key: cord-0114673-lmoawnyq
authors: Mathas, Andrew; Tubbenhauer, Daniel
title: Cellularity for weighted KLRW algebras of types $B$, $A^{(2)}$, $D^{(2)}$
date: 2022-01-06
journal: nan
DOI: nan
sha: 7ca0b775553b90006095fed4800317cc53dfbd62
doc_id: 114673
cord_uid: lmoawnyq

This paper constructs homogeneous affine sandwich cellular bases of weighted KLRW algebras in types $B_{mathbb{Z}_{geq 0}}$, $A^{(2)}_{2cdot e}$, $D^{(2)}_{e+1}$. Our construction immediately gives homogeneous sandwich cellular bases for the finite dimensional quotients of these algebras. Since weighted KLRW algebras generalize KLR algebras, we also obtain bases and cellularity results for the (finite dimensional) KLR algebras.

KLR algebras, or quiver Hecke algebras, are certain graded infinite dimensional algebras attached to a quiver. These algebras arose in categorification and categorical representation theory, see e.g. [KL09] , [KL11] , [Rou08] or [Rou12] . They admit finite dimensional quotients, called cyclotomic KLR algebras. Both, KLR algebras and their cyclotomic quotients, play crucial roles in modern representation theory and have attracted a lot of attention since their introduction.

A crucial problem is to determine whether these algebras are (graded) cellular in the sense of [GL96] , or some variation of cellularity such as affine cellular [KX12] . It turns out that they tend to be graded affine cellular in the infinite dimensional case, see e.g. [KL15] , and graded cellular in the finite dimensional case, see e.g. [HM10] . Even better, in both cases one can write down explicit homogeneous (affine) cellular bases. However, to the best of our knowledge and modulo weighted KLRW algebras, there is no clear relationship between the bases for the infinite dimensional algebras and the bases for their finite dimensional quotients.

Weighted KLRW algebras are generalizations of KLR algebras that were introduced by Webster. See, for example, [Web19] , [Web17] , [Bow17] or [MT21] . Similar to KLR algebras, these algebra also admit finite dimensional quotients. With appropriate choices, the KLR algebras are idempotent subalgebras of the weighted KLRW algebras. This, for example, means that there is no immediate relationship between the simple modules of weighted KLRW and KLR algebras, and care must be taken when comparing these algebras.

In the AC types, that is, types A Z , A

(1)

e , we showed [MT21] that one of the key properties of these weighted KLRW algebras is that in the they have homogeneous affine cellular bases constructed in the style of low dimensional topology, and these bases automatically descend to the finite dimensional quotients. As a result, we obtained homogeneous (affine) cellular bases for the corresponding KLR algebras and their finite dimensional quotients. Even better, all of these bases work over any commutative integral domain R (for example R = Z).

It is natural to ask whether these constructions can be extended to other types. In this paper we answer this question affirmatively for types B Z ≥0 , A (2) 2·e , D (2) e+1 : the BAD types. That is, we construct homogeneous affine sandwich cellular bases for weighted KLRW algebras that descend to the finite dimensional quotients. Again, we obtain bases for the corresponding KLR algebras. The sandwich part of our bases corresponding to the finite dimensional quotients is given by copies of the dual numbers R[X]/(X 2 ), which explains the powers of 2 appearing in the dimension formulas of the finite dimensional KLR algebras. For completeness, we note that the weighted KLRW algebras and their finite dimensional quotients of BAD types are actually (affine) cellular; see Remark 2A.7 for a more precise statement. As a consequence we obtain the usual results such as a construction of the graded simple modules and that the decomposition matrix is unitriangular. This paper is a sequel to [MT21] , so we assume some familiarity to the definitions and results of that paper. In sequels to this paper we hope to discuss weighted KLRW algebras of other types and their simple modules.

Remark 1.1. The colors used in this paper are not essential and all strings are distinguishable by their thickness and if they are solid or dashed.

Remark 2A.1. Strictly speaking [KX12] sandwich a polynomial ring or a quotient of a polynomial ring. In particular, as we will see, for weighted KLRW algebras of BAD types we only need affine cellularity. However, we want to reserve the notion affine for the case of honest polynomial rings, so we tend to say sandwich cellular instead.

The following is a slight reformulation of [MT21, Definition 6B.5 ], see also [TV21, Section 2] .

Definition 2A.2. Let R be a commutative ring with a unit. Let A be a locally unital graded R-algebra. A graded sandwich cell datum for A is a quintuple (P, T, S, C, deg), where:

• P = (P, ≤) is a poset (the middle set),

• T = λ∈P T (λ) is a collection of finite sets (the bottom/top sets), • S = λ∈P S λ is a direct sum of graded algebras S λ (the sandwiched algebras) such that B λ is a homogeneous basis of S λ (we write deg for the degree function on S λ ),

ST is an injective map (the basis), • deg : λ∈P T (λ) −→ Z is a function (the degree), such that: (AC 1 ) For b ∈ B λ , S, T ∈ T (λ) and b ∈ B λ , λ ∈ P the element C b ST is homogeneous of degree deg(S) + deg(b) + deg(T ).

(AC 2 ) The set {C b ST | λ ∈ P, S, T ∈ T (λ), b ∈ B λ } is a basis of A. (AC 3 ) For all x ∈ A there exist scalars r SU ∈ R that do not depend on T or on b, such that

where A >λ is the R-submodule of A spanned by {C c U V | µ ∈ P, µ > λ, U, V ∈ T (µ), ∈ B µ }. (AC 4 ) Let A(λ) = A ≥λ /A >λ , where A ≥λ is the R-submodule of A spanned by {C c U V | µ ∈ P, µ ≥ λ, U, V ∈ T (µ), c ∈ B µ }. Then A(λ) is isomorphic to ∆(λ) ⊗ S λ ∇(λ) for free graded right and left S λ -modules ∆(λ) and ∇(λ), respectively. The algebra A is a graded sandwich cellular algebra if it has a graded sandwich cell datum.

Assume that there is an antiinvolution ( − ) : A → A such that:

. In this case we call (P, T, V, C, deg, ( − ) ) involutive.

