Path combinatorics and light leaves for quiver Hecke algebras

We recast the classical notion of standard tableaux in an alcove-geometric setting and extend these classical ideas to all reduced paths in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias--Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the Bott--Samelson truncation of the quiver Hecke algebra.

theory (in type A) entirely within the context of the group algebra of the symmetric group, without the need for calculating intersection cohomology groups, or working with parity sheaves, or appealing to the deepest results of 2-categorical Lie theory. In this paper, we specialise Theorem A by making certain path-theoretic choices which allow us to reconstruct Elias-Williamson's generators entirely within the quiver Hecke algebra itself, using our language of paths.
Theorem B. The Bott-Samelson truncation of the Hecke algebra f n,σ (H σ n /H σ n y h H σ n )f n,σ is generated by horizontal and vertical concatenation of the elements e Pα , fork αø αα , spot ø α , hex βαβ αβα , com γβ βγ , e P ∅ , and adj ∅α α∅ (this notation is defined in Section 3) for α, β, γ ∈ Π such that α and β label an arbitrary pair of non-commuting reflections and β and γ label an arbitrary pair of commuting reflections.
The paper is structured as follows. In Section 1 we construct a "classical-type" cellular basis of H σ n /H σ n y h H σ n in terms of tableaux but using a slightly exotic dominance ordering -the proofs in this section are a little dry and can be skipped on the first reading. In Section 2, we upgrade this basis to a "light leaf type" construction and prove Theorem A. Finally, in Section 3 of the paper we illustrate how Theorem A allows us to reconstruct the precise analogue of the light leaves basis for the Bott-Samelson endomorphism algebras for regular blocks of quiver Hecke algebras (as a special case of Theorem A, written in terms of the generators of Theorem B). We do this in the exact language used by Elias, Libedinsky, and Williamson in order to make the construction clear for a reader whose background lies in either field. This paper is a companion to [BCH], but the reader should note that the results here are entirely self-contained (although we refer to [BCH] for further development of ideas, examples, etc).

A tableaux basis of the quiver Hecke algebra
We let S n denote the symmetric group on n letters and let : S n → Z denotes its length function. We let denote the (strong) Bruhat order on S n . Given i = (i 1 , . . . , i n ) ∈ (Z/eZ) n and s r = (r, r + 1) ∈ S n we set s r (i) = (i 1 , . . . , i r−1 , i r+1 , i r , i r+2 , . . . , i n ).
Definition 1.1 ([BK09, KL09,Rou]). Fix e > 2. The quiver Hecke algebra (or KLR algebra), H n , is defined to be the unital, associative Z-algebra with generators {e i | i = (i 1 , ..., i n ) ∈ (Z/eZ) n } ∪ {y 1 , ..., y n } ∪ {ψ 1 , ..., ψ n−1 }, subject to the relations e i e j = δ i,j e i i∈(Z/eZ) n e i = 1 Hn y r e i = e i y r ψ r e i = e sr(i) ψ r y r y s = y s y r (R1) for all r, s, i, j and ψ r y s = y s ψ r for s = r, r + 1 ψ r ψ s = ψ s ψ r for |r − s| > 1 (R2) y r ψ r e i = (ψ r y r+1 − δ ir,i r+1 )e i y r+1 ψ r e i = (ψ r y r + δ ir,i r+1 )e i (R3) for all permitted r, s, i, j. We identify such elements with decorated permutations and the multiplication with vertical concatenation, •, of these diagrams in the standard fashion of [BK09, Section 1] and as illustrated in Figure 1. We let * denote the anti-involution which fixes the generators.
Definition 1.2. Fix e > 2 and σ ∈ Z . The cyclotomic quiver Hecke algebra, H σ n , is defined to be the quotient of H n by the relation y {σm|σm=i 1 ,1 m } 1 e i = 0 for i ∈ (Z/eZ) n . (1.1) As we see in Figure 1, the y k elements are visualised as dots on strands; we hence refer to them as KLR dots. Given p < q we set w p q = s p s p+1 . . . s q−1 w q p = s q−1 . . . s p+1 s p ψ p q = ψ p ψ p+1 ...ψ q−1 ψ q p = ψ q−1 ...ψ p+1 ψ p . and given an expression w = s i 1 . . . s ip ∈ S n we set ψ w = ψ i 1 . . . ψ ip ∈ H n . We let denote the horizontal concatenation of KLR diagrams. Finally, we define the degree as follows, deg(e i ) = 0 deg(y r ) = 2 deg(ψ r e i ) = 1.1. Box configurations, partitions, residues and tableaux. For a fixed n ∈ N and 1 we define a box-configuration to be a subset of {[i, j, m] | 0 m < , 1 i, j n} of n elements, which we call boxes, and we let B (n) denote the set of all such box-configurations. We refer to a box [i, j, m] ∈ λ ∈ B (n) as being in the ith row and jth column of the mth component of λ. Given a box, [i, j, m], we define the content of this box to be ct[i, j, m] = σ m + j − i and we define its residue to be res[i, j, m] = ct[i, j, m] (mod e). We refer to a box of residue r ∈ Z/eZ as an r-box.
Given λ ∈ B (n), we define a λ-tableau to be a filling of the boxes of [λ] with the numbers {1, ..., n}. For λ ∈ B (n), we say that a λ-tableau is row-standard, column-standard, or simply standard if the entries in each component increase along the rows, increase along the columns, or increase along both rows and columns, respectively. We say that a λ-tableau S has shape λ and write Shape(S) = λ. Given λ ∈ B (n), we let Tab(λ) denote the set of all tableaux of shape λ ∈ B (n). Given T ∈ Tab(λ) and 1 k n, we let T −1 (k) denote the box ∈ λ such that T( ) = k. We let RStd(λ), CStd(λ), Std(λ) ⊆ Tab(λ) denote the subsets of all row-standard, column-standard, and standard tableaux, respectively. We let Std(n) = ∪ λ∈P (n) Std(λ) for n ∈ N. Definition 1.3. We define the reverse cylindric ordering, , as follows. Let 1 i, i , j, j n and 0 m, m < . We write [i, j, m] [i , j , m ] if i < i , or i = i and m < m , or i = i and m = m and j < j . For λ, µ ∈ B (n), we write λ µ if the -minimal box ∈ (λ ∪ µ) \ (λ ∩ µ) belongs to µ.
Definition 1.4. We define the dominance ordering, , as follows. Let 1 i, i , j, j n and 0 m, m < . We write [i, j, m] [i , j , m ] if m < m , or m = m and i < i , or i = i and m = m and j < j . Given λ, µ ∈ B (n), we write λ µ if the -maximal box ∈ (λ ∪ µ) \ (λ ∩ µ) belongs to λ.
