally acts on the subalgebra $e_\chi \mathbb{\tilde {T}} e_\chi $ by permuting groups of red strands (or equivalently, dots on those red strands).

This is a canonical deformation of the algebra $\tilde {T}^{{\underline {\boldsymbol {\lambda }}}}$ of [Reference WebsterWeb17b] attached to the sequence of dominant weights ${\underline {\boldsymbol {\lambda }}}=(g_1\omega _{m-1},\dots , g_k\omega _{m-1})$ , where $g_1,\dots , g_k$ are the sizes of the blocks of consecutive equal entries in $\chi $ , i.e., $S_\chi =S_{g_1}\times \cdots \times S_{g_k}$ .

Note that this algebra breaks up into a sum of subalgebras where we fix the number of strands with each label; as usual, we let $v_i$ denote the number with label i. We’ll be particularly interested in the case when $v_1\leq v_2\leq \dots \leq v_m$ . This is the condition that the corresponding weight of $\mathfrak {sl}_m$ is dominant; in our usual correspondence, it corresponds to the n-tuple $\nu $ where $(1,\dots , 1,2,\dots , 2,\dots ,)$ where i appears $v_i-v_{i-1}$ times. This same algebra is considered in [Reference Khovanov, Lauda, Sussan and YonezawaKLSY18, Section 4] in the case $m=2$ and is denoted $W(\mathbf {g},v_1)$ (using our $g_*$ and $v_*$ as above).

From its realization as a weighted KLR algebra, the algebra $\mathbb{\tilde {T}}$ inherits a polynomial representation.

Definition 2.7 The polynomial representation of $\mathbb{\tilde {T}}$ is the vector space

with sum running over total orders on $\Omega $ satisfying (2.1). The action is given by the rules:

• • $e({\mathbf {i}})$ acts by projection to the corresponding summand;

• • a dot on the kth strand from the left acts by multiplication by $Y_k$ ;

• • a crossing of the kth and $k+1$ st strands with $\mathbf {i}$ at the bottom and $\mathbf {i}'$ at top acts by:

  1. – If $i_k=i_{k+1}$ , the divided difference operator

     $$\begin{align*}fe_{\mathbf{i}}\mapsto \frac{f^{(k,k+1)}-f}{Y_{k+1}-Y_k}e_{\mathbf{i}'}.\end{align*}$$

  2. – If $i_k+1=i_{k+1}$ , the permutation $(k,k+1)$ followed by a multiplication

     $$\begin{align*}fe_{\mathbf{i}}\mapsto (Y_{k+1}-Y_k) f^{(k,k+1)}e_{\mathbf{i}'}.\end{align*}$$

  3. – Otherwise, the permutation $(k,k+1)$

     $$\begin{align*}fe_{\mathbf{i}}\mapsto f^{(k,k+1)}e_{\mathbf{i}'}.\end{align*}$$

The polynomial representation $\mathscr {P}^\chi $ for $\mathbb{\tilde {T}}^\chi $ is given by $(e_\chi \mathscr {P})^{S_\chi }$ , where $S_\chi $ acts by permuting red dots as usual.

2.2 Violating quotients

The algebra $\mathbb{T}^\chi $ is not precisely the algebra $T^\chi $ defined in [Reference WebsterWeb17b], but a deformation of it which we’ve considered in several contexts, in particular, in [Reference WebsterWebc, Section 4]. This deformation is flat, since it is a special case of deforming the polynomials defining the KLRW algebra (as discussed in [Reference WebsterWeb19, Proposition 2.23]); in the case $m=2$ , this is the redotted algebra discussed by Khovanov–Sussan in [Reference Khovanov and SussanKS18]. The algebra $T^\chi $ is the quotient of $\mathbb{T}^\chi $ by all red dots. Since the red dots are central, and the polynomial ring is graded local, every gradable simple $\mathbb{T}^\chi $ -module factors through $T^\chi $ , and so the Grothendieck group of $T^\chi \operatorname {-wgmod}$ agrees with the Grothendieck group of $\mathbb{T}^\chi \operatorname {-wgmod}$ . From [Reference WebsterWebc, Section 4], we have the following theorem.

2.3 Ladder bimodules

In our notation, we identify the dominant weight $\chi $ with a weight of $\mathfrak {sl}_\infty $ by ${\mu _{\chi }=\sum _{i=1}^n \epsilon _{\chi _i}}$ . This is an injective map, but is far from surjective, since it only hits weights where the coefficients of the $\epsilon _i$ ’s are positive and sum to n (in the usual parlance, they are level n). In particular,

$$\begin{align*}\alpha_j^\vee(\mu_{\chi})=\#\{j\mid \chi_j=i+1\}-\#\{j \mid \chi_j=i\}.\end{align*}$$

These operations are uniquely characterized by the following fact.

Proof If $\chi ^{+ i}$ exists, then for some k, we have $i=\chi _k<\chi _{k+1}$ . In this case, the dominant weight

$$\begin{align*}\chi^{+ i}=\chi+\epsilon_k=(\chi_1,\dots, \chi_k+1,\dots,\chi_n)\end{align*}$$

satisfies

$$\begin{align*}\alpha_j^\vee(\chi+\epsilon_k)=\begin{cases} \alpha_j^\vee(\mu_{\chi})-1 & j=\chi_k\pm 1,\\ \alpha_j^\vee(\mu_{\chi})+2 & j=\chi_k,\\ \alpha_j^\vee(\mu_{\chi}) & \text{otherwise}. \end{cases} \end{align*}$$

Thus, we have that $\mu _{\chi +\epsilon _k}=\mu _\chi +\alpha _{i}$ as desired. The second half of the result follows from the fact that $(\chi ^{+ i})^{-i}=\chi $ .

Assume that i appears at least a times in $\chi $ . Let $\chi '=\chi ^{+i^a}$ be the dominant weight obtained by changing a instances of i to $i+1$ . Let $S_{\chi ,\chi '}=S_{\chi }\cap S_{\chi '}$ .