Note however that, in general, we do not have do we have such a factorization. So this pictures has to be interpreted with care.

We use the following terminology for special cases:

(a) A graded affine sandwich cell datum for A is a graded sandwich cell datum such that, for all λ ∈ P and for some n(λ) ∈ Z ≥0 , we have S λ ∼ = R[X 1 , ..., X n(λ) ].

(b) A graded cell datum for A is a graded sandwich cell datum such that S λ ∼ = S for all λ ∈ P.

The image of C in A is an homogeneous sandwich cellular basis for A. Similarly, we refer to affine sandwich cellular bases etc.

The following Clifford-Munn-Ponizovskiȋ theorem parametrizes the simple modules of these types of algebras. To state this result we need some notion.

For each λ ∈ P there exists a cell module ∆(λ) and a cellular pairing φ λ on ∆(λ), see [TV21, Section 2B]. The pairing φ λ is a symmetric bilinear form. Let P =0 ⊂ P be the subset of those λ for which φ λ is nonzero. The illustration to keep in mind is (stolen from [TV21, Section 2B]), the details can be copied from [KX12, Section 2.2]:

Theorem 2A.4. Let R be a field, and let A be a graded sandwich cellular algebra.

(a) All (graded) simple A-modules are uniquely associated to a λ ∈ P =0 , called their apex.

(b) Assume that S λ is unital Artinian or commutative. For a fixed apex λ ∈ P =0 there exists a 1:1correspondence

(c) Assume that S λ is unital graded Artinian or commutative. For a fixed apex λ ∈ P =0 there exists a 1:1-correspondence

Even for non-commutative S λ the assumption of being unital (graded) Artinian can be avoided under certain conditions, see [TV21, Section 2C] for details.

Proof. The proof is not much different from the general theory as in [GL96] , [KX12] , [AST18] , [ET21] , [GW15] or [TV21] . In particular the above is just a (graded) reformulation of [GW15, Theorem 3] and [TV21, Section 2C]. Details are omitted.

Remark 2A.5. The bijection in Theorem 2A.4 can be made explicit: the simple A-modules for apex λ ∈ P =0 can be constructed as the simple heads of the cell modules.

Remark 2A.6. The formulation of Theorem 2A.4 is strongly inspired by Green's theory of cells (Green's relations) [Gre51] , and the Clifford-Munn-Ponizovskiȋ theorem of semigroup theory, see e.g. [GMS09] for a modern formulation.

Remark 2A.7. If A is involutive and all sandwiched algebras have an (affine sandwich) cellular datum compatible with the sandwich structure on A, then A also has a cell datum that can be constructed by refining the sandwich cell datum on A in an appropriate sense. However, refining the datum can make the natural sandwich datum cumbersome with very little gain. In our case all algebras have (copies of) polynomial rings and dual numbers sandwiched, and both are affine sandwich cellular algebras, and all of our algebras have a refined (affine sandwich) cell datum. Since these are easy algebras, refining the sandwich cell datum for the weighted KLRW algebras as in this paper appears to be unnecessary.

Notation 2A.8. From now on all of our algebras are assumed to be involutive, and we will omit the use of this word. We have separated the involutive condition (AC 5 ) in Definition 2A.2 from the other axioms because being involutive is not necessary for Theorem 2A.4 to hold.

We recall the basic constructions and statements regarding weighted KLRW algebras. As we assume some familiarity with [MT21] , we will be brief.

Notation 3.1. The following conventions used throughout the paper.

(a) We work over a commutative integral domain R, the ground ring.

(b) Graded algebra or module will always mean a Z-graded algebra or module.

(c) We use the same diagrammatic conventions as in [MT21] :

In particular, left actions and left modules are given by acting from the top. Modules will always be left modules.

3A. Weighted KLRW algebras in a nutshell. The weighted KLRW algebras are diagram algebras consisting of weighted KLRW diagrams (diagrams for short). These diagrams have three types of strings: solid, ghost and red strings. All of these strings are labeled, and we will illustrate these as solid :

where the label under the solid and red strings and over the ghost strings. The labels on the strings are also called residues.

The weighted KLRW algebras depend on the following input: (b) Let u and v be indeterminates over R. For i, j ∈ I define Q-polynomials

These are Q-polynomials as in [Rou08, Section 3.2.3], [Rou12] or [Web19, Section 2.1]. Recall also that Q i,j,i (u, v, w) =

(c) Non-negative integers n, ∈ Z ≥0 and a tuple ρ = (ρ 1 , ..., ρ ) ∈ I . These are the number of solid strings, the number of red strings (called the level ) and the red labels, respectively.

(d) Various types of data determining the positions of the strings. That is, a solid positioning x = (x 1 , ..., x n ) ∈ R n , a ghost shift σ = (σ ∈ R =0 ) ∈E which is a weighting of Γ, a charge κ = (κ 1 , ..., κ ) ∈ R such that κ 1 < ... < κ . The x, σ and κ are used to determine the boundary points of solid, ghost and red strings in the diagrams, and are such that there are no overlapping strings.

(e) A set X of loadings, i.e. endpoints for the various strings. We stress that the rank of the weighted KLRW algebras strongly depends on the choice of X. Note that these two ghosts strings are different strings that behave differently because the relations above depend on the edges and not on the residues. In the example above the two ghost 1-strings play different roles. For example, one of them has nontrivial Reidemeister II relations with the 0-strings and the other has nontrivial Reidemeister II relations with the 2-strings.

Remark 3A.6. Note that [MT21] mostly works with W ρ β (X) for β fixing the labels for the solid strings. The difference is not important since W ρ n (X) = β∈Q + n W ρ β (X) with Q + n corresponding to the set of all possible labels of the n solid strings.

Finally, the cyclotomic weighted KLRW algebra R ρ n (X) is the finite dimensional quotient of W ρ n (X) by the two-sided ideal generated by all diagrams that factor through an unsteady diagram. A diagrams is unsteady if it contains a solid string that can be pulled arbitrarily far to the right when the red strings are bounded by X. For example, we have Unsteady : Note that the ghost string mimics its parent solid string, so the solid i-string in the left diagram does indeed pull freely to the right.