Given S, we write S↓ k or S↓ {1,...,k} (respectively S↓ k ) for the subtableau of S consisting solely of the entries 1 through k (respectively of the entries k through n). Given λ ∈ B (n), we let T λ denote the λ-tableau in which we place the entry n in the minimal -node of λ, then continue in this fashion inductively. Given λ ∈ B (n), we let S λ denote the λ-tableau in which we place the entry n in the minimal -node of λ, then continue in this fashion inductively. Finally, given S, T two λ-tableaux, we let w S T ∈ S n be the permutation such that w S T (T) = S.
Given h ∈ N , we let P h (n) (respectively C h (n)) denote the subsets of -multipartitions (respectively -multicompositions) with at most h m columns in the mth component.
Lemma 1.8. Let λ ∈ P h (n). For any box [i, j, m], the multiset of residues of the boxes in λ ∩ Gar [i, j, m] is multiplicity-free (i.e. no residue appears more than once).
Proof. This follows immediately from the definitions since λ ∈ P h (n).

Let
be an ordering on B (n). Given 1 k n, we let A T (k), (respectively R T (k)) denote the set of all addable res(T −1 (k))-boxes (respectively all removable res(T −1 (k))-boxes) of the boxconfiguration Shape(T↓ {1,...,k} ) which are less than T −1 (k) in the -order. We define the ( )-degree of T ∈ Std(λ) for λ ∈ P (n) as follows, We let res(T) denote the residue sequence consisting of res(T −1 (k)) for k = 1, . . . , n in order. We set e T := e res(T) ∈ H σ n . We set (1.2) For λ ∈ P (n), the element y λ was first defined in [HM10,Definition 4.15]. We remark that y λ = e T λ for λ ∈ P h (n). Given S, T ∈ Std(λ) and w any fixed reduced word for w S T we let ψ S T := e S ψ w e T .
Definition 1.9. We set Y n = e i , y k | i ∈ (Z/eZ) n , 1 k n .
1.2. The quotient. A long-standing belief in modular Lie theory is that we should (first) restrict our attention to fields whose characteristic, p, is greater than the Coxeter number, h, of the algebraic group we are studying. This allows one to consider a "regular block" of the algebraic group in question. What does this mean on the other side of the Schur-Weyl duality relating GL h (k) and kS n ? By the second fundamental theorem of invariant theory, the kernel of the group algebra of the symmetric group acting on n-fold h-dimensional tensor space is the 2-sided ideal generated by the element g∈S h+1 Sn sgn(g)g ∈ kS n Modulo "more dominant terms" this element can be rewritten in the form we introduced in equation (1.2), as follows by [HM10,4.16 Corollary]. The simples of this algebra are indexed by partitions with at most h columns. Given σ ∈ Z , we let h = (h 0 , . . . , h −1 ) ∈ N be such that h m σ m+1 −σ m for 0 m < −1 and h −1 < e + σ 0 − σ −1 . We define to be the higher level analogue of the element in (1.3).
Definition 1.10. Given weakly increasing σ ∈ Z , we let h = (h 0 , . . . , h −1 ) ∈ N be such that h m σ m+1 − σ m for 0 m < − 1 and h −1 < e + σ 0 − σ −1 . We define H σ n := H σ n /H σ n y h H σ n . In level = 1, the condition of Definition 1.10 is equivalent to h 0 < e (and P h (n) is the set of partitions of n with strictly fewer than e columns).
1.3. Generator/partition combinatorics. Our cellular basis will provide a stratification of H σ n in which each layer is generated by an idempotent correspond to some multipartition. Whence we wish to understand the effect of multiplying a generator of a given layer in the cell-stratification by a KLR "dot generator". This leads us to define combinatorial analogues of the dot generators as maps on the set of box configurations.
Definition 1.11. Let λ ∈ B (n) and let [i, j, m] ∈ λ. We say that [i, j, m] is left-justified if either j e or there exists some [i, j − p, m] ∈ λ for 1 p e. Definition 1.12. Let λ ∈ B (n). For α ∈ λ an r-box, we define Y α (λ) = λ − α ∪ β where the r-box β ∈ λ is such that β α, it is left-justified, and is minimal in with respect to these properties (if such a box exists). If such a box does not exist then we say that Y α (λ) and β are both undefined.
We write λ µ if λ = Y α (µ) for some α ∈ µ and we then extend to a partial ordering on B (n) by taking the transitive closure. Suppose that {[i k , j k , m k ] | 0 < k p} is a set of r-boxes and that We remark that λ µ implies that λ µ and so is a coarsening of .
Given an idempotent generator, e j for j ∈ (Z/eZ) n , of the KLR algebra, we wish to identify to which layer of our stratification our idempotent belongs. To this end we make the following definition.
1.4. A tableaux theoretic basis. We are now ready to construct our first basis of H σ n . This basis will serve as the starting point for our light-leaves bases of Theorem A. The combinatorics of this basis will be familiar to anyone who has studied symmetric groups and cyclotomic Hecke algebras (but with respect to the, less familiar, ( )-ordering).
Definition 1.16. Let λ ∈ P h (n). We define the Garnir adjacency set of an r-box α = [i, j, m] ∈ λ to be the set of boxes, γ ∈ λ ∩ Gar (α) such that |res(γ) − r| 1 and denote this set by Adj-Gar (α). We set res(Adj-Gar (α)) = {res(γ) | γ ∈ Adj-Gar (α)}. Remark 1.18. We are endeavouring to construct a 2-sided chain of ideals of H σ n , ordered by , in which each 2-sided ideal is generated by an idempotent e T λ for λ ∈ P h (n). Equation (1.6) and (1.7) of Theorem 1.19 will allow us to rewrite any element of Y n in the required form by moving a given box through the partition λ one row at a time along the ordering until it comes to rest at some point λ ∪ ∈ P h (n). For J the tableau in Figure 3, then the eight steps involved in rewriting e J as an element of H (3,2 2 ,1 6 ) are illustrated in Figure 4.
• If λ ∪ α ∈ P h (n), then we have that (the cases are detailed in the proof ). • If λ ∪ α ∈ P h (n), then we have that (c) For j ∈ (Z/eZ) n , using the notation of Definition 1.15, we have that: if Shape(J) = 0 (1.8) whereψ J Tν is obtained from ψ J Tν by possibly adding some dot decorations along the strands. Thus if is any total refinement of then the Z-algebra H σ n has a chain of two-sided ideals (1.9) Remark 1.20. In the proof, we will often relate ideals in smaller and larger algebras using horizontal concatenation of diagrams, this is made possible by the definition of the reverse cylindric ordering (which distinguishes between box configurations based on the first discrepancy upon reading a pair of box configurations backwards).