These are generalizations of the ladder bimodules defined in [Reference Khovanov, Lauda, Sussan and YonezawaKLSY18, Section 5.2]. We draw these by pinching together the red strands in a single equivalence class under the $\chi $ -parabolic preorder giving an idempotent into a single red strand. Thus, elements of this bimodule look like:

(2.3)

These bimodules have a “representation” as well. By a representation of a bimodule B over algebras $A_1$ and $A_2$ , we mean a representation of the Morita context:

$$\begin{align*}A_B=\begin{bmatrix} A_1 & B \\ 0& A_2\end{bmatrix}\end{align*}$$

with the obvious matrix multiplication. That is, a left module $V_i$ of $A_i$ , and a bimodule map . In our case, we will use the polynomial representations $\mathscr {P}^\chi $ of $\mathbb{\tilde {T}}^\chi $ . Diagrams other than the split and join of red strands act by the formulas in Definition 2.7. The formulas for splitting and joining are the same as in [Reference Stroppel and WebsterSW]:

• • split corresponds to the inclusion $\mathscr {P}^\chi \hookrightarrow \mathscr {P}^{\chi ,\chi '}$ , where the latter is the invariants under $S_{\chi ,\chi '}\subset S_{\chi }$ ;

• • merge corresponds to the divided difference operator $\mathscr {P}^{\chi ,\chi '}\to \mathscr {P}^{\chi '}$ given by

  $$\begin{align*}f\mapsto \sum_{\sigma\in S_{\chi'}/S_{\chi,\chi'}}\frac{\displaystyle f^{\sigma}}{\Delta_{+i^{(a)}}^\sigma},\end{align*}$$
  where $\Delta _{+i^{(a)}}$ is the product of $Y_k-Y_\ell $ , where k ranges over the red strands with $\lambda _{n,j}=i+1$ , and $\ell $ over the red strands in the “rung” of the ladder. This corresponds to the operator in equivariant cohomology which integrates a $P_{\chi ,\chi '}$ -equivariant class over $P_{\chi '}/P_{\chi ,\chi '}$ to give a $P_{\chi '}$ -equivariant class. In terms of the nilHecke algebra, this corresponds to the diagram:
  

Proof Here, we use the fact that $\mathbb{E}_i$ by definition is the subbimodule of $e_{\chi '}\mathbb{\tilde {T}} e_{\chi }$ invariant under $S_{\chi ,\chi '}$ . This embedding corresponds to taking a diagram as in (2.3), and simply expanding red strands.

This does not precisely match the operators above, but it does after we add a crossing of the red strands that joined at the top:

The action of this in the usual polynomial representation of the KLR algebra of $A_{m}$ matches the formulas we have given. The fact that this is a bimodule map follows from the faithfulness of the polynomial representations.

3.1 Quiver representations

As in the previous section, we fix $m,n$ , and $\chi $ . We can think of this as giving a cocharacter into $GL_n$ , and let $P_\chi \subset GL_n$ be the parabolic whose Lie algebra is the non-positive weight space for this cocharacter. This is the standard Borel if $\chi _i\neq \chi _j$ for all $i,j$ , and in general is block upper-triangular matrices, with blocks corresponding to the consecutive $\chi _i$ which are equal.

Fix a dimension vector $\mathbf {v}=(v_1,\dots , v_{m-1})\in \mathbb{Z}_{\geq 0}^{m-1}$ as in the previous section. Consider the representation

$$\begin{align*}V=\mathrm{Hom}({\mathbb{C}}^{v_1},{\mathbb{C}}^{v_2})\oplus \mathrm{Hom}({\mathbb{C}}^{v_2},{\mathbb{C}}^{v_3})\oplus \cdots \oplus \mathrm{Hom}({\mathbb{C}}^{v_{m-2}},{\mathbb{C}}^{v_{m-1}})\oplus \mathrm{Hom}({\mathbb{C}}^{v_{m-1}},{\mathbb{C}}^n)\end{align*}$$

of the group $G=G_{\chi }=P_\chi \times GL_{v_1}\times \cdots \times GL_{v_{m-1}}$ . Recall that we introduced the notation $G^0=GL_{v_1}\times \cdots \times GL_{v_{m-1}}$ in the introduction. In the notation popular with physicists, this corresponds to the following quiver:

Attached to the space V with the action of G, we have an equivariant derived category $D^b_G(V)$ as introduced in [Reference Bernstein and LuntsBL94]; in modern terminology, we would think of this as the derived category of constructible sheaves on the quotient stack $V/G$ . There are various other avatars of this category, such as strongly equivariant D-modules, but we will only use a few basic facts about this category, such as the decomposition theorem and the computation of Ext-algebras as Borel–Moore homology familiar from [Reference Chriss and GinzburgCG97, Section 8.6].

First, we simply need to classify the orbits of $P_\chi $ in V. Recall that a segment in $[1,m]$ is a list of consecutive integers $(k,k+1,\dots , \ell )$ , and a multi-segment is a multi-set of segments. The dimension vector of a segment is the vector $(0,\dots , 1,\dots , 1,\dots , 0)\in \mathbb{Z}^{m}$ with 1 in every position in $[k,\ell ]$ and 0 in all others, and the dimension vector of a multi-segment is the sum of those for the constituent segments. That is, it is the vector that records how many times an index $i\in [1,m]$ appears in the constituent segments.

These are relevant because of the following fact from the appendix.

We view the subset $\{(i,1),\dots ,(i,v_i)\}\subset \Omega $ as corresponding to an ordered basis $\{b_{(i,1)},\dots , b_{(i,v_i)}\}$ of ${\mathbb{C}}^{v_i}$ . Giving this copy of ${\mathbb{C}}^{v_i}$ degree i, we can view

$$\begin{align*}{\mathbb{C}}^\Omega \cong {\mathbb{C}}^{v_1}\oplus \cdots \oplus {\mathbb{C}}^{v_{m}}\end{align*}$$

as a graded vector space, and we can view a degree 1 map $f\colon {\mathbb{C}}^\Omega \to {\mathbb{C}}^\Omega $ as an element of V, that is, of quiver representation of $A_{m}$ with dimension vector $(v_1,\dots , v_{m})$ .