3B. Duality and partners. We use the usual diagrammatic antiinvolution ( − ) given by (the R-linear extension of) flipping diagrams on their heads. This antiinvolution is the one we use for the homogeneous (affine) sandwich cellular basis. An illustration of the diagrammatic antiinvolution is:

There is a different kind of duality on diagrams, which exists because the relations (3A.2)-(3A.4) come in mirrored pairs. This could mean that they are horizontal mirrors or are obtained by changing the roles of solid and ghost strings, both potentially up to scalars. Most of the relations we will use have this type of duality, and we will only illustrate one of them and call the others partner relations. (Note that the partner relations can have different scalars, however this will not play a role for us.)

3C. Some diagrams that we need. We need certain types of diagrams:

(a) Idempotent diagrams are diagrams with no dots and no crossings, and fixed x-coordinates for each strings. Example 3C.1. Examples of these types of diagrams are: idempotent:

.

These illustrate (dotted) idempotents, (dotted) straight line and permutation diagrams. 3

Notation 3C.2. Let 1 x,i be the idempotent diagram with bottom boundary given by (x, i), for x ∈ X and i ∈ I n . We use the notation p1 x,i , where p ∈ R[y 1 , ..., y n ] is a polynomial in the indeterminates y 1 , ..., y n to put dots on 1 x,i so that y k corresponds to a dot on the kth solid string. Additionally, let S n = s 1 , ..., s n−1 be the symmetric group on {1, ..., n} with s i = (i, i + 1). For all w ∈ S n fix a reduced expression and let D(w) be the associated permutation diagram. As is usual in the KLR world, the diagram D(w) is only well-defined up to some care that needs to be taken, see [MT21, Definition 3B.1] for details.

Finally, as in any idempotented algebra, we only need to (and will) indicate the idempotent diagrams once in any expression. Example 3C.3. In pictures:

Here we have not drawn red strings. 3

. Given a straight line diagram S and some 0 < ε 1, then there exists an idempotent diagram L(S), called the left justification of S, such that S factors through L(S), and L(S) has its strings as far to the left as possible while its coordinates are within the interval defined by X and strings are at least ε apart. (The ε > 0 is only needed to make L(S) well-defined, and we omit it in the following.) See [MT21, Section 6D] for a detailed account.

Example 3C.5. Here is an example of a left justification:

In general, the left justification is obtained by considering a collar neighborhood of a horizontal cut of S.

In this neighborhood one inductively pulls the strings to the left. 3

In a dotted straight line diagram two strings are close if you can pull them arbitrarily close to one another using only (bilocal) isotopies. We use close and to the left/right in the evident way.

Example 3C.6. Let us repeat [MT21, Example 6D.10]:

The solid i-string is close to the solid j-string in the left but not in the right diagram. 3

One of the other crucial points about weighted KLRW algebras is that they generalize KLR algebras, see [MT21, Section 3F]. The construction given therein uses a KLRW positioning, which is a certain x ∈ X, and then the KLR algebra is graded isomorphic to 1 x W ρ β (X)1 x where 1 x = i∈I 1 x,i . We return to this point in Section 5G below. 3D. A basis and a faithful module. The following standard basis proposition works only in the infinite dimensional case. One of the main features of our approach is that we can use it to prove cellularity of the finite dimensional quotients as well.

Proposition 3D.1. The algebra W ρ n (X) is free as an R-module with homogeneous basis B β = {D(w)y a1 1 ...y an n 1 x,i | a 1 , ..., a n ∈ Z ≥0 , w ∈ S n , x, y ∈ X, i ∈ I n }. Proof. See [MT21, Proposition 3B.12] for details.

Remark 3D.2. The proof of Proposition 3D.1 in [MT21] uses a faithful action on

We will not recall the action here as it is quite standard in the field, see [MT21, Section 3C] for details.

As we now recall, our strategy to construct homogeneous (affine) sandwich cellular bases for the weighted KLRW algebras is the same as in [MT21, Remark 6.1].

Remark 4A.1. A crucial ingredient is the idea of using a certain form of minimality :

(a) First, construct an idempotent diagram 1 λ by placing strings inductively as far to the right as possible, called placing strings to the right. Here we use that (3A.3) only allows strings to be pulled to the right in certain situations, such as when they carry a dot. In this way we think of previously placed strings as keeping a new one in check. This ensures that 1 λ is minimal with respect to placing the strings to the right. In fact, this strategy is a greedy algorithm, as it is designed to be locally minimal but it produces a globally minimal diagram.

(b) By (3A.3) again, the diagram 1 λ stays minimal when dots are put on certain strands, but putting dots on other strings allows the strings or the dots to move further to the right. So without violating minimality we can place more dots on some strings in 1 λ to obtain minimal diagrams of the form y a y f y λ 1 λ , which form the middle of the homogeneous (affine) sandwich cellular basis.

(c) The (sandwich) cellular basis is then obtained by a standard construction for diagram algebras, which in our case means putting semistandard permutation diagrams above and below y a y f y λ 1 λ .

(d) The basis itself is minimal, by construction, and it is not hard to prove that it is indeed a homogeneous (affine) sandwich cellular basis. For example, putting additional dots on the basis elements allows one pull strings and jump dots to the right, making the result bigger. This gives an inductive way of proving results.

This strategy works perfectly in types A Z and A

(1) e , and requires some small adjustments in other types.

The following lemmas are the crucial diagrammatic relations that we need to pull strings and jump dots to the right. Here we are pulling the leftmost string to the right or jumping the leftmost jump dot to the right. We highlight the strings where the action happens by coloring them.

Lemma 4A.2. For any quiver and any choice of Q-polynomials we have the following, plus partner relations:

For i → j edges and the choice of Q-polynomials in (3A.1) we have the following, plus partner relations:

The right relation and its partner also hold for i ⇐ j.

For i ⇒ j edges and the choice of Q-polynomials in (3A.1) we have the following, plus partner relations: Proof. We first use (4A.3) to create a dot on the now middle ghost i-string. If i ⇐ j are not connected, then either a plain Reidemeister II relation applies, or either of (4A.4) and (4A.5) apply. In any case, we can pull the leftmost solid i-string to the right.