Proof of Theorem 1.19. Part (a), let α = [1, j, 0] for some j 1. Claim (ii) follows by applying case 3 of R4 a times, followed by the commuting KLR relations and the cyclotomic relation. The proof of claim (i) is similar. Thus (a) follows. For parts (b) and (c), we assume that equation (1.6) to (1.9) all hold for rank n − 1. We further assume that equation (1.6) and (1.7) have been proven for all ν = µ ∪ α with µ ∈ P h (n − 1) such that µ λ; thus leaving us to prove equation (1.6) and (1.7) for ν = λ ∪ α for λ ∈ P h (n − 1) and equation (1.8) for all j ∈ (Z/eZ) n . Equation (1.9) follows immediately from equation (1.8) and the idempotent decomposition of the identity in relation R1. By Definition 1.12, res(α) = res(β) and we set this residue equal to r ∈ Z/eZ.
Proof of equation (1.6) for a given λ and α. We include a running example for e = 5 and = 1 and λ = (2 3 , 1 6 ). We assume that λ ∪ α ∈ P h (n). There are four cases to consider, depending on the residue of α := T −1 λ∪α (a − 1). We assume that α is in the same row of the same component as α (as otherwise y T λ∪α = y T λ∪β by definition). We let β be the box determined by (i) Suppose α has residue r ∈ Z/eZ and so y T λ∪α = e T λ∪α by Lemma 1.8. By application of relations R3 and R4, we have that e T λ∪α = ψ a−1 y a−1 e T λ∪α ψ a y a−1 − y a ψ a−1 y a−1 e T λ∪α ψ a−1 . (1.10) An example of the visualisation of the idempotents on the righthand-side of equation (1.10) is given in the first step of Figure 4; the corresponding righthand-side of equation (1.10) is depicted in Figure 5. Now, we have that and so by our inductive assumption for equation (1.7) for rank a − 1 < n, we have that where we have implicitly used the following facts: Substituting this back into equation (1.10), we obtain as required (as in the second case in equation (1.6)). An example is depicted in Figure 5 (although we remark that the error terms belonging to H (ii) Now suppose α = [x, y, z] has residue r + 1 ∈ Z/eZ. We have two subcases to consider. We first consider the easier subcase, in which [x, y, z] = [i, 1, m] and so b = a − 1. We have that By relation R4, we have that We have that y a−1 y T λ ∈ H λ∪β −α n−1 and so the former term is of the required form by our assumption for λ ∪ β − α = µ λ. Now, if y > 1 then the (a−2)th, (a−1)th and ath strands have residues r, r+1, and r respectively. We have that where the first equality follows from Lemma 1.8, the second from relation R5 and the third follows from relations R3 and R4. We set ξ = Shape(T λ ↓ <a−2 ). The two terms in equation (1.12) factor through the elements respectively.
• We first consider the latter term on the righthand-side of equation (1.12) (which we will see, is the required non-zero term). We note that [x, y − 1, z] and α have the same residue and so By our inductive assumption that equation (1.7) holds for rank a − 2 < n, we have that (1.14) Substituting this back into the second term of equation (1.13) we obtain and then substituting into the second term of equation (1.12) we obtain • We now consider the former term of equation (1.12) (which, we will see, is zero modulo the ideal).
We have that by our inductive assumption that equation (1.6) holds for rank a − 2 < n. Concatenating, we have that e T λ ↓ <a−2 e r+1 y 1 e r,r e T λ∪α ↓ >a ∈ ±ψ a−2 c y T λ∪γ∪α−[x,y,z] ψ c a−2 + H ( (λ−[x,y,z]∪γ))∪α n and we note that the idempotent on the righthand-side belongs to the ideal H ( (λ−[x,y,z]∪γ))∪α n and so the result follows by our assumption for λ ∪ γ − [x, y, z] = µ λ. (iii) Now suppose α has residue d ∈ Z/eZ such that |d − r| > 1. We set ξ = Shape(T λ ↓ <a−1 ). By case 2 of relation R4, we have that for k ∈ {0, 1}. By the inductive assumption for rank a − 1 < n of equation (1.6), we have that As in the case (ii) above, we concatenate to deduce the result. Two examples of the visualisation of the righthand-side of equation (1.15) are given in the third and fourth steps of Figure 4; the corresponding elements are depicted in Figure 6. (iv) Suppose α = [x, y, z] has residue r −1 ∈ Z/eZ (thus [x, y, z] = [i, j −1, m] by residue considerations) and that [i − 1, j, m] ∈ λ (if [i − 1, j, m] ∈ λ, this implies that λ ∪ α ∈ P h (n) and so the process would terminate). We remark that res(Adj-Gar(α)) = {r − 1} and this is the unique case of the proof for which this holds. Let γ = [i − 1, j − 1, m] and we set c = T λ∪α (γ) and let ξ = Shape(T λ ↓ <a−1 ) (see Figure 7 for an example). By Lemma 1.8, e T λ∪α = y T λ∪α . We have that where the first and third equalities follow from the commuting case 2 of relation R4 and Lemma 1.8, and the second equality follows from case 1 of relation R5; the fourth equality follows from R3; the fifth is either trivial or follows from case (3) of R4 (in the latter case, the error term is zero by our inductive assumption for rank c − 1 < n of equation (1.7)). For our continuing example, the righthand-side of equation (1.19) is depicted in Figure 9; the box-configurations labelling the idempotents on the left and righthand-sides of equation (1.18) are depicted in Figure 7. We have moved the 1-box using case (iii) and this leaves us free to move the 2-boxes up their corresponding diagonals. The righthand-side is a box configuration which is strictly higher than λ ∪ β in the -ordering.
We now consider the second term on the righthand-side of equation (1.18). We have that By our inductive assumption for ranks a − 2, a − 1 < n for equation (1.6), we have that: . Given π (ρ ∪ γ), we can left justify π ∪ [i, j + e, m] to obtain π ∪ α. We note that Gar (α) ∩ π contains no nodes of residue r or r ± 1. Therefore . In this case α = α 5 and β = α 7 as in the first case of equation (1.5).
Proof of equation (1.7) for a given λ and α. We assume that equation (1.7) holds for all λ ∈ P h (n − 1). We set ν = λ ∪ α ∈ P h (n) and we assume that α is of residue r ∈ Z/eZ. We have that e T λ∪α = y T λ∪α . We have that In the first case, this follows from case 3 of relation R4 and the commutativity relations, to see this note that the (a − 1)th strand has residue r − 1 ∈ Z/eZ. In the second case, the statement follows from Lemma 1.8 and the fact that b = a. Letting β be as in Subsection 1.4, we note that and so the first case of equation (1.21) is of the required form by our assumption for λ ∪ β − α = µ λ.
Proof of equation (1.8). Let j = (j 1 , . . . , j n−1 , r) ∈ (Z/eZ) n . We can assume that Shape(J n−1 ) = λ ∈ P h (n − 1) as otherwise e j e jn = 0 e r = 0 by induction. Thus it remains to show that We can associate this rightmost r-strand to the (unique) left-justified r-box α such that α for all ∈ λ. (For example, = [9, 6, 0] for λ = (2 3 , 1 6 ), see Figure 4.) Thus every strand in the diagram is labelled by a box. We pull the strand labelled by α through the centre of the diagram (which is equal to the idempotent e Tν ) one row at a time using equation (1.6). We can utilise equation (1.6) precisely when λ ∪ α ∈ P h (n). Therefore this process terminates when we reach the smallest addable r-box of λ under the -ordering, namely J −1 (n). Thus equation (1.8) follows.