3.2 Quiver flag varieties and the equivariant derived category

As before, consider a total preorder $\preceq $ satisfying (2.1) and (2.2). This choice of preorder induces a flag $F^{\prec }_\bullet $ , with each subspace given by the formula:

$$\begin{align*}F^{\prec}_{(i,k)}=\operatorname{span}(\{b_{(j,\ell)}\mid (j,\ell)\preceq (i,j)\}.\end{align*}$$

If the preorder is not an order, equivalent elements give the same subspace, so this flag will have some redundancies in it. We say that a flag on ${\mathbb{C}}^\Omega $ indexed by the equivalence classes has type $\preceq $ or type $\mathbf {i}$ if it is conjugate to a flag of this form under $G^0$ , and let $\operatorname {Fl}(\mathbf {i})$ be the set of such flags. Note that $G^0$ acts on this space transitively, with the stabilizer of $F^{\prec }_\bullet $ given by a parabolic $P_0$ , which only depends on the equivalence relation of $\preceq $ ; in particular, for any total order, we get the same Borel $B_0$ . Consider the $G^0$ -space

$$\begin{align*}X(\mathbf{i})=\{ (f,F_\bullet)\in V\times \operatorname{Fl}(\mathbf{i}) \mid f(F^{\prec}_{(i,k)})\subset F^{\prec}_{(i,k)}\}.\end{align*}$$

Proof Assume that $\mathbf {i}$ is $\chi $ -parabolic. Thus, we have that if $\chi _k=\chi _{k+1}$ , then $F^{\prec }_{(m,k+1)}=F^{\prec }_{(m,k)}+{\mathbb{C}}\cdot b_{m,k+1}$ . By degree considerations, $f(b_{m,k+1})=0$ , so $f(F^{\prec }_{(m,k)})\subset F^{\prec }_{(m,k)}$ , then automatically, we have

$$\begin{align*}f(F^{\prec}_{(m,k+1)})=f(F^{\prec}_{(m,k)})\subset F^{\prec}_{(m,k)}\subset F^{\prec}_{(m,k+1)}.\end{align*}$$

Thus, for a $\chi $ -parabolic flag, we only need to check that $f(F^{\prec }_{(i,k)})\subset F^{\prec }_{(i,k)}$ for $i<m$ or when $i=m$ and $\chi _k\neq \chi _{k-1}$ .

Consider $(f,F_\bullet )\in X(\mathbf {i})$ and $g\in P_\chi $ . Consider the map $gf\colon {\mathbb{C}}^{\Omega } \to {\mathbb{C}}^{\Omega }$ . This is again a quiver representation, which is compatible with the flag $gF^{\prec }_{(i,k)}$ . Since $g\in P_\chi $ , we have that $gF^{\prec }_{(i,k)}=F^{\prec }_{(i,k)}$ if $i<m$ or if $i=m$ and $\chi _k\neq \chi _{k-1}$ . Thus, by our observation above, our original flag is still compatible with $gf$ . The fact that this commutes with $G^0$ is clear.

For each segment $(k,\dots , \ell )$ , let $\mathbf {i}_{(k,\dots , \ell )}=(\ell , \ell -1,\dots , k)$ be the word where we list the entries in reverse order.

Example 2 Let $n=2,m=3$ and consider the multi-segment

$$\begin{align*}\mathbf{Q}=\{(1),(2),(2,1), (3,2,1) ,(3,2)\}.\end{align*}$$

If $\chi _1>\chi _2$ , a flavoring of this multi-segment is a bijection between the sets $\{(3,2,1) ,(3,2)\}$ and $\{\chi _1,\chi _2\}$ . In this case, good words for this multi-segment with the flavorings for the order we’ve written the sets above and its opposite are, respectively:

$$\begin{align*}(1,2,1,2,3,2,1,3,2)\qquad (1,2,1,2,3,2,3,2,1).\end{align*}$$

On the other hand, if $\chi _1=\chi _2$ , there is only one possible flavoring with

$$\begin{align*}(1,2,1,2,3,3,2,2,1).\end{align*}$$

Let $\pi ^{\mathbf {i}}\colon X(\mathbf {i})\to V$ be the projection map, and $\mathscr {F}_{\mathbf {i}}=\pi _*^{\mathbf {i}}{\mathbb{C}}_{X(\mathbf {i})}[\dim X(\mathbf {i})]$ be the pushforward of the sheaf of locally constant ${\mathbb{C}}$ -valued functions on $X(\mathbf {i})$ .

Recall that for each $G_{\chi }$ -orbit, there is a unique $G_{\chi }$ -equivariant perverse sheaf $\mathbf {IC}_{\mathbf {Q}}$ which extends the trivial local system on the orbit.

3.3 Graded lifts

Let $\mathscr {F}_{\chi }=\oplus _{\mathbf {i}}\mathscr {F}_{\mathbf {i}}$ , where $\mathbf {i}$ ranges over totalizations of $\chi $ -parabolic words and ${X_{\chi }=\sqcup _{\mathbf {i}} X(\mathbf {i})}$ . As usual, by [Reference Chriss and GinzburgCG97], we have that

$$\begin{align*}A_\chi=H^{BM,G_{\chi}}_*(X_{\chi}\times_V X_{\chi})\cong \operatorname{Ext}^{\bullet}_{D^b_{G_{\chi}}(V)}(\mathscr{F}_{\chi},\mathscr{F}_{\chi}) \end{align*}$$

as dg-algebras, where we give the left-hand side a non-standard grading.

The sheaves $\mathscr {F}_{\chi }$ have an additional structure which is hard to see directly on the sheaf level: they are the solutions of complexes of D-modules $\mathsf {F}_{\chi }$ which carry a mixed Hodge structure. See [Reference SaitoSai] for more details on mixed Hodge structures. We don’t need much of the full power of this mixed Hodge structure, but as in [Reference WebsterWebb, Section 2.8], this gives us a natural dgg (differential-graded and graded) structure on $A_{\chi }$ . A dgg-algebra A is a dg-algebra in the category of graded vector spaces; that is, A has a homological grading and an internal grading, with the differential having degree $(1,0)$ . Given a dgg-algebra A, we let $A\operatorname {-dggmod}$ be the category of representations of A, again, in the category of graded vector spaces, that is, dg-modules over A equipped with an additional grading. These form a graded dg-category where the morphism complexes are the degree $(*,0)$ part of the usual morphism complexes of dg-modules – again, this just means that we only use degree 0 morphisms between graded vector spaces.