The next example should be compared with Remark 5C.7 below.

Example 4A.7. The following close configurations and their partners are stuck for i ⇒ j respectively for i ⇐ j, so that Lemma 4A.2 and Lemma 4A.6 do not apply:

These configurations will appear whenever i correspond to a leaf of Γ. 3

We now explain the main constructions of this paper. In Remark 6A.10 we summarize the parts of the arguments that are general and those that depend on the underlying quiver.

5A. Some notation. We fix some conventions.

Notation 5A.1. We will use affine red strings in diagrams, illustrated by:

Note that affine red strings are not part of the diagrams and they are drawn only as a visual aid.

Notation 5A.2. Unless we are in specific example, we fix arbitrary n, ∈ Z ≥0 , e ∈ Z ≥2 , κ ∈ Z , with κ 1 < ... < κ , and ρ ∈ I for the duration. We let = + n(e + 1) be the affine level. More generally, we will use the underline notation to indicate definitions that only play a role for the affine case.

Notation 5A.3. Our constructions given in this section work for the quivers below.

(a) We use Kac's notation [Kac90] for Dynkin quivers (but we mirrored the quivers left-to-right). The main quivers of study in this paper are:

2·e :

e+1 :

where e ∈ Z ≥2 . Here we orient the simply laced edges i → (i + 1). We will omit B Z ≥0 from the discussion: this type can be viewed as D (2) e+1 for e → ∞, but doing so needs some (harmless) adjustments of the exposition since e.g. the affine level would be infinite.

4n be a small shift. We let the ghost shifts for edges in BAD types be 1 with the exception of the edge 0 ⇐ 1 in type D (2) e+1 where the ghost shift is 1 − ε 2 . Definition 5A.5. A sink is a vertex of a Dynkin quiver Γ that is a (graph-theoretical) sink. A multisink is a sink with only adjacent multi-laced edges.

Note that, if i is a sink, then the solid i-strings do not have ghosts.

Example 5A.6. The vertex e is a multisink for BAD types, and the vertex 0 is additionally a multisink for type D (2) e+1 . With contrast, in type C

(1) e no vertex is a multisink, but some vertices are sinks.

3

In type D

(2) e+1 the solid 1-string has two ghosts that are very close to one another by our choice of ghost shifts. We display these two 1-ghosts as doubled lines:

We stress that these are two different ghost 1-strings, cf. Remark 3A.5.

Notation 5A.7. We will draw diagrams that are supposed to make sense in any type, but the reader may need to remove or double some ghost strings to obtain the required diagram for a particular type.

Definition 5A.8. Define the affine charge κ = (κ 1 , ..., κ ) ∈ Z and the affine red labels ρ = (ρ 1 , ..., ρ ) ∈ I by

We call κ m and ρ m the position and residue of a red string for m ≤ , and the position and residue of an affine red string for m > .

Note that the coordinates of the (affine) red strings κ are always integers.

Example 5A.9. Take n = 3, e = 2 and = 1, so = 1 + 3 · (2 + 1) = 10. If κ = (2) and ρ = (1), then κ = (2, 8, 14, 20, 26, 32, 38, 44, 50, 56) and ρ = (1, 0, 0, 0, 1, 1, 1, 2, 2, 2). All entries, of κ and of ρ, except the first are affine. 3

5B. Partition combinatorics. Before coming to our main definitions, we introduce the tableaux combinatorics that arise in BAD types. For the standard tableaux combinatorics that appears in the context of KLR algebras we refer the reader to [HM10, Section 3.3].

Remark 5B.1. The partition combinatorics that we use is motivated by e.g. [AP14] and [AP16] . The associated weighted KLRW diagram combinatorics is a slight modification of the combinatorics of type C

(1) e as in [MT21, Section 7] . The reader should be careful because, as we will see, the partition combinatorics depends on ρ. In particular, the combinatorics from [AP14] and [AP16] only applies for cyclotomic KLR algebras for the fundamental weights associated to multisink vertices.

Identify the vertices of I with Z/(e+1)Z. We use (usual) partitions of n, following the same conventions as in [MT21, Section 6A]. We also use shifted partitions of n, that is, partitions λ with strictly decreasing components λ = (λ 1 > ... > λ k ). Let |λ| be the size of a partition or shifted partition.

Definition 5B.2. The set of ρ-partitions is

We identify a ρ-partition λ with its shifted ρ-Young diagram, which is the set of nodes

Notation 5B.3. We use the (shifted) English convention to illustrate the associated (shifted) ρ-Young diagrams. That is, we illustrate these partitions by drawing them as boxes in the plane, with rows ordered from top to bottom, and columns left to right, and where the rth row is shifted r positions to the right for That is, if ρ i corresponds to a multisink vertex (i.e. the vertices e in BAD types and additional the vertex 0 in type D (2) e+1 ), then we consider shifted partitions in the ith entry of λ, and usual partitions otherwise. We make similar definitions as above for ρ-partitions. The nodes for m > are called affine nodes. Identify P ρ ,n consisting of the ρ that do not contain any affine nodes.

Definition 5B.4. Let a = 1 for type A

2·e and a = 2 for type D

e+1 . Given an integer k ∈ Z we need the remainder of division by 2e + a (viewed as an element in {0, ..., 2e + a} ⊂ Z ≥0 ) which we denote by Mod(k, 2e + a). Define the residue function r : Z −→ I by

2e + 1 − Mod(k, 2e + a) + (e + 1)Z if e < Mod(k, 2e + a) < 2e + a.

The (ρ-)residue of the node (m, r, c) is res ρ (m, r, c) = r(c − r) + ρ m + (e + 1)Z.

In illustrations we will often fill nodes with their residues.