Proposition 1.22 ([BKW11, Lemma 2.4 and Proposition 2.5]). We let w, w be any two choices of reduced expression for w ∈ S n and let v be any non-reduced expression for w. We have that Proposition 1.23. Let k be an integral domain. The k-algebra H σ n has spanning set Proof. Let d ∈ e i H σ n e j for some ı,  ∈ (Z/eZ) n . By equation (1.8), we can rewrite e j (or equivalently e i ) so that d = x,y∈Sn e i a x e T λ a y e j for some a x , a y which are linear combinations of KLR elements tracing out some bijections x, y ∈ S n respectively (but are possibly decorated with dots and need not be reduced). It remains to show that a x , a y ∈ H σ n can be assumed to be reduced and undecorated. We establish this by induction along the Bruhat order, by working modulo the span of elements It remains to show that a spanning set is given by the elements T has a pair of crossing strands from 1 i < j n to 1 and T −1 λ (j) = [r, c + 1, m] are in the same row and in particular so that i = j − 1. It suffices to show that ψ x e T λ ψ y belongs to the ideal H λ n for a preferred choice of y; we choose y = s i w (for some w ∈ S n such that s i w = y). Thus it remains to show that e T λ ψ s i w ∈ H λ n . However, this immediately follows from equation (1.6) because e T λ ψ s i = ψ s i e s i (T λ ) and we have that Given any T ∈ RStd(λ)\Std(λ) we let k be minimal such that T↓ <k ∈ Std(µ) for some µ ∈ P h (k−1) and Shape(T↓ k ) = ν for some ν ∈ P h (k). We have that e T = e T k e T >k where Shape(e T k ) = ν ∈ C h (k) \ P h (k) and so e T k ∈ H ν k by equation (1.6) and so e T ∈ H λ n by concatenation and the definition of . This implies that e T λ ψ T λ T = ψ T λ T e T ∈ H λ n , as required. Theorem 1.24. Let k be an integral domain. The k-algebra H σ n is graded cellular with basis anti-involution * and the degree function deg : Std → Z. For k a field, H σ n is quasi-hereditary.
Since λ ∈ P h (n), we observe that [1, 1, m + 1] is the unique box in λ of residue s m+1 ∈ Z/eZ in which we can place the integer h m + 1 (or any integer smaller than h m + 1) without violating the standard condition, by Lemma 1.8. The presentation of the Specht module in [KMR12, Definition 5.9] implies that (i) ψ w y λ ∈ H λ n for any w = w S S λ for some S ∈ Std(λ) with λ ∈ P h (n) (since every ( )-Garnir belt has fewer than e boxes) and (ii) y k y λ ∈ H λ n for any 1 k n. We are now ready to prove the claim. Using equation (1.24), we move the dot at the top of y hm+1 ψ S S λ y λ down the (h m + 1)th strand to obtain a linear combination of undecorated diagrams (in which we have undone some number of crossings s m -strands) and ψ S S λ y S λ (1,1,m+1) y λ . By our above observation, all of these undecorated diagrams are labelled by non-standard λ-tableaux. Therefore all of these terms (and hence y hm+1 ψ S S λ y λ ) are zero, by (i) and (ii). The claim and result follow. Let k be an integral domain. We define the standard or Specht modules of H σ n as follows, for λ ∈ P h (n). We immediately deduce the following corollary of Theorem 1.24.
Corollary 1.25. The module S k (λ) is the module generated by e T λ subject to the following relations: • e i e T λ = δ i,res(T λ ) e T λ for i ∈ (Z/eZ) n ; • y k e T λ = 0 for 1 k n; • ψ k e T λ = 0 for any 1 k < n such that s k (T λ ) is not row standard; • ψ S T λ e T λ = 0 for S ∈ RStd(λ) \ Std(λ).
Proof. We have already checked that all of these relations hold (and so one can define a homomorphism from the abstractly defined module with this presentation to S k (λ)) it only remains to check that these relations will suffice (i.e. the homomorphism is surjective). We know that S k (λ) has a basis indexed by standard tableaux and so the result follows.
We now recall that the cellular structure allows us to define bilinear forms, for each λ ∈ P h (n), there is a bilinear form , λ on S k (λ), which is determined by for any S, T ∈ Std(λ). Let k be a field of arbitrary characteristic. Factoring out by the radicals of these forms, we obtain a complete set of non-isomorphic simple H σ n -modules D k (λ) = S k (λ)/rad(S k (λ)), λ ∈ P h (n).
Proposition 1.26. Let λ ∈ P h (n) and let A 1 A 2 · · · A z denote the removable boxes of λ, totally ordered according to the -ordering. The restriction of S k (λ) has an H σ n−1 -module filtration 0 = S z+1,λ ⊂ S z,λ ⊂ · · · ⊂ S 1,λ = Res H σ n−1 (S k (λ)) (1.33) given by S x,λ = k{ψ S T λ | Shape(S n−1 ) = λ − A y for some z y x}. For each 1 r z, we have that (1.34) Proof. On the level of k-modules, this is clear. Lifting this to H σ n−1 -modules is a standard argument which proceeds by checking the relations of Corollary 1.25 in a routine manner.

General light leaves bases for quiver Hecke algebras
The principal idea of categorical Lie theory is to replace existing structures (combinatorics, bases, and presentations of Hecke algebras) with richer structures which keep track of more information. In this section, we replace the classical tableaux combinatorics of symmetric groups (and quiver Hecke algebras) with that of paths in an alcove geometry. This will allow us to construct "light leaves" bases of these algebras, for which p-Kazhdan-Lusztig is baked-in to the very definition. The light leaves bases of S k (λ) are constructed in such a way as to keep track of not just the point λ ∈ E h (or rather the single path, T λ , to the point λ) but of the many different ways we can get to the point λ by a reduced path/word in the alcove geometry. This extra generality is essential when we wish to write bases in terms of "2-generators" of the algebras of interest.