One useful observation is that a dgg-algebra where the internal grading coincides with the homological grading is necessarily formal, since it is quasi-isomorphic to a graded $A_{\infty }$ algebra, and all the higher operations are trivial for grading reasons (see [Reference WebsterWebb, Theorem 2.38] and [Reference Cirici and Sopena-GilboyCSG, Proposition 5.4]).

We can use this dgg-algebra structure to construct a graded lift.

By [Reference WebsterWeb19, Theorem 4.5], we have the following theorem.

Theorem 3.10 For any preorders $\mathbf {i},\mathbf {j}$ satisfying (2.1) and (2.2), we have an isomorphism

$$\begin{align*}\operatorname{Ext}^{\bullet}(\mathscr{F}_{\mathbf{i}},\mathscr{F}_{\mathbf{j}})\cong e'(\mathbf{i})\mathbb{\tilde{T}}^\chi e'(\mathbf{j})\end{align*}$$

matching convolution product with multiplication in $\mathbb{\tilde {T}}^\chi $ , the red dots with the cohomology ring $H^*(BP_\chi )$ and the black dots with Chern classes of the tautological bundles on $X(\mathbf {i})$ . In particular, we have an isomorphism $A_\chi \cong \mathbb{\tilde {T}}^\chi $ .

It might seem strange that in the definition of the algebra $A_\chi $ , we used only honest orders, not preorders, when the first sentence of the theorem covers preorders. This is needed to match the usual definition of $\mathbb{\tilde {T}}^\chi $ . If we incorporated preorders, the result would be a larger, Morita equivalent algebra which contained thick strands as in the extended graphical calculus of [Reference Khovanov, Lauda, Mackaay and StošićKLMS12, Reference Stroppel and WebsterSW]. It is much more difficult to give a nice presentation of this larger algebra.

These results combine to show that the algebra $\mathbb{\tilde {T}}^\chi $ controls the derived category $D^b_{G_{\chi }}(V)$ .

Theorem 3.11 The object $\mathscr {F}_{\chi }$ is a compact generator of $D^b_{G_{\chi }}(V)$ . Thus, the functor $\operatorname {Ext}(\mathscr {F}_{\chi },-)$ induces equivalences:

$$\begin{align*}D^b_{G_{\chi}}(V)\cong \mathbb{\tilde{T}}^\chi\operatorname{-dg-mod}\qquad D^{b,\operatorname{mix}}_{G_{\chi}}(V)\cong\mathbb{\tilde{T}}^\chi\operatorname{-dggmod}\cong D^b(\mathbb{\tilde{T}}^\chi\operatorname{-gmod}).\end{align*}$$

The functor of forgetting graded structure corresponds to the functor considering a complex of graded modules as a dg-module by collapsing gradings.

3.4 The categorical action

Consider the 2-category $\mathsf {Perv}$ whose:

• • objects are dominant integral weights $\chi =(\chi _1\leq \cdots \leq \chi _n)$ ;

• • 1-morphisms $\chi \to \chi '$ are sums of shifts of semi-simple $P_{\chi '}\times P_{\chi }$ -equivariant perverse sheaves on $GL_n$ , or equivalently, $GL_n$ -equivariant perverse sheaves on $GL_n/P_{\chi '}\times GL_n/P_{\chi '}$ with composition given by convolution;

• • 2-morphisms in the equivariant derived category.

As discussed in [Reference WebsterWeb17a, Theorem 6], this equivalent to the category $\mathsf {Flag}$ whose:

• • objects are dominant integral weights $\chi =(\chi _1\leq \cdots \leq \chi _n)$ ;

• • 1-morphisms $\chi \to \chi '$ are sums of singular Soergel $H^*(BP_{\chi '})\operatorname {-}H^*(BP_{\chi })$ -bimodules;

• • 2-morphisms are degree 0 homomorphisms of bimodules.

via the functor that takes hypercohomology of a sum of shifts of equivariant perverse sheaves. For technical reasons, we add to these categories an object $\emptyset $ , which has Hom with any other object given by the zero category.

Consider the fiber product

$$\begin{align*}P_{\chi'}\setminus GL_n/P_{\chi}\times_{BP_{\chi}}V/G_{\chi}=\frac{GL_{n}\times V}{P_{\chi'}\times G_{\chi}}\end{align*}$$

where the action is via $(p',p)\cdot (g,v)=(p'gp^{-1},pv)$ . This is, of course, equipped with an action map

$$\begin{align*}a\colon P_{\chi'}\setminus GL_n/P_{\chi}\times_{BP_{\chi}}V/G_{\chi}\to V/G_{\chi'}, \end{align*}$$

and as usual, we define convolution of sheaves $\mathcal {F}\in D^b_{P_{\chi '}\times P_\chi }(GL_n)$ and $\mathcal {G}\in D^b_{G_{\chi }}(V)$ as $\mathcal {F}\star \mathcal {G}=a_* (\mathcal {F}\boxtimes \mathcal {G}).$

The category $\mathsf {Flag}$ carries a well-known categorical action defined by Khovanov and Lauda [Reference Khovanov and LaudaKL10, Section 6]. We define $\mathcal {U}$ to be the 2-category categorifying the quantum group of $\mathfrak {sl}_\infty $ (following the definition of [Reference BrundanBru16, Definition 1.1] which unifies all previous definitions). In this category:

• • objects are integral weights of $\mathfrak {sl}_\infty $ , that is, finite sums of the unit vectors $\epsilon _k$ for $k\in \mathbb{Z}$ ;

• • 1-morphisms are generated by formal symbols

  

• • 2-morphisms are given by certain string diagrams modulo the relations in [Reference BrundanBru16].

As discussed in [Reference WebsterWeb17a, Section 2.2], we thus have a 2-functor $\Phi _{\mathsf {P}}\colon \mathcal {U}\to \mathsf {Perv}$ sending $\mu _\chi \to \chi $ for weights of the correct form, and to $\emptyset $ otherwise.