Example 5B.5. For λ = (10, 9, 8, 7, 6, 5, 4) and e = 3, starting with 0, i.e. ρ = (3), we get: Note that colored/shaded residues, for multisinks, are doubled. In words, the residues increase along rows and columns until they hit e, this value is doubled, and then the residues bounces back until they hit 0, this value is doubled for D 

2·3 has usual and type D

3+1 shifted partition combinatorics. We emphasize that, in general, there are more usual partitions than shifted partitions so there are more of these diagrams when ρ i is not a multisink 3

The coloring/shading from Example 5B.5 also distinguishes between usual and shifted partition combinatorics: we are in the case of the shifted combinatorics if and only if the first node is colored/shaded. Example 5B.6. We continue with Example 5A.9 and fix type A We get P ρ ,

The set P ρ,all ,n is much bigger, and we will not list it here. Note however that the affine components of P ρ,all ,n have either usual or shifted partition combinatorics. Precisely, since ρ = (ρ, 0, 0, 0, 1, 1, 1, 2, 2, 2) we get that the first six affine components of P ρ,all ,n have usual and the last three have shifted partition combinatorics.

3

Remark 5B.7. In examples and also proofs below we will mostly focus on the shifted partition combinatorics. The situation for the usual partition combinatorics is then a slight adjustment of that used for type C 5C. The (dotted) idempotent. We now introduce some crucial definitions. For any r + (e + 1)Z ∈ I with r ∈ {0, ..., e} we sometimes abuse notation and identify (r + (e + 1)Z) for the associated real number in r ∈ R. Definition 5C.2. Let λ ∈ P ρ,all ,n . The coordinate of (m, r, c) ∈ λ with p λ (m, r, c) = k, is

The coordinates x κ (λ) of λ is the ordered tuple of the coordinates of its nodes listed in row reading order.

Definition 5C.2 looks more complicated than it actually is. The coordinate function simply places strings in order, following the strategy outlined in Remark 4A.1. (d) −kε ensures that nodes move a little bit to the left as we read along rows.

Note that in type D (2) e+1 the solid 0-strings and the solid 2-strings have roughly the same coordinates. This is important because the solid 0-strings do not have associated ghosts, but the solid 1-strings have two associated ghosts.

Example 5C.4. We continue with Example 5B.6. Fix from now on κ = (0). We have the following coordinates of ν 1 and ν 2 :

We have illustrated the coordinates for the nodes in the shifted Young diagrams, and also what happens in the limit ε → 0. Note hat we have omitted the additional shift of m = 1 10 . 3

We write x κ (f ) for the maximum of x κ on P ρ ,n , and call coordinates with x κ (m, r, c) > x κ (f ) affine. Lemma 5C.5. The nodes with affine coordinates are precisely the affine nodes.

Proof. Easy and omitted.

We now define the (dotted) idempotent diagrams 1 λ and 1 y λ associated to λ ∈ P ρ,all ,n .

Definition 5C.6. For λ ∈ P ρ,all ,n let 1 λ be the idempotent diagram given by (a) placing red strings with labels given by ρ at coordinates given by κ, and, (b) n solid strings with labels res ρ (m, r, c) at coordinates x κ (m, r, c), for (m, r, c) ∈ λ.

As mentioned already, in diagrams we will also draw affine red strings at positions κ m for < m ≤ .

Remark 5C.7. In AC types the crucial illustrations that we used to show that our bases span are [MT21, (6A.10), (6A.11), (7A.8), (7A.9) and (7A.10)]. These diagrams identify local configurations of nodes in Young diagrams with local configurations of strings in diagrams. Their analogs, illustrating the residues in the nodes, are as follows.

Assuming that the middle node is not 0 or e and we do not have i − 2 = 0, i = 0 or i + 2 = 0 in type D (2) e+1 , we have:

The special cases in Definition 5C.1 for type D That the two cases in (5C.8) and (5C.9) look different is an artifact of our conventions for ghost strings. However, this can not be avoided (meaning that one always gets special behavior) in type D (2) e+1 as there is no way to orient the quiver from left to right, or right to left.

When the middle residue is 0 or e we have: 3) annihilates the diagram. In Definition 5C.12 below, we will place a dot on the strands to avoid this.

Finally (note that |i − j| ≤ 1 in these pictures):

the solid/ghost k-string is close to a ghost/solid (k ± 1)-string otherwise,

where the j-string is only illustrated in the top diagram, and in the last case the i and k-strings do not need to be close (neither the solids nor the ghosts). Note that in the shifted Young diagram k = ρ k .

Definition 5C.12. Suppose λ ∈ P ρ,all ,n . Set p λ (m, r, c) = k and p λ (m, r , c ) = k + 1. If ρ m is not a multisink (so usual partitions), then a k = 1 if res ρ (m, r, c) = res ρ (m, r + 1, 1), 0 otherwise. (5C.13) If ρ m is a multisink (so shifted partitions), then a k = 1 if res ρ (m, r, c) = res ρ (m, r , c ) or (r > 1, c = 1), 0 otherwise. (5C.14)

The dotted idempotent associated to λ is 1 y λ = y λ 1 λ , where y λ = y a1 1 ...y an n ∈ R[y 1 , ..., y n ].

In other words, (5C.13) places a dot whenever the kth and the (k + 1)th string are close and have the same residue, and takes care of new rows as in (5C.11), e.g.:

In the case of usual partitions the dot placement is the same as in type C

(1) e from [MT21, Section 7], see also (5C.19) below. Note that 1 y λ has zero or one dot on each strand. For example, the top diagram in (5C.11) gets two dots, one on the middle and one on the rightmost string. Proof. One direction is immediate, so let us assume that λ = µ. If res ρ (λ) = res ρ (µ), then 1 y λ = 1 y µ follows using the faithful polynomial module from Remark 3D.2. Moreover, a different dot placement on the same 1 λ can also be distinguished by Remark 3D.2, so it remains to argue that λ = µ implies a different dot placement if res ρ (λ) = res ρ (µ). To see this assume that the kth node is the first node that is different for λ and µ. Without loss of generality, we can assume that the kth node of µ is in a new row when compared to the kth node of λ. There are three cases to check now, depending on the residue j of the kth string and the residue i of the (k − 1)th string. When i = j is a multisink the local diagrams for λ and µ are: = (a 1 , ..., a n 

For 1 ≤ k ≤ n define c k (λ) = 1 if the kth string is close as in (5D.1) and does not already have a dot in 1 y µ for some µ ∈ P ρ,all ,n with 1 y λ = 1 y µ , and otherwise set c k (λ) = 0. Define the set of finite dots to be

The set of sandwiched dots is S y (λ) = A y (λ) ∪ F y (λ). Note that a k = 0 can only happen for affine coordinates x κ (λ) k > x κ (f ).