2.1. The alcove geometry. For ease of notation, we set H m = h 0 + · · · + h m for 0 m < , and h = h 0 + · · · + h −1 . For each 1 i n and 0 m < we let ε i,m := ε (h 0 +···+h m−1 )+i denote a formal symbol, and define an h-dimensional real vector space and E h to be the quotient of this space by the one-dimensional subspace spanned by We have an inner product , on E h given by extending linearly the relations ε i,p , ε j,q = δ i,j δ p,q for all 1 i, j n and 0 p, q < , where δ i,j is the Kronecker delta. We identify λ ∈ C h (n) with an element of the integer lattice inside E h via the map where (−) T is the transpose map. We let Φ denote the root system of type A h−1 consisting of the roots {ε i,p − ε j,q : 0 p, q < , 1 i h p , 1 j h q , with (i, p) = (j, q)} and Φ 0 denote the root system of type A h 0 −1 × · · · × A h −1 −1 consisting of the roots We choose ∆ (respectively ∆ 0 ) to be the set of simple roots inside Φ (respectively Φ 0 ) of the form ε t − ε t+1 for some t. Given r ∈ Z and α ∈ Φ we define s α,re to be the reflection which acts on E h by The group generated by the s α,0 with α ∈ Φ (respectively α ∈ Φ 0 ) is isomorphic to the symmetric group S h (respectively to S f := S h 0 × · · · × S h −1 ), while the group generated by the s α,re with α ∈ Φ and r ∈ Z is isomorphic to S h , the affine Weyl group of type A h−1 . We set α 0 = ε h − ε 1 and Π = ∆ ∪ {α 0 }. The elements S = {s α,0 : α ∈ ∆} ∪ {s α 0 ,−e } generate S h .
Notation 2.1. We shall frequently find it convenient to refer to the generators in S in terms of the elements of Π, and will abuse notation in two different ways. First, we will write s α for s α,0 when α ∈ ∆ and s α 0 for s α 0 ,−e . This is unambiguous except in the case of the affine reflection s α 0 ,−e , where this notation has previously been used for the element s α,0 . As the element s α 0 ,0 will not be referred to hereafter this should not cause confusion. Second, we will write α = ε i − ε i+1 in all cases; if i = h then all occurrences of i + 1 should be interpreted modulo h to refer to the index 1.
We shall consider a shifted action of the affine Weyl group S h on E h,l by the element that is, given an element w ∈ S h , we set w ·x = w(x+ρ)−ρ. This shifted action induces a well-defined action on E h ; we will define various geometric objects in E h in terms of this action, and denote the corresponding objects in the quotient with a bar without further comment. We let E(α, re) denote the affine hyperplane consisting of the points E(α, re) = {x ∈ E h | s α,re · x = x}. Note that our assumption that e > h 0 + · · · + h −1 implies that the origin does not lie on any hyperplane. Given a hyperplane E(α, re) we remove the hyperplane from E h to obtain two distinct subsets E > (α, re) and E < (α, re) where the origin lies in E < (α, re). The connected components of The connected components of E h \ (∪ α∈Φ,r∈Z E(α, re)) are called alcoves, and any such alcove is a fundamental domain for the action of the group S h on the set Alc of all such alcoves. We define the fundamental alcove A 0 to be the alcove containing the origin (which is inside the dominant chamber). We have a bijection from S h to Alc given by w −→ wA 0 . Under this identification Alc inherits a right action from the right action of S h on itself. Consider the subgroup The dominant chamber is a fundamental domain for the action of S f on the set of chambers in E h .
We let S f denote the set of minimal length representatives for right cosets S f \ S h . So multiplication gives a bijection S f × S f → S h . This induces a bijection between right cosets and the alcoves in our dominant chamber. Under this identification, alcoves are partially ordered by the Bruhat-ordering on S h which is a coarsening of the opposite of the order . If the intersection of a hyperplane E(α, re) with the closure of an alcove A is generically of codimension one in E h then we call this intersection a wall of A. The fundamental alcove A 0 has walls corresponding to E(α, 0) with α ∈ ∆ together with an affine wall E(α 0 , e). We will usually just write E(α) for the walls E(α, 0) (when α ∈ ∆) and E(α, e) (when α = α 0 ). We regard each of these walls as being labelled by a distinct colour (and assign the same colour to the corresponding element of S). Under the action of S h each wall of a given alcove A is in the orbit of a unique wall of A 0 , and thus inherits a colour from that wall. We will sometimes use the right action of S h on Alc. Given an alcove A and an element s ∈ S, the alcove As is obtained by reflecting A in the wall of A with colour corresponding to the colour of s. With this observation it is now easy to see that if w = s 1 . . . s t where the s i are in S then wA 0 is the alcove obtained from A 0 by successively reflecting through the walls corresponding to s 1 up to s h . We will call a multipartition regular if its image in E h,l lies in some alcove; those multipartitions whose images lies on one or more walls will be called singular.
Given λ ∈ C h (n) we let Path(λ) denote the set of paths of length n with shape λ. We define Path h (λ) to be the subset of Path(λ) consisting of those paths lying entirely inside the dominant chamber; i.e. those P such that P(i) is dominant for all 0 i n. We let Path h (n) = ∪ λ∈P h (n) Path h (λ).
Given a path T defined by such a map p of length n and shape λ we can write each p(j) uniquely in the form ε p(j) = ε m j ,c j where 0 m j < and 1 c j h j . We record these elements in a tableau of shape λ T by induction on j, where we place the positive integer j in the first empty box in the c j th column of component m j . By definition, such a tableau will have entries increasing down columns; if λ is a multipartition then the entries also increase along rows if and only if the given path is in Path h (λ), and hence there is a bijection between Path h (λ) and Std(λ). For this reason we will sometimes refer to paths as tableaux, to emphasise that what we are doing is generalising the classical tableaux combinatorics for the symmetric group.
In other words the paths P and Q agree up to some point P(s) = Q(s) which lies on E(α, re), after which each Q(t) is obtained from P(t) by reflection in E(α, re). We extend ∼ by transitivity to give an equivalence relation on paths, and say that two paths in the same equivalence class are related by a series of wall reflections of paths and given S ∈ Path h (n) we set [S] = {T ∈ Path h (n) | S ∼ T}.
This definition of a reduced path is easily seen to be equivalent to that of [BCH, Section 2.3].
Figure 12. The first and second paths have degrees −1 and +1 respectively. The third and fourth paths have degree 0. Here we take the convention that the origin is below the pink hyperplane.
There exist a unique reduced path in each ∼-equivalence class (and, of course, each reduced path belongs to some ∼-equivalence class and so ∼-classes and reduced paths are in bijection). We remark that T µ , the maximal path in the reverse cylindric ordering , is an example of a reduced path. Given S ∈ Path h (n), we let min[S] denote the minimal path in the ∼-equivalence class containing S. Given a reduced path P λ ∈ Path h (λ), we have that decomposes (in a unitriangular fashion) as a sum of projective indecomposable modules for some generalised p-Kostka coefficients k µ P λ ∈ k. In general, we have H σ n e P λ ∼ = H σ n e Q λ for reduced paths P λ , Q λ ∈ Path h (λ) and so the choice of reduced path does matter. (This is not surprising, the auxiliary steps in Soergel's algorithm for calculating Kazhdan-Lusztig polynomials produces a different pattern depending on the choice of reduced expression.) However, they do agree modulo higher terms under , as we shall soon see (and indeed, after the cancellations in Soergel's algorithm one obtains that the Kazhdan-Lusztig polynomials are independent of choices of reduced expressions).
