Degenerate 0-Schur Algebras and Nil-Temperley-Lieb Algebras

In Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014) constructed 0-Schur algebras, using double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver presentation naturally gives rise to a new class of algebras, which are introduced and studied in this paper. That is, these algebras are defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters t. In particular, when all the entries of t are 1, we recover 0-Schur algebras. When all the entries of t are zero, we obtain a class of basic algebras, which we call the degenerate 0-Schur algebras. We prove that the degenerate algebras are both associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014), and show how the centralizer algebras are related to nil-Hecke and nil-Temperley-Lieb algebras.


Introduction
It is well known that the classical Schur algebras are specialisations of q-Schur algebras (see [5] and [6]) at q = 1. Analogously, 0-Schur algebras are specialisations of q-Schur algebras at q = 0. The 0-Schur algebras have been studied by Donkin [6, §2.2] in terms of 0-Hecke algebras of symmetric groups, by Krob and Thibon [13] in connection with noncommutative symmetric functions, and by Deng and Yang on their presentations and representation types [3,4].
A new approach towards 0-Schur algebras was investigated by Jensen and Su [11], by considering their monoid structure. Inspired by Beilinson, Lusztig and MacPherson's geometric construction of q-Schur algebras [1] and Reineke's work on a monoid structure of Hall algebras [16], Su defined a generic multiplication in the positive part of q-Schur algebras [19]. The generic multiplication was then generalized by Jensen and Su [11] to give a global geometric construction of 0-Schur algebras. This geometric construction produces a monoid structure, simplifies the multiplication and provides a new approach to studying the structure of 0-Schur algebras. In [12] we gave a construction of indecomposable projective modules and studied homomorphism spaces between projective modules.
The nature of 0-Schur algebras exposed in [11] leads to several interesting related algebras. First of all, we can modify the generating relations of 0-Schur algebras, relying on multiple parameters t. In particular, when all the parameters are 1, we recover 0-Schur algebras and when all parameters are 0, we obtain a class of basic algebras. We prove that similarly to 0-Schur algebras, these newly defined algebras have reduced paths as basis (see [12]), which are in one-to-one correspondence with GL(V )-orbits in double flag varieties or certain sets of integral matrices. So they have the same dimension as the corresponding 0-Schur algebras and they are proved to be degenerations of 0-Schur algebras. We further show that they are isomorphic to quotients of 0-Schur algebras by naturally defined ideals and to the associated graded algebras of 0-Schur algebras, which notably have a natural structure of filtered algebras. We also investigate relations between their centralizer algebras, Nil-Hecke and Nil-Temperley-Lieb algebras.
The remainder of this paper is organised as follows. In Section 2, we give a brief background on q-Schur and 0-Schur algebras and discuss how to consider 0-Schur algebras as filtered algebras. In Section 3, we prove some preliminary results on a family of idempotent ideals. In Section 4, we construct a class of algebras D t (n, r) using quivers and modified relations of 0-Schur algebras and prove an isomorphism theorem between different D t (n, r). In Section 5, we first construct the associated graded algebras DS 0 (n, r) of 0-Schur algebras and give them a geometric interpretation. We then show that D t (n, r) for t = 0 is a degenerate 0-Schur algebra and prove our main result that the three algebras, DS 0 (n, r), D 0 (n, r) and the quotient of S 0 (n, n + r) modulo a natural idempotent ideal are isomorphic. In Section 6, we discuss relations between centralizer algebras of the degenerate algebras, nil-Hecke and nil-Temperley-Lieb algebras.
2 Background on the Algebra S 0 (n, r)

The Algebra S q (n, r)
We first recall the construction of quantised Schur algebras given by Beilinson-Lusztig-MacPherson [1]. Let k be a field and let V be a k-vector space of dimension r. Let F = F(V ) be the set of n-step flags The natural action of GL(V ) on the vector space V induces a diagonal action of GL(V ) on Let (n, r) be the set of n × n-matrices A = (a ij ) i,j with a ij nonnegative integers and 1≤i,j ≤n a ij = r. Then there is a bijection from F × F/GL(V ) to (n, r) sending the orbit of (f, f ) to A = (a i,j ) i,j with such that for all finite fields k, where |X| denotes the cardinality of a set X. Following a remark by Du [7], the q-Schur algebra studied by Dipper and James in [5] (see also [6]) can now be defined as follows.
Definition 2.1 [1] The quantised Schur algbra S q (n, r) is the free Z[q]-module with basis {e A | A ∈ (n, r)} and multiplication given by e A · e B = C∈ (n,r) g A,B,C (q)e C , for all A, B ∈ (n, r).
For an n × n-matrix A = (a ij ), define the row and column vectors of A by Note that if g A,B,C (q) = 0, then ro(A) = ro(C), co(A) = ro(B) and co(B) = co(C).

Definition of S 0 (n, r)
In [11], Jensen and Su defined a generic multiplication of orbits in where V is defined over an algebraically closed field k and e C is the unique open orbit in π −1 (e A × e B ). This defines an associative Z-algebra G(n, r) with basis (n, r) and in fact G(n, r) is isomorphic to the 0-Schur algebra (Theorem 7.2.1 in [11]) where Z is viewed as the Z[q]-module Z[q]/(q). As the multiplication in G(n, r) is much simplified (e.g. the multiplication of two orbits is either 0 or again an orbit), in the rest of this paper, we will take G(n, r) as the 0-Schur algebra S 0 (n, r).