Example 5D.3. We continue with Example 5C.17.

The condition x κ (λ) k ≤ x κ (f ) implies that A y (λ 1 ) = A y (λ 2 ) = A y (λ 3 ) = {(0, 0, 0)}. Moreover, all strings associated to the second and third component have affine coordinates so A y (µ 1 ) = {(0, 0, n) | n ∈ Z ≥0 } and A y (µ 2 ) = {(0, m, n) | m, n ∈ Z ≥0 }.

For the finite dots we have F y (λ 1 ) = F y (λ 3 ) = {(0, 0, 0), (0, 0, 1)} and F y (λ 2 ) = F y (µ 1 ) = F y (µ 2 ) = {(0, 0, 0)}. We also have F y (λ 1 ) = {(0, 0, 0), (0, 0, 1)} for type D (c) T (m, r, c) < T (m, r, c − 1) + 1 for all (m, r, c), (m, r, c − 1) ∈ λ.

Let SStdκ(λ, µ) be the set of semistandard λ-tableaux of type µ and set SStdκ(λ) = µ SStdκ(λ, µ).

Definition 5E.2. For T ∈ SStdκ(λ, µ) define the permutation w T ∈ S n by requiring that

This defines the permutation diagram D T = D(w T ) from x κ (µ) to x κ (λ), as in Notation 3C.2.

In other words, a semistandard λ-tableau T of type µ is a filling of the nodes of λ with the type A coordinates of µ together with an anchor condition and such that the fillings decrease along rows and columns with an offset of 1. The associated permutation has top points defined by x κ (λ), bottom points by x κ (µ) and permutes them according to the entries of T . 

.

The tableau S is the canonical λ-tableau where all nodes are filled with their coordinates. Its associated permutation diagram is the identity. The permutation diagram D T is build from the permutation illustrated above, which connects x A κ (µ) to x A κ (λ), using x κ (µ) at the bottom and x κ (λ) at the top. 3

5F. Basis diagrams. We consider the following set of endpoints X.

Definition 5F.1. Let P ,n be the set defined by the condition that λ ∈ P ρ ,n only if λ = µ ∪ α, where µ ∈ P ρ ,n−1 and α is an addable i-node of µ such that: whenever β is an addable i-node of µ with x κ (β) < x κ (α), then x κ (β) ≤ x κ (f ).

Finally, let X be the set of all coordinates x κ (λ) for all λ ∈ P ρ ,n .

Remark 5F.2. In general, only a handful of the elements of P ρ,all ,n belong to P ρ ,n as the rule in Definition 5F.1 allows only the addition of affine nodes as far to the left as possible.

We are ready for our main definition:

for a = (a 1 , ..., a n ) ∈ Z n ≥0 , f = (f 1 , ..., f n ) ∈ {0, 1} n , S ∈ SStdκ(λ, ν), T ∈ SStdκ(λ, µ).

We call S a,f λ = y a y f 1 y λ the sandwiched part, y a the affine part, and y f the finite part of D a,f ST . The following are the bases that we consider:

,n , S, T ∈ SStd κ (λ), a ∈ A y (λ), f ∈ F y (λ)}. (5F.6)

Remark 5F.8. Note that we use D a,f ST in (5F.4) below to distinguish it from the abstract definition. Of course, these elements are the C b ST in Definition 2A.2.

Remark 5F.9. The relevant picture for (5F.6) is: This is the same as in AC types, which one crucial difference: in AC types the finite part is trivial.

5G. Homogeneous (affine) sandwich cellular bases. The order is as in [MT21, Definition 7C.1]:

Definition 5G.1. Let λ, µ ∈ P ρ ,n . Then λ dominates µ, written λ µ, if there exists a bijection d : λ → µ such that x κ (α) ≥ x κ (d(α)) and the solid string in 1 λ at position x κ (α) has at least as many dots as the solid string in 1 µ at position x κ (d(α)), for all α ∈ λ. Write λ µ if λ µ and λ = µ.

We are now ready to define (involutive) bases for W ρ n (X) and R ρ n (X). Recall that we are working with BAD types. We also use the Q-polynomials as in (3A.1).

The cell datum C = (P ρ ,n , T, S, B W ρ n (X) , deg, ( − ) * ) that we use is:

• we take B W ρ n (X) from (5F.6), viewed as a map, as our basis, • the degree is S → deg D S , • the antiinvolution is the diagrammatic antiinvolution ( − ) . The proof of the following theorem is postponed to Section 6 below.

Theorem 5G.2. The datum C is a graded affine sandwich cell datum for W ρ n (X). In particular, (5F.6) is a homogeneous affine sandwich cellular basis for W ρ n (X).

with B R ρ n (X) , then Theorem 5G.2 and comparing the definitions directly implies: Corollary 5G.3. The datum C = (P ρ ,n , T, S c , B R ρ n (X) , deg, ( − ) * ) is a graded affine sandwich cell datum for R ρ n (X). In particular, (5F.7) is a homogeneous sandwich cellular basis for R ρ n (X). Define a semistandard tableaux S ∈ SStd κ (λ) to be standard if it is of type ω = (n|0|...|0). Let Std(λ) be the set of standard λ-tableaux. Let W ρ n = 1 ω W ρ n (X)1 ω and R ρ β = 1 ω R ρ n (X)1 ω be the associated KLR and cyclotomic KLR algebra, respectively. This terminology is justified at the end of Section 3C. We use E instead of D to refer to the basis elements of the idempotent truncations. We will not highlight the cell datum below.

Proposition 5G.4. The set B W ρ n = E a,f st λ ∈ P ρ ,n , S, T ∈ Std(λ), a ∈ A y (λ), f ∈ F y (λ) is a homogeneous affine sandwich cellular basis of W ρ n . Proof. Apply [MT21, Proposition 3F.1 and Example 6A.11].