Lemma 2.4. Given λ ∈ P h (n), let P λ , Q λ , S λ be any triple of reduced paths in Path h (λ). The element e P λ generates H λ /H λ and moreover Proof. Let R λ be any reduced path in Path h (λ). Two paths have the same residue sequence if and only if they belong to the same ∼-class. If S ∼ R λ then either S = R λ or S terminates at a point µ λ. Thus, we have that This implies that e R λ ∈ H λ /H λ and therefore generates H λ /H λ and belongs to the simple head of the Specht module; the result follows.

Branching coefficients.
We now discuss how one can think of a permutation as a morphism between pairs of paths in the alcove geometries of Subsection 2.1. Let λ ∈ C h (n). Given a pair of paths S, T ∈ Path(λ) we write the steps in S and T in sequence along the top and bottom edges of a frame, respectively. We can now reinterpret the element w S T ∈ S n (of Section 1) as the unique step-preserving permutation with the minimal number of crossings.
In the following (running) example we label our paths by P ø (= T (3 3 ) ) and P α . For this section, we do not need to know what inspires this notation; however, all will become clear in Section 3.
Definition 2.7. Fix (S, T) an ordered pair of paths which both terminate at some point λ ∈ P h (n). We now inductively construct a reduced expression for w S T . We define the branching coefficients d p (S, T) = w p q where q = |{1 i p | w S T (i) w S T (p)}| and Υ p (S, T) = (−1) {p<k q|i k =ip} ψ dp(S,T) for 1 p n. These allow us to fix a distinguished reduced expression, w S T , for w S T as follows, w S T = d 1 (S, T) . . . d n (S, T). and we set Example 2.8. We continue with the assumptions of Example 2.6. We have that for each p = 1, 2, 3, 4, 5, 6, 9 because w Pø P α (p) i for all 1 i p. We have that d 7 (P ø , P α ) = w 7 3 d 8 (P ø , P α ) = w 8 6 and so our reduced word is depicted in Figure 13.
We can think of the branching coefficients as "one step morphisms" which allow us to mutate the path S into T via a series of n steps (as each branching coefficient moves the position of one step in the path) and so this mutation proceeds via n + 1 paths S = S 0 , S 1 , S 2 , . . . , S n = T see Figure 13 for an example. We now lift these branching coefficients to the KLR algebra. ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 3 ε 3 ε 3 Figure 13. The reduced word, w P α Pø (see also Examples 2.6 and 2.8).
Remark 2.9. The "sign twist" in Definition 2.7 is of no consequence in this paper as we are mostly concerned with constructing generators and bases of quiver Hecke algebras and their truncations. However, in order to match-up our relations with those of Elias-Williamson, this sign twist will be necessary and so we introduce it here for the purposes of consistency with [BCH]. Now, let's momentarily restrict our attention to pairs of paths of the form (S, T λ ). In this case, the branching coefficients actually come from the "branching rule" for restriction along the tower · · · ⊂ H σ n−1 ⊂ H σ n ⊂ . . . . To see this, we note that w S T λ = w 1 (S, T λ )w 2 (S, T λ ) . . . w n (S, T λ ) where w n (S, T λ ) = w n T λ ( ) for some removable box ∈ Rem(λ) and where w 1 (S, T λ )w 2 (S, T λ ) . . . w n−1 (S, T λ ) = w(S n−1 , T λ− ) ∈ S n−1 S n . (2.3) By Proposition 1.26, we have that Thus the branching coefficients above provide a factorisation of the cellular basis of Theorem 1.24 which is compatible with the restriction rule. At each step in the restriction along the tower, there is precisely one removable box of any given residue and so the restriction is, in fact, a direct sum of Specht modules.
We wish to modify the branching coefficients above so that we can consider more general (families of) reduced paths P λ in place of the path T λ . Given S ∈ Path h (λ), we can choose a reduced path vector as follows P S = (P S,0 , P S,1 , . . . , P S,n ) such that Shape(P S,k ) = Shape(S k ) for each 0 k n. In other words, we choose a reduced path P S,p for each and every point in the path S. For 0 p n and Shape(S <p ) + ε ip = Shape(S p ), we define the modified branching coefficient, d p (S, P S ) = Υ P S,p−1 P ip P S,p and we hence define Υ S P S = 1 p n d p (S, P S ).
Here we have freely identified elements of algebras of different sizes using the usual embedding H σ n−1 → H σ n given by d → d ( i∈Z/eZ e(i)). We set Υ Remark 2.11. For symmetric groups there is a canonical choice of reduced path vector coming from the coset-like combinatorics which has historically been used for studying these groups. For the light leaves construction of Bott-Samelson endomorphism algebras, Libedinsky and Elias-Williamson require very different families of reduced path vectors whose origin can be seen as coming from a basis which can be written in terms of their 2-generators [Lib08, EW16].
Example 2.12. Continuing with Example 2.10 we have already noted that P ø = T (3 3 ) . We choose to take the sequence T µ for µ = Shape((P α ) k ) for k 0 as our reduced path vector P S . Having made this choice, we have that Υ Pø (this holds more generally, see the Corollary 2.14 and the discussion immediately prior). We record this in tableaux format to help the reader transition between the old and new ways of thinking. The light leaves basis will be given in terms of products Υ S P S Υ P T T for S, T ∈ Path h (λ) and "compatible choices" of P S and P T . Here the only condition for compatibility is that P S,n = Q λ = P T,n for some fixed choice of reduced path Q λ ∈ Path h (λ), in other words the final choices of reduced path for each of S and T coincide. We remark that if P S,n = P T,n then the product is clearly equal to zero (by idempotent considerations) and so this is the only sensible choice to make for such a product. In light of the above, we let Q λ be a reduced path and we say that a reduced path vector P S terminates at Q λ if P S,n = Q λ .
Theorem 2.13 (The light leaves basis). Let k be an integral domain and h = (h 0 , . . . , h −1 ) ∈ N be such that h m σ m+1 − σ m for 0 m < − 1 and h −1 < e + σ 0 − σ −1 . For each λ ∈ P h (n) we fix a reduced path Q λ ∈ Path h (λ) and for each S ∈ Path h (λ), we fix an associated reduced path vector P S terminating with Q λ . The k-algebra H σ n is a graded cellular algebra with basis {Υ S P S Υ P T T | S, T ∈ Path h (λ), λ ∈ P h (n)} anti-involution * and the degree function deg : Path h → k.