If $\chi '=\chi ^{+i^a}$ , then the $P_{\chi }\times P_{\chi '}$ -orbits on G are in bijection with orbits of double cosets for $(S_\chi ',S_\chi )$ . In particular, there is a unique closed orbit, given by the product $P_{\chi ',\chi }=P_{\chi '}P_{\chi }$ . The cohomology $H^*_{P_{\chi '}\times P_{\chi }}(P_{\chi ',\chi })$ of this orbit gives the action of [Reference Khovanov and LaudaKL10, Section 6]. As discussed below the proof of Theorem 6 in [Reference WebsterWeb17a, Section 2.2], we have the following lemma.

For us, the important new ingredient here is that since we have an algebraic manifestation of the category $D^b_{P_\chi }(V)$ given by the algebra $\mathbb{\tilde {T}}^\chi $ , the action of the 2-category $\mathcal {U}$ has a similar manifestation.

For every 1-morphism $\mathcal {F}\colon \chi \to \chi '$ in the 2-category $\mathsf {Perv}$ , we can define a bimodule

$$\begin{align*}B_{\mathcal{F}}=\operatorname{Ext}_{D^b_{G_{\chi'}}(V)}(\mathscr{F}_{\chi'}, \mathcal{F}\star \mathscr{F}_{\chi}).\end{align*}$$

By Theorem 3.11, we have a commutative diagram

It follows immediately that the functor of tensor product with $B_{\mathcal {F}}$ is exact. It also follows that

Proof Of course, we only need to check one idempotent at a time. That is, let $\mathbf {i}$ be a $\chi $ -parabolic word, and $\mathbf {i}'$ a $\chi '$ parabolic. We only need to show that

$$\begin{align*}e(\mathbf{i}')\mathbb{E}_i^{(a)}e(\mathbf{i})\cong e(\mathbf{i}')B_{{\mathbb{C}}_{\chi',\chi}}e(\mathbf{i}) =\operatorname{Ext}_{D^b_{G_{\chi}}(V)}(\mathscr{F}_{\mathbf{i}'}, {\mathbb{C}}_{\chi',\chi}\star \mathscr{F}_{\mathbf{i}}).\end{align*}$$

Note that this isomorphism is only correct up to grading shift due to the need to shift ${\mathbb{C}}_{\chi ',\chi }$ to make it Verdier-self-dual. Since ${\mathbb{C}}_{\chi ^{\prime },\chi }\star -$ is the composition of restriction of $\mathscr {F}_{\mathbf {i}}$ to the $P_\chi \cap P_{\chi ^{\prime }}$ -equivariant derived category, with the induction to the $P_{\chi '}$ -equivariant derived category, we can use adjunction to show that

(3.1) $$ \begin{align} \operatorname{Ext}_{D^b_{G_{\chi}}(V)}(\mathscr{F}_{\mathbf{i}'}, {\mathbb{C}}_{\chi',\chi}\star \mathscr{F}_{\mathbf{i}})\cong \operatorname{Ext}_{D^b_{G_{\chi}\cap G_{\chi'}}(V)}(\mathscr{F}_{\mathbf{i}'},\mathscr{F}_{\mathbf{i}})\cong\operatorname{Ext}_{D^b_{B\times G^0}(V)}(\mathscr{F}_{\mathbf{i}'},\mathscr{F}_{\mathbf{i}})^{S_{\chi',\chi}}.\nonumber\\ \end{align} $$

Of course, by Theorem 3.10 in the case where $P_\chi =B$ , we have

(3.2) $$ \begin{align} \operatorname{Ext}_{D^b_{B\times G^0}(V)}(\mathscr{F}_{\mathbf{i}'},\mathscr{F}_{\mathbf{i}})\cong e(\mathbf{i}')\mathbb{\tilde{T}} e(\mathbf{i}). \end{align} $$

Thus, combining (3.1) and (3.2), we find that:

$$\begin{align*}\kern-12pt e(\mathbf{i}')\mathbb{E}_i^{(a)}e(\mathbf{i})=(e(\mathbf{i}')\mathbb{\tilde{T}} e(\mathbf{i}))^{S_{\chi',\chi}}=\operatorname{Ext}_{D^b_{B\times G^0}(V)}(\mathscr{F}_{\mathbf{i}'},\mathscr{F}_{\mathbf{i}})^{S_{\chi',\chi}}=e(\mathbf{i}')B_{{\mathbb{C}}_{\chi',\chi}}e(\mathbf{i}).\\[-37pt] \end{align*}$$

Combining Lemma 3.15 and Theorem 3.16, we find the following.

This generalizes [Reference Khovanov, Lauda, Sussan and YonezawaKLSY18, Theorem 9.1], which is the special case where $m=2$ , and avoids the long computation required by the direct proof given in that paper.

3.5 Restriction to the flag variety

Assume now that $v_1\leq v_2\leq \cdots \leq v_n$ . In this case, the stack $V/G^0$ contains an open subset where the compositions $f_{i;1}=f_i\dots f_1$ are injective for all i. On this open subset,

$$\begin{align*}\operatorname{image}(f_1)\subset \operatorname{image}(f_{2;1})\subset \cdots \subset \operatorname{image}(f_{n;1})\subset {\mathbb{C}}^n\end{align*}$$

gives a partial flag. As before, we let $\nu =(1,\dots , 1,2, \dots , 2,\dots )\in \mathbb{Z}^n$ be the unique increasing sequence with i occuring $v_i-v_{i-1}$ times. We can then describe our space of partial flags as exactly $GL_n/P_{\nu }.$ Thus, we obtain a functor

$$\begin{align*}\operatorname{Res}\colon D^{b,\operatorname{mix}}_{G_{\chi}}(V)\to D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_{\nu})\end{align*}$$

by restriction to this open set. This is, of course, compatible with the natural action of $\mathsf {Perv}$ .