As before we obtain:

is a homogeneous sandwich cellular basis of R ρ n . Let us now discuss the upshot of Theorem 5G.2 and Corollary 5G.3 for simple modules. To this end, let a(λ) and f (λ) be the number of possible non-zero positions of A y (λ) and F y (λ), respectively.

. Proof. The first claim, regarding the X i , follows by using Proposition 3D.1. For the second claim, regarding the Y i , we inductively pull strings and jump dots to the right. That is, if one of the strings corresponding to possible non-zero positions of F y (λ) carries two dots we can use (4A.5) and the claim follows inductively.

Proposition 5G.7. The algebra R ρ n (X) is free of rank

Proof. Directly from Lemma 5G.6 an the respective corollaries above.

By the above, it is also easy to write down the graded dimensions of these algebras. (1) e .) We will ignore the affine part in this example. Fix = 1, κ = (0) and ρ = (0) and take β = (0, 1, 2) ∈ I 3 . We are first looking for all 1-partitions of 3 that have β as their residue sequence, that is, λ ∈ P ρ ,4 whose nodes have residues β in row reading order. We get the following 1-partitions:

2·2 , D 2 . Thus, we get by using Theorem 5G.2 that

Here we have listed the graded dimension (using the usual q-notation indicating the degree) of the idempotent truncation of R ρ β (X) determined by β. We also listed the relevant numbers for β = (2, 1, 0) where ρ = (2). These numbers match [HS21, Theorem 1.1], which is expected as this case is the cyclotomic KLR algebra of the respective types.

3

Proposition 5G.10. Suppose that R is a field, and let (P ρ ,n ) =0 or (P ρ ,n ) =0 denote the sets of apexes. (a) For a fixed apex λ ∈ (P ρ ,n ) =0 there exist a 1:1-correspondence between simple W ρ n (X)-modules with apex λ and R a(λ) . Moreover, up to isomorphism, there exists exactly one graded simple W ρ n (X)module of that apex.

(b) For a fixed apex λ ∈ (P ρ ,n ) =0 there exists exactly one simple, and one graded simple, R ρ n (X)-module of that apex up to isomorphism.

Proof. This is a combination of Theorem 2A.4 and the results from this section. For example, the explicit parametrization of the simple modules for fixed apexes follows from Lemma 5G.6.

We are now ready to prove Theorem 5G.2.

Remark 6A.1. As before, the combinatorics below is separated into usual and shifted partition combinatorics. The former is very similar to type C (1) e , which was covered in [MT21, Section 7], so we focus on the shifted partition combinatorics.

As in [MT21, Section 7E] the most important notion that we need is that of Young equivalence. To define it we need some preliminary notions.

Definition 6A.2. For i, j ∈ I, a close (i, i, i)-triple, respectively a close (i, j, k)-triple or a close (i, j, j, i)-quadruple, is a collection of close strings as in the following local configurations: We also need the following, which should be compared with Remark 5C.7. Here we consider the two ghost 1-strings in type D (a) i j, there are no dots on the i or j-strings, and the ghost i-string is close and to the left of the solid j-string;

(b) i j, there are no dots on the i or j-strings, and the solid i-string is close and to the left of the ghost j-string;

(c) i = j is a multisink, the i-string carries a dot, and the solid i-string is close and to the right of the solid j-string; A pseudo row equivalence class is a row equivalence class if there are no close (i, i, i)-triples, the only close (i, j, k)-triples are of the form (i, i + 1, i + 2) or (i, i − 1, i − 2), or (1, 0, 1) in type A (2) 2·e , and if (i, j, j, i) is a close quadruple, then j is a multisink and either i j or i j. These should be compared with (5C.8) and (5C.10). Note that (5C.9) is also included in the description since pseudo row equivalence classes only take two strings per step into account.

Definition 6A.5. Assume that we in the case of shifted partitions. A Young equivalence class Y of solid strings in S is a disjoint union of row equivalence classes R 1 ∪ ... ∪ R z such that: (a) The first string in R 1 has no dot and is close to an (affine) red string of the same residue;

(c) the first string in R a+1 is an i-string, and close to a dotted solid i-string of the same residue in R a or there is a j-string in R a that satisfies one of closeness conditions in (a) and (b) of Definition 6A.4 with respect to this string. For usual partitions we use the analog of [MT21, Definition 7E.6]. That is, (b) above is replaced with |R 1 | ≥ |R 2 | ≥ ... ≥ |R z | and (c) mimics (5C.11) with a dot on the i-string in the first two cases therein.

Recall that L(S) is the left-justification of the dotted straight line diagram S as e.g. in Example 3C.5. Proof. By construction, the solid strings in 1 y λ are a disjoint union of Young equivalence classes cf. Remark 5C.7.

To prove the converse, given a dotted straight line diagram S, we construct an -partition λ by inductively associating the solid strings in each Young equivalence class Y to nodes of a component λ (m) of λ. We explain the situation of shifted partitions, that is, when ρ m is a multisink. The case of usual partitions can be proven similarly, following [MT21, Lemma 7E.8].

By Definition 6A.5.(a), the first string of Y is left adjacent to an (affine) red string of the same residue. If this is the mth red string, then identify the solid string with the node (m, 1, 1). By induction we now assume that the kth solid i-string in Y corresponds to the node (m, r, c) ∈ λ. There are two cases to consider.

Case 1. First, if i is not the last string in its row equivalence class, then (5C.8), (5C.9) and (5C.10) correspond to (a)-(c) of Definition 6A.4 and the condition on close (i, j, k)-triples and close (j, i, i, j)-quadruples, with the correct dot placement. Moreover, no other configurations can appear, i.e. there are no close (i, i, i)triples or any other triples or quadruples due to (a)-(c) of Definition 6A.4. Hence, the (k + 1)st solid j-string corresponds to the node (m, r, c + 1).

Case 2. If on the other hand i is the last string in its row equivalence class, then we observe that (5C.11) corresponds to Definition 6A.5.(c), and the (k + 1)st solid j-string corresponds to the node (m, r + 1, c).