Proof. By Theorem 1.24, we have that } provides a k-basis of H λ /H λ . By Lemma 2.4, we have that Υ T λ Q λ e Q λ Υ Q λ T λ = ke T λ for some k ∈ k \ {0} modulo higher terms under and so By equation (2.3) and (2.4), we have that provides a k-basis of H λ /H λ . By Proposition 1.26, we have that generates a left subquotient of H λ /H λ which is isomorphic to S k (λ − ε i ). Now, for each pair ε i ∈ Rem(λ) and s ∈ Std(λ−ε i ), we fix a corresponding choice of reduced path P S,n−1 ∈ Path h (λ−ε i ). By Lemma 2.4 and Proposition 1.26, we have that the set of all as we vary over all s ∈ Path h (λ−ε i ), ε i ∈ Rem(λ), and T ∈ Path h (λ) provides a k-basis of H λ /H λ . Re-bracketing the above, we have that the set of all as we vary over all s ∈ Path h (λ−ε i ), ε i ∈ Rem(λ), and T ∈ Path h (λ) provides a k-basis of H λ /H λ . Finally, simplifying using Proposition 1.22 we obtain that is a k-basis of H λ /H λ where we note that the middle term in the KLR-product is our modified branching coefficient. Repeating n times, we have that is a k-basis of H λ /H λ ; repeating the above for the righthand-side, the result follows.
In particular, we can set P S = (Q λ ↓ k ) k 0 and obtain the following corollary, which specialises to Theorem 1.24 for Q λ = T λ .
Corollary 2.14. For each λ ∈ P h (n) we fix a reduced path Q λ ∈ Path h (λ). The k-algebra H σ n is a graded cellular algebra with basis , λ ∈ P h (n)} anti-involution * and the degree function deg : Path h → Z.

Light leaf generators for the principal block
We now restrict our attention to the principal block and illustrate how the constructions of previous sections specialise to be familiar ideas from Soergel diagrammatics. In particular, we provide an exact analogue of Libedinsky's and Elias-Williamson's algorithmic construction of a light leaves basis for such blocks. In order to do this, we provide a short list of path-morphisms which we will show generate the algebra f n,σ (H σ n /H σ n y h H σ n )f n,σ (thus proving Theorem B).
3.1. Alcove paths. When passing from multicompositions to our geometry E h,l , many non-trivial elements map to the origin. One such element is δ = ((h 1 ), ..., (h )) ∈ P h (h). (Recall our transpose convention for embedding multipartitions into our geometry.) We will sometimes refer to this as the determinant as (for the symmetric group) it corresponds to the determinant representation of the associated general linear group. We will also need to consider elements corresponding to powers of the determinant, namely δ k = ((h k 1 ), ..., (h k )) ∈ P h (kh). We now restrict our attention to paths between points in the principal linkage class, in other words to paths between points in S h · 0. Such points can be represented by multicompositions µ in S h · δ k for some choice of k.
Definition 3.1. We will associate alcove paths to certain words in the alphabet where s ∅ = 1. That is, we will consider words in the generators of the affine Weyl group, but enriched with explicit occurrences of the identity in these expressions. We refer to the number of elements in such an expression (including the occurrences of the identity) as the degree of this expression. We say that an enriched word is reduced if, upon forgetting occurrences of the identity in the expression, the resulting word is reduced.
Given a path P between points in the principal linkage class, the end point lies in the interior of an alcove of the form wA 0 for some w ∈ S h . If we write w as a word in our alphabet, and then replace each element s α by the corresponding non-affine reflection s α in S h to form the element w ∈ S h then the basis vectors ε i are permuted by the corresponding action of w to give ε w(i) , and there is an isomorphism from E h,l to itself which maps A 0 to wA 0 such that 0 maps to w · 0, coloured walls map to walls of the same colour, and each basis element ε i map to ε w(i) . Under this map we can transform a path Q starting at the origin to a path starting at w · 0 which passes through the same sequence of coloured walls as Q does.
If there is a unique such w then we may simply write P ⊗ Q. If w = s α we will simply write P ⊗ α Q.
We now define the building blocks from which all of our distinguished paths will be constructed. We begin by defining certain integers that describe the position of the origin in our fundamental alcove.
We can now define our basic building blocks for paths.
Given all of the above, we can finally define our distinguished paths for general words in our alphabet. There will be one such path for each word in our alphabet, and they will be defined by induction on the degree of the word, as follows.
Definition 3.7. We now define a distinguished path P w for each word w in our alphabet S ∪ {1} by induction on the degree of w. If w is s ∅ or a simple reflection s α we have already defined the distinguished path in Definition 3.6. Otherwise if w = s α w then we define If w is a reduced word in S hl , then the corresponding path P w is a reduced path.
Remark 3.8. Contextualised concatenation is not associative (if we wish to decorate the tensor products with the corresponding elements w). As we will typically be constructing paths as in Definition 3.7 we will adopt the convention that an unbracketed concatenation of n terms corresponds to bracketing from the right: We will also need certain reflections of our distinguished paths corresponding to elements of Π.
Definition 3.9. Given α ∈ Π we set .., +ε h ) bα (ε i ) bα the path obtained by reflecting the second part of P α in the wall through which it passes.
Example 3.10. We illustrate these various constructions in a series of examples. In the first two diagrams of Figure 14, we illustrate the basic path P α and the path P α and in the rightmost diagram of Figure 14, we illustrate the path P ∅ . A more complicated example is illustrated in Figure 11, where we show the distinguished path P w for w = s ε 3 −ε 1 s ε 2 −ε 1 s ε 3 −ε 2 s ε 3 −ε 1 s ε 2 −ε 1 s ε 3 −ε 2 as in Figure 11. The components of the path between consecutive black nodes correspond to individual P α s. Figure 14. The leftmost two diagrams picture the path P α walking through an α-hyperplane in E + 1,3 , and the path P α which reflects this path through the same α-hyperplane. The rightmost diagram pictures the path P ∅ in E + 1,3 . We have bent the paths slightly to make them clearer.
3.2. The principal block of H σ n . We now restrict our attention to regular blocks of H σ n . In order to do this, we first recall that we consider an element of the quiver Hecke algebra to be a morphism between paths. The easiest elements to construct are the idempotents corresponding to the trivial morphism from a path to itself. Given α a simple reflection or α = ∅, we have an associated path P α , a trivial bijection w Pα Pα = 1 ∈ S bαh , and an idempotent element of the quiver Hecke algebra e Pα := e res(Pα) ∈ H σ bαh . More generally, given any w = s α (1) s α (2) . . . s α (k) , we have an associated path P w , and an element of the quiver Hecke algebra e Pw := e res(Pw) = e P α (1) ⊗ e P α (2) ⊗ · · · ⊗ e P α (k) ∈ H σ hb α (1) +···+hb α (k) We let Std n,σ (λ) be the set of all standard λ-tableaux which can be obtained by contextualised concatenation of paths from the set We let P h (n, σ) = {λ ∈ P h (n) | Std n,σ (λ) = ∅}. We let Std n,σ = ∪ λ∈P h (n,σ) Std n,σ (λ). For example, the path in Figure 11 is equal to P α ⊗ P γ ⊗ P β ⊗ P α ⊗ P γ ⊗ P β . We define f n,σ = S∈Stdn,σ(λ) λ∈P h (n,σ) e S (3.1) and the remainder of this paper will be dedicated to understanding the algebra In fact, we will provide a concise list of generators for this truncated algebra (in the spirit of [EW16]) and rewrite the basis of Theorem 2.13 in terms of these generators.