Theorem 3.18 We have an equivalence of categories

$$\begin{align*}\beta=\operatorname{Ext}_{D^b_{P_{\chi}}(GL_n/P_\nu)}(\operatorname{Res}(\mathscr{F}_{\chi}), -)\colon D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_\nu)\cong D^b(\mathbb{T}^\chi_\nu\operatorname{-gmod})\end{align*}$$

such that we have a commutative diagram:

Proof This is essentially a restatement of [Reference WebsterWeb17c, Corollary 5.4], but let us be a bit more careful about the details of this point. We have a map $\mathbb{\tilde {T}}^\chi \to \operatorname {Ext}^*(\operatorname {Res}(\mathscr {F}_{\chi }), \operatorname {Res}(\mathscr {F}_{\chi }))$ induced by the functor $\operatorname {Res}$ .

If $\mathbf {i}$ is violating, then $(i_1,1)\prec (m,1)$ . For any point in $\operatorname {Fl}(\mathbf {i})$ , by assumption, we have $f_{m-1;i_1}F_{\preceq (i_1,1)}=0$ , and so one of $f_k$ is not injective. This shows that $\operatorname {Res}(\mathscr {F}_{\mathbf {i}})=0$ . On the other hand, if $\operatorname {Res}(\mathbf {IC}_{\mathbf {Q}})=0$ , then $\mathbf {Q}$ must be a multi-segment that includes a segment without m, so the corresponding good word $\mathbf {i}_{\mathbf {Q}}$ is violating. Thus, ${\operatorname {Res}(\mathbf {IC}_{\mathbf {Q}})=0}$ if and only if $\mathbf {IC}_{\mathbf {Q}}$ is a summand of $\mathscr {F}_{\mathbf {i}}$ .

The usual formalism of recollement shows that $D^{b,\operatorname {mix}}_{P_\chi }(GL_n/P_\nu )$ (after suitable enhancement) is a dg-quotient of $D^{b,\operatorname {mix}}_{G_{\chi }}(V)$ by the subcategory of objects killed by $\operatorname {Res}$ ; by the paragraph above, this is exactly the subcategory generated by $\mathscr {F}_{\mathbf {i}}$ for $\mathbf {i}$ violating. Since $\mathbb{\tilde {T}}^\chi $ is quasi-isomorphic to $\operatorname {Ext}^*(\mathscr {F}_{\chi }, \mathscr {F}_{\chi })$ as a formal dg-algebra, taking dg-quotient by the projectives $\mathbb{\tilde {T}}^\chi e(\mathbf {i})$ for $\mathbf {i}$ violating has the effect of simply modding out by the corresponding 2-sided ideal. Thus, the map above induces an isomorphism

$$\begin{align*}\mathbb{T}^\chi=\operatorname{Ext}^*_{D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_\nu)}(\operatorname{Res}(\mathscr{F}_{\chi}), \operatorname{Res}(\mathscr{F}_{\chi})).\end{align*}$$

By dg-Morita theory, this completes the proof.

Of course, we can also interpret $D^b_{P_{\chi }}(GL_n/P_\nu )$ as a Hom-category in the equivalent 2-categories $\mathsf {Flag}\cong \mathsf {Perv}$ , and this identification intertwines the left regular action on $\mathsf {Perv}$ with the action we’ve discussed on $D^b_{P_{\chi }}(GL_n/P_\nu )$ .

To fix this equivalence algebraically, we need to describe the image of the identity. This corresponds to the constant sheaf on $P_\nu /P_\nu \subset GL_n/P_{\nu }$ . Consider the flavored multi-segment where we take the segment $(i,\dots ,m)$ exactly $v_i-v_{i-1}$ times, and flavor these with i.

The corresponding word is

$$\begin{align*}\mathbf{i}_{\nu}=(m^{(v_1)},(m-1)^{(v_1)},\dots, 1^{(v_1)},m^{(v_2-v_1)},(m-1)^{(v_2-v_1)},\dots, 2^{(v_2-v_1)},\dots).\end{align*}$$

This generalizes the main theorem of [Reference Khovanov and SussanKS18]. Of course, the category of Soergel bimodules has a “reversing functor” where one swaps the left and right actions; this is an equivalence $\mathrm {Hom}_{\mathsf {Flag}}(\chi ,\nu )\cong \mathrm {Hom}_{\mathsf {Flag}}(\nu ,\chi )$ which is anti-monoidal. Applying Corollary 3.20, we find the following.

Of course, this Morita equivalence is realized by a bimodule over these two algebras. It should be possible to capture this equivalence via a diagrammatic bimodule, but the author has not yet been successful in producing a workable description of it. One important special case is that relating $\mathbb{T}^\nu _\nu $ to itself. This is, of course, true tautologically, but there are two ways of relating the ring $e(\mathbf {i}_{\nu })\mathbb{T}^\nu _\nu e(\mathbf {i}_{\nu })$ to the ring $H^*_{P_{\nu }\times P_{\nu }}(P_\nu )$ . The one naturally arising from the left action of $P_{\nu }$ is given by polynomials on the red strands, which are symmetrized on each strand. In the diagram, we insert symmetric polynomials in alphabets with sizes given by $v_1,v_2-v_1,\dots v_m-v_{m-1}$ at each point marked by a $*$ on the red strands:

That arising from the right action is given by the same polynomials on the rightmost strand with each label:

Geometrically, these correspond to polynomials in the Chern classes of tautological bundles: in the first case, of the associated bundles for the action of $P_{\nu }$ on simple subquotients of ${\mathbb{C}}^n$ under the left action of $P_{\nu }$ ; in the second case on ${\mathbb{C}}^{v_k}/f({\mathbb{C}}^{v_{k-1}})$ . These are the same, since as discussed in the proof of Lemma 3.19, the corresponding vector bundles are isomorphic.

4.1 Presentation of the category $\mathcal {GT}_{\chi }$

A choice of integral $\lambda \in \operatorname {MaxSpec}_{\mathbb{Z},\chi }(\Gamma )$ can be used to give a total preorder $\prec $ on the set $\Omega $ as in previous sections for the dimension vector $v_i=i$ for $m=n$ . We set $(i,k) \preceq (j,\ell )$ if and only if $\lambda _{i,k}< \lambda _{j,\ell }$ or $\lambda _{i,k}= \lambda _{j,\ell }$ and $i\geq j$ . Note that the resulting preorder satisfies (2.1) by assumption and (2.2) since if $(i,k) \approx (j,\ell )$ , then we must have $\lambda _{i,k}=\lambda _{j,\ell }$ and $i=j$ .