Finally, note that the condition in Definition 6A.5.(b) ensures that the result is a shifted -partition.

Proposition 6A.7. Suppose that D ∈ W ρ n (X) and that D factors through the dotted idempotent diagram S. Then there exists λ ∈ P ρ ,n such that D factors through 1 y λ and λ L(S). Proof. Without loss of generality we can assume that D = S. If D = 1 y λ , then there is noting to prove by Lemma 6A.6. So assume that Lemma 6A.6 is not satisfied, i.e. that D is not a disjoint union of Young equivalence classes.

We can assume that all strings are within [min X, max X + 1] × [0, 1], the region defined by X. Let s be the rightmost solid string in D that is not in any Young equivalence class. We want to argue that we can pull s to the right, jump dots on s further to the right, or we can attach s to a Young equivalence class, which implies the claim by induction. There are a few cases which we need to discuss. We only consider shifted partition combinatorics since the arguments for usual partition combinatorics are mutatis mutandis as in [MT21, Proposition 7E.9 ].

Case 1a. s is the rightmost string in the sense that we can pull it arbitrarily far to the right, and s does not have a dot. In this case we can park it next to an affine red string of the same residue, and it is now part of a Young equivalence class by Definition 6A.5.(a).

Case 1b. As in Case 1a, but now s carries a dot. After pulling the dot to the top of the diagram, the same argument as in Case 1a works.

We will now assume that we are not in Cases 1a and 1b. Then, up to isotopy, s or its ghost is close and to the right of a solid, ghost or red string t. We focus on the situation when s is close to t, where we again have several cases. The cases where the ghost of s is close to t follow mutatis mutandis and are omitted. We also assume that s does not carry a dot. If it does, then there is an additional extra argument one needs to make as explained in [MT21, Proof of Proposition 7E.9, Case 5] (this argument works mutatis mutandis in the BAD types), but in the end the relation used below allow us to continue with the induction.

Case 2a. Assume that s is not in the Young equivalence class of t because the close (i, j, k)-triple condition is not satisfied for s being the leftmost string in (6A.3). In this case the right-hand relation in (4A.4) applies.

(This works unless we in a close (1, 0, 1)-triple situation in type A (2) 2·e , where we do not want to pull s further.) Case 2b. Similarly, assume that s is not in the Young equivalence class of t because the close (i, j, j, i)quadruple condition is not satisfied for s being the leftmost string in (6A.3). Then we can use Lemma 4A.6 to pull s further to the right.

Case 2c. We now assume that s is not in the Young equivalence class of t because the condition |R 1 | > |R 2 | > ... > |R z | is not satisfied. For i, j ∈ I and i j or i j, the crucial configurations are There are a few cases, but for all these we can use (4A.3) or (4A.4) to pull s to the right. Assume now that we are not in any of the cases above. Case 3a. If t is an (affine) red string, then a Reidemeister II move pulls s further to the right. We can apply such a move since the case where s has the same residue as t is covered above.

Case 3b. If t is a solid string, then a Reidemeister II applies unless s has the same residue as t. In this latter case (4A.3) applies.

Case 3c. Finally, if t is a ghost string, then we can use a Reidemeister II relation to pull s rightwards. To see this note that the assumption that s and t are not in a Young equivalence class imply that s and t are not as in (a) and (b) of Definition 6A.4, or as in Definition 6A.5.(c).

Hence, the result follows by induction.

The rest of the proof of Theorem 5G.2 is essentially the same as in [MT21, Section 7E] . That is, applying dots or crossings to 1 y λ gives a linear combination of bigger elements. To this end, recall the definition of the finite dots from Definition 5D.2.

Lemma 6A.8. Suppose that λ ∈ P ρ ,n and 1 ≤ m ≤ n. Then y m y cm(λ) m 1 y λ ∈ W λ n .

Proof. We can use Proposition 6A.7 so that y m y cm(λ) m 1 y λ factors through 1 y µ for µ λ. Consider λ as a composition, and let S λ be the associated Young subgroup of S n . Lemma 6A.9. Suppose that λ ∈ P ρ ,n and w ∈ S λ . Then D λ (w)1 λ , 1 λ D λ (w) ∈ W λ n . Proof. As in [MT21, Lemma 6D.17].

Proof of Theorem 5G.2. The arguments given in [MT21, Sections 6D and 7E] , which are the analogous statements for the AC types, apply in BAD types as well. In fact, these arguments are general and use only Proposition 3D.1 and Remark 3D.2, as well as the analogs of the results proven above.

Remark 6A.10. If one agrees with the strategy in Remark 4A.1, then the construction of the homogeneous (affine) sandwich cellular basis and proof of Theorem 5G.2 splits into several parts:

(a) Because the bottom and top will be given by permutation diagrams, the first step is to find tableaux combinatorics associated to the quiver under study. For a general quiver this is potentially hopeless, but for a lot of quivers an answer is already in the literature.

(b) The construction of the middle is then crucial. This part in noncanonical, although mostly dictated by Remark 4A.1 and we hope to explain a more general approach in future work. Note that additional dots might be necessary to prevent basis elements being annihilated by (3A.3) and to have the analog of Lemma 5C.18.

(c) From here onwards the arguments are general and do not depend on the quiver anymore: Proposition 6A.7 follows by analyzing the combinatorics of the string placement of 1 λ , and this proposition in turn directly implies Lemma 6A.8 and Lemma 6A.9. Once these two lemmas have been established the proof of cellularity Theorem 5G.2 is formal. Linear independence follows using the faithful polynomial module in Remark 3D.2, spanning using Lemma 6A.8 and Lemma 6A.9 and the standard basis in Proposition 3D.1. The latter arguments are independent of the underlying quiver.

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The University of Sydney, School of Mathematics and Statistics F07

Acknowledgments. Want to thank Chris Bowman for a helpful zoom discussion that made us realize several questions related to (affine) sandwich cellular algebras.Both authors were supported, in part, by the Australian Research Council. In these COVID-19 infested times, we thank the first author's office for sponsoring us for the one hour where the main bulk of the mathematics in this paper was discovered.