Remark 2.2
We will see below (Lemma 2.3, 2.4 and 2.7) that the multiplication can be described combinatorially, independent of the choice of k.
Let (n, r) be the set of compositions of r into n parts. For each λ = (λ 1 , . . . , λ n ) ∈ (n, r), let diag(λ) denote the diagonal matrix diag(λ 1 , . . . , λ n ) and write k λ = e diag(λ) . By definition, for each A ∈ (n, r), Thus, λ∈ (n,r) k λ is the identity of S 0 (n, r). Let E ij be the elementary n × n matrix with a single nonzero entry 1 at (i, j ). Denote by e i,λ (resp. f j,λ ) the basis element of S 0 (n, r) corresponding to the matrix that has column vector λ and the only nonzero off diagonal entry is 1 at (i, i + 1) (resp. (j + 1, j)).

The Fundamental Multiplication Rules
Note that S 0 (n, r) is generated by e i,λ , f i,λ and k λ , where 1 ≤ i ≤ n − 1 and λ ∈ (n, r) (see Lemma 6.9 in [11]). Let This says that only one term remains in the product and the same holds for e A e i , e A f i and f i e A . The following are the fundamental multiplication rules in S 0 (n, r), which describe the action of generators on basis elements. [11,Lemma 6.11] Let e A ∈ S 0 (n, r) with ro(A) = λ.

Lemma 2.3
Symmetrically, there are the following formulas.

Presenting S 0 (n, r) by Quiver with Relations
Let α i = (0, . . . , 0, 1, −1, 0, . . . , 0) ∈ Z n , where the only nonzero entries 1 and −1 are at the ith and (i + 1)th positions, respectively. We define a quiver (n, r) with vertices corresponding to λ ∈ (n, r) and arrows and let J be the ideal of the path algebra Z (n, r) generated by the binomial relations i e j − e i e j e i for i = j − 1, −e i e j e i + e j e 2 i for i = j + 1, e i e j − e j e i otherwise, Note that the ideal J is not admissible, as the relations C ij involve idempotents. The quotient algebra Z (n, r)/J is isomorphic to S 0 (n, r), by an isomorphism that maps the arrows e i,λ , f i,λ and vertices k λ to the corresponding elements in S 0 (n, r) (Theorem 7.1.2 in [11]). The relations P ij,λ , N ij,λ are called the Serre relations of S 0 (n, r). The relations C ij,λ are commutative relations if i = j or λ i λ i+1 = 0, idempotent relations if exactly one of λ i , λ i+1 is zero, and empty relations otherwise.

Bases of Reduced Paths and S 0 (n, r) as a Filtered Algebra
We say that a path in (n, r) is reduced if it is not equal to a path of shorter length in (n, r) modulo J . Equivalence classes of reduced paths form a multiplicative basis for S 0 (n, r), which coincides with the basis e A , A ∈ (n, r). There are in general many paths equal to e A , modulo J , but by the fundamental multiplication rules and the presentation of S 0 (n, r), the numbers of occurrences of e i and f i in any reduced path are

Lemma 2.5 For any
Proof The generating relations P ij and N ij do not change the length of a path. The length of a path either remains unchanged or decreases by 2, when we apply the relation C ii,λ . It decreases if and only if λ i = 0 or λ i+1 = 0 (but not both), as in these cases the relation C ii,λ becomes k λ f i e i k λ = k λ if λ i = 0 and λ i+1 = 0; or k λ e i f i k λ = k λ if λ i+1 = 0 and λ i = 0. In this situation, the numbers of the occurrences of e i and f i both decrease by 1. Therefore

Corollary 2.6 S 0 (n, r) is a filtered algebra with degree function e A → (E(e A ), F (e A )).
We give two explicit descriptions of reduced paths equal to a basis element e A , analogous to the monomial and PBW bases of S q (n, r), respectively.
Proof The formulas follow by repeatedly applying the fundamental multiplication rules in Lemma 2.3. In both (1) and (2), the numbers of occurrences of e i and f i are E i (e A ) and F i (e A ). So the paths are reduced.
We explain by an example on how to achieve the above formulas.
Note that e B = k co(A) . Using the fundamental multiplication rules, we have the following. That is, 13 1 · e (a 13 +a 23 ) 2 · e a 12 1 · f That is, In both cases, the lower triangular parts/upper triangular parts are created column-wise. The main differences are, for instance in the lower triangular parts, in (2) each step creates an entry a ij at a time, starting from the lowest entry and then moving upwards, while in (1) it goes downwards and in each step we apply the maximal times of f i so that afterwards we have an exactly right entry in row i.

Idempotents and Idempotent Ideals of S 0 (n, r)
The set of compositions (n, r) (i.e. the vertices in the quiver (n, r)) can be drawn on an (n − 1)-simplex, where compositions with zero entries lie on the boundary. We call an idempotent k λ boundary if λ lies on the boundary of the simplex, and interior otherwise. In this section, we are interested in the ideal generated by all boundary idempotents. We obtain a