Note that [Reference WebsterWeba, Proposition 5.4] shows that any maximal ideals giving the same preorder on $\Omega $ give isomorphic functors on $\mathcal {GT}_\chi $ . If we change the order by swapping pairs $(i,k) $ and $(j,\ell )$ , this is called a neutral swap, and more generally, neutral swaps don’t change the functor given by two maximal ideals. For example, two Gelfand–Tsetlin patterns with the same $\chi $ don’t necessarily give the same order, but they differ by neutral swaps, and indeed they give isomorphic functors.

Example 3 The action of $\Gamma $ on the trivial representation gives the weight $\lambda _{i,k}=k$ . Thus, the resulting word is

$$\begin{align*}(n,n-1,\dots, 2,1,n, n-1, \dots, 3,2, n,n-1,\dots, 3, \dots n,n-1,n).\end{align*}$$

As mentioned before, every Gelfand–Tsetlin pattern gives an ordering that differs from this one by neutral swaps.

On the other hand, a weight like $\lambda _3=(1,2,3), \lambda _2=(4,4), \lambda _1=(1)$ gives $(3,1,3,3,2^{(2)})$ , whereas if $\lambda _2=(4,5)$ , then we have $(3,1,3,3,2,2).$ This is an example of a maximal ideal of the Gelfand–Tsetlin subalgebra that appears in no finite-dimensional module.

Using the techniques of [Reference Kamnitzer, Tingley, Webster, Weekes and YacobiKTW⁺19, Reference WebsterWeba, Reference WebsterWebb], we can give an algebraic presentation of the category $\mathcal {GT}_{\chi }$ . This is accomplished by presenting the algebra of natural transformations of the functors Let $\widehat {\mathbb{\tilde {T}}}_\chi $ be the completion of the algebra $\mathbb{\tilde {T}}_{\chi }$ with respect to its grading. See [Reference WebsterWeb20, Section 2] for more discussion of the induced topology (in particular, the continuity of multiplication in it).

Theorem 4.8 We have an isomorphism compatible with multiplication

This is extremely closely related to [Reference Kamnitzer, Tingley, Webster, Weekes and YacobiKTW⁺19, Theorem 5.2], but slightly different, since that result is for Gelfand–Tsetlin modules with honest central character, which corresponds in $\widehat {\mathbb{\tilde {T}}}_\chi $ to setting red dots to 0. Under this isomorphism, the red dots give the nilpotent parts of certain (complicated) elements of the center of $U(\mathfrak {gl}_n).$ In [Reference Silverthorne and WebsterSW24, Lemma 3.3 and Proposition 3.4], we give a more direct algebraic proof of this theorem.

Together with Theorem 3.10, this establishes Theorem 1.

Let us record how this equivalence matches various properties of interest for Gelfand–Tsetlin modules.

In particular, the modules over $\mathbb{\tilde {T}}^\chi _\nu $ which factor through the quotient by all red dots and by violating strands is the category $\mathcal {O}^{\prime }_\chi $ of modules locally finite under $U(\mathfrak {n})$ and semi-simple for the action of the center with central character $\chi $ ; such a module is automatically a generalized weight module, but not necessarily an honest weight module, and so not necessarily in category $\mathcal {O}$ .

On the other hand, the modules that factor through the quotient of $\mathbb{T}^{\chi }_\nu $ by the symmetric polynomials $h_{i,1}$ for all i, are those which lie in the usual category $\mathcal {O}$ . Note that we showed in [Reference WebsterWeb17b, Corollary 9.10] that $T^{\nu }_\chi \operatorname {-mod}$ was equivalent to $\mathcal {O}_\chi $ with the translation functors matched with the induction–restriction action. Thus, this is related to our result using the Morita equivalence of Corollary 3.21 between $T^{\nu }_\chi $ and this quotient of $\mathbb{T}^\chi _\nu $ .

4.2 Translation functors

Now, let us consider how this perspective interacts with translation functors. This requires studying the bimodule ${\mathbb{C}}^n\otimes U$ .

Fix $\chi \in \operatorname {MaxSpec}(Z_n)$ , and assume that $\chi ^{+i}$ exists. Consider

$$\begin{align*}\lambda\in \operatorname{MaxSpec}_{\mathbb{Z},\chi}(\Gamma)\qquad \lambda'\in \operatorname{MaxSpec}_{\mathbb{Z},\chi^{+i}}(\Gamma).\end{align*}$$

Theorem 4.13 We have an isomorphism compatible with bimodule structure:

$$\begin{align*}e'(\lambda') \mathbb{E}_i e'(\lambda)=\varprojlim U'/(\mathsf{m}_{\lambda'}^NU'+U'\mathsf{m}_{\lambda}^N).\end{align*}$$

Proof Note that $e_n$ commutes with $Z_k$ for $k<n$ , embedded in $U(\mathfrak {gl}_{n+1})$ via the standard inclusion. Thus, $e_n$ only has non-zero image if $\lambda _{i,k}=\lambda ^{\prime }_{i,k}$ for all $i<n$ . We therefore can describe the desired map by specifying the image of $e_n$ in this case.

As in [Reference Kamnitzer, Tingley, Webster, Weekes and YacobiKTW⁺19, Reference WebsterWebb], we can read this off by carefully studying the polynomial representation of $U(\mathfrak {gl}_{n+1})$ , induced by its realization as an orthogonal Gelfand–Zetlin algebra.Footnote ¹ This was originally given in [Reference MazorchukMaz99], though it might be easier to follow the notation in [Reference WebsterWeba, Section 5.1]. Let $\Gamma _{(n+1)}$ be the Gelfand–Tsetlin subalgebra of $U(\mathfrak {gl}_{n+1})$ , which we identify with the space of symmetric polynomials in the alphabets $\{x_{i,1},\dots , x_{i,i}\}$ for $i=1,\dots , i+1$ .