In this section, we use our concrete branching coefficients to define the "Soergel 2-generators" of f n,σ (H σ n /H σ n y h H σ n )f n,σ explicitly. In the companion paper [BCH], we will show that these generators are actually independent of these choices of reduced expressions (however, this won't be needed here -we simply make a note, again, for purposes of consistency with [BCH]).
3.3. Generator morphisms in degree zero. We first discuss how to pass between paths P w and P w which are in different linkage classes but for which w and w have the same underlying permutation. Fix two such paths with α (1) , . . . , α (k) , β (1) , . . . , β (k) ∈ Π ∪ {∅}. We suppose, only for the purposes of this motivational discussion, that both paths are reduced. In which case, we have that w ∈ S h and so the expressions w and w differ only by applying Coxeter relations in of S h and the trivial "adjustment" relation s i 1 = 1s i (made necessary by our augmentation of the Coxeter presentation). Moreover, w and w are both reduced expressions and so we need only apply the "hexagon" relation s i s i+1 s i = s i+1 s i s i+1 and the "commutation" relation s i s j = s j s i for |i − j| > 1. The remainder of this subsection will be dedicated to lifting these path-morphisms to the level of generators of the KLR algebra. We stress that one can apply these adjustment/hexagon/commutator path-morphisms to any paths (not just reduced paths) but the reduced paths provide the motivation.
3.3.2. The KLR hexagon diagram. We wish to pass between the two distinct paths around a vertex in our alcove geometry which lies at the intersection of two hyperplanes labelled by non-commuting reflections. To this end, we let α, β ∈ Π label a pair of non-commuting reflections. Of course, one path around the vertex may be longer than the other. Thus, we have two cases to consider: if b α b β then we must pass between the paths P αβα and P ø−ø ⊗ P βαβ and if b α b β then we pass between the paths P ø−ø ⊗ P αβα and P βαβ , where here ø − ø := ∅ bα−b β . Figure 17. We let h = 3, = 1, e = 5 and α = ε 3 − ε 1 and β = ε 1 − ε 2 and γ = ε 2 − ε 3 . The paths P αβα , P βαβ , P γβγ and P βγβ are pictured.
We define the KLR-hexagon to be the element hex αβα βαβ := Υ P αβα P ø−ø ⊗P βαβ or hex αβα βαβ := Υ P ø−ø ⊗P αβα P βαβ for b α b β or b α b β respectively. Two such pairs of paths are despited in Figure 17. For the latter pair, the corresponding KLR-hexagon element is depicted in Figure 18.

3.4.
Generator morphisms in non-zero degree. We have already seen how to pass between S, T ∈ Std n,σ (λ) any two reduced paths. We will now see how to inflate a reduced path to obtain a non-reduced path. Given S, T ∈ Std n,σ (λ), we suppose that the former is obtained from the latter by inflating by a path through a single hyperplane α ∈ Π. Of course, since S and T have the same shape, this inflation must add an P α at some point (and will involve removing an occurrence of T ø in order to preserve n). There are two ways which one can approach a hyperplane: from above or from below. Adding an upward/downward occurrence of P α corresponds to the spot/fork Soergel generator.
which is of degree +1 (corresponding to the unique step of off the α-hyperplane). We have already constructed an example of an element spot ø α in great detail over the course of Examples 2.6, 2.8 and 2.10 and Figure 13. 3.4.2. The fork morphism. We wish to understand the morphism from P α ⊗ P α to P ø ⊗ P α . We define the KLR-fork to be the elements fork øα αα := Υ Pø⊗Pα Pα⊗P α The element fork øα αα is of degree −1.

3.5.
Light leaves for the Bott-Samelson truncation. We now rewrite the truncated basis of Theorem 2.13 in terms of the Bott-Samelson generators (thus showing that these are, indeed, generators of the truncated algebra). Of course, the idempotent of equation (3.1) is specifically chosen so that the truncated algebra f n,σ (H σ n /H σ n y h H σ n )f n,σ has basis indexed by the (sub)set of alcove-tableaux (and this basis is simply obtained from that of Theorem 1.24 by truncation). It only remains to illustrate how the reduced-path-vectors can be chosen to mirror the construction of paths in Std n,σ (λ) via concatenation.
one another by some iterated application of hexagon, adjustment, and commutativity permutations. We let rex Pv Pw denote the corresponding path-morphism in the algebras H σ n /H σ n y h H σ n (so-named as they permute reduced expressions). In the following construction, we will assume that the elements c S T exist for any choice of reduced path S . We then extend S using one of the U 0 , U 1 , D 0 , and D 1 paths (which puts a restriction on the form of the reduced expression) but then use a "rex move" to obtain cellular basis elements "glued together" along an idempotent corresponding to an arbitrary reduced path.
Definition 3.11. Suppose that λ belongs to an alcove which has a hyperplane labelled by α as an upper alcove wall. Let T ∈ Std n,σ (λ). If T = T ⊗ P α then we inductively define c T P = (c T P ⊗ e Pα )rex P ⊗Pα P .
Proof. Suppose that Q, U ∈ Std k,σ (ν) with Q reduced and k < n divisible by h. By induction, we may assume that c Q U = Υ P U U for some reduced path vector P U such that P U = (P U,0 , P U,1 , . . . , P U,k ) with P U,k = Q. By Theorem 2.13 and our inductive assumption, the result holds for all k < n divisible by h. Now suppose that λ ∈ P h (n, σ) and that λ belongs to an alcove, A λ , which has a hyperplane labelled by α and that µ = λ·s α . We now reconstruct the element c P T in terms of the basis of modified branching coefficients (as in Theorem 2.13) with P := P λ reduced and T equal to either U ⊗ P α or U ⊗ P α . This amounts to defining a reduced path vector, P T = (P U , P T,k+1 , P T,k+2 , . . . , P T,n ) for which c P T = Υ P T T . To do this, we simply set (Q ⊗ P α ) ↓ j if T = U ⊗ P α and k < j < n (Q ⊗ P α ) ↓ j if T = U ⊗ P α and k < j < n P if j = n.
To summarise: we incorporate the "rex" move into the final branching coefficient (and all other branching coefficients are left unmodified). Choosing the reduced path vectors in this fashion, we obtain the required basis as a special case of Theorem 2.13 .
We have shown that we can write a basis for our algebra entirely in terms of the elements e Pα , fork αø αα , spot ø α , hex βαβ αβα , com γβ βγ , e P ∅ , and adj ∅α α∅ for α, β, γ ∈ Π such that α and β label an arbitrary pair of non-commuting reflections and β and γ label an arbitrary pair of commuting reflections. Thus we deduce the following: Corollary 3.13. Theorem B of the introduction holds.