Let $\varphi _{i,j}$ be the translation on all polynomials in these alphabets satisfying

$$\begin{align*}\varphi_{i,j}(x_{k,\ell})=(x_{k,\ell}+\delta_{ik}\delta_{j\ell}) \varphi_{i,j}.\end{align*}$$

We then have a representation of $\mathfrak {gl}_{n+1}$ , where $E_i$ and $F_i$ act by the operators

$$\begin{align*}X^\pm_i=\mp\sum_{j=1}^{i}\frac{\displaystyle\prod_{k=1}^{v_{i\pm 1}} (x_{i,j}-x_{i\pm 1,k})}{\displaystyle\prod_{k\neq j} (x_{i,j}-x_{i,k})}\varphi_{i,j}^{\pm 1}.\end{align*}$$

Thus, $U'$ embeds into the bimodule of operators $\Gamma _{(n+1)}$ when this space is made into a module over $U(\mathfrak {gl}_{n})$ as the subspace generated by the operator

$$\begin{align*}X^+_n=-\sum_{j=1}^{n}\frac{\displaystyle\prod_{k=1}^{{n+ 1}} (x_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{k\neq j} (x_{n,j}-x_{n,k})}\varphi_{n,j}.\end{align*}$$

Consider $P'=\varprojlim U'/U'\mathsf {m}_{\lambda }^N$ . This is a left module over U, and clearly pro-Gelfand–Tsetlin, since for any element of $u\in U'$ , the sub-bimodule $\Gamma u\Gamma $ lies in the left $\Gamma $ module generated by finitely many translations (this is a version of the Harish-Chandra property discussed in [Reference Drozd, Futorny and OvsienkoDFO94]). This means that $\Gamma u\Gamma \otimes _{\Gamma }\Gamma /\mathsf {m}_{\lambda }^N$ is finite-dimensional so $U'/U'\mathsf {m}_{\lambda }^N$ is Gelfand–Tsetlin. Thus, $P'$ decomposes into pieces $P'=\oplus _{i\in \mathbb{Z}} P_i'$ on which $Z_n$ acts by $\chi ^{+i}$ for i ranging over the values of $\chi _g$ ; note that $P_i'=E_i(\varprojlim U/U\mathsf {m}_{\lambda }^N)$ .

Consider the image of $X^+_n$ in $P^{\prime }_i$ . If we let $I_i=\{j\mid \lambda _{n,j}=i\}$ , then we have that this image is given by

$$\begin{align*}\sum_{j\in I_{i+1}} \frac{\displaystyle\prod_{k=1}^{{n+ 1}} (x_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{k\in [1,n]\setminus \{j\}} (x_{n,j}-x_{n,k})} \varphi_{n,j}.\end{align*}$$

Note that if $k\notin I_{i+1}$ , then the corresponding factor in the denominator is invertible in the local ring $\varprojlim \Gamma _{n}/\Gamma _{n}\mathsf {m}_{\lambda '}^N$ , using the geometric series

$$\begin{align*}\frac{1}{x_{n,j}-x_{n,k}}=\frac{1}{\lambda_{n,j}'-\lambda^{\prime}_{n,k}}+\frac{x_{n,k}-\lambda^{\prime}_{n,k}-x_{n,j}+\lambda^{\prime}_{n,j}}{(\lambda_{n,j}'-\lambda^{\prime}_{n,k})^2}+\frac{(x_{n,k}-\lambda^{\prime}_{n,k}-x_{n,j}+\lambda^{\prime}_{n,j})^2}{(\lambda_{n,j}'-\lambda^{\prime}_{n,k})^3}+\cdots\end{align*}$$

Thus, we can write the image of $X^+_n$ as a sum of operators of the form

$$\begin{align*}X_{i}^{(p,s)}=-\sum_{j\in I_{i+1}} \frac{\displaystyle\prod_{k=1}^{{n+ 1}} (x_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{p\in I_{i+1}\setminus \{j\}} (x_{n,j}-x_{n,p})} (x_{n,j}-i-1)^sp(x_{n,1},\dots, x_{n,n})\varphi_{n,j},\end{align*}$$

where p is a polynomial symmetric in the alphabets $I_i$ .

We can identify p with a polynomial in the red dots at the bottom of the diagram, with the shifted variables $\{x_{n,j}-i\}_{j\in I_i}$ sent to the red dots on the corresponding strand with $i=\chi _j$ . We then send $X_{i}^{(p)}$ to the element of $\mathbb{E}_i$ , where we place the polynomial p applied to the red dots at the bottom of the diagram and “sandwich” s red dots on the middle strand of the ladder, with all black strands are straight vertical at positions determined by $\lambda $ . That is,

(4.2)

We claim that this defines a map

$$\begin{align*}\varprojlim U'/(\mathsf{m}_{\lambda'}^NU'+U'\mathsf{m}_{\lambda}^N)\to e'(\lambda') \mathbb{E}_i e'(\lambda).\end{align*}$$

First, let us match the completion with the completed polynomial ring $\varprojlim \Gamma /\Gamma \mathsf {m}_\lambda ^N$ by matching $Y_r$ with $x_{p,j}-\lambda _{p,j}$ if the rth strand from the left is the jth (from the left) with label p; let us simplify notation by writing $Y_{p,j}:=Y_r$ in this case. Under this isomorphism, $X_i^{(p,s)}$ transforms to

$$\begin{align*}\mathscr{X}_{i}^{(p,s)}=-\sum_{j\in I_{i+1}} \frac{\displaystyle\prod_{k=1}^{{n+ 1}} (Y_{n,j}+\lambda_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{p\in I_{i+1}\setminus \{j\}} (Y_{n,j}-Y_{n,p})} (Y_{n,j})^sp(Y_{n,1}+\lambda_{n,1},\dots, Y_{n,n}+\lambda_{n,1})\varphi_{n,j}.\end{align*}$$

This is exactly $C=\prod _{k=1}^{{n+ 1}} (Y_{n,j}+\lambda _{n,j}-x_{n+ 1,k})$ times the action in the polynomial representation of $\mathbb{E}_i$ given the diagram on the left-hand side of (4.2) in Lemma 2.13. Multiplication by C commutes with the left and right actions on this bimodule, so this shows that the representations of bimodules match.

This shows that (4.2) defines a homomorphism, which must be injective by the faithfulness of both representations. Obviously, every bare ladder is in the image of this map, so it must be surjective as well.

Together with Theorem 3.16, this establishes Theorem 2, completing the main results of the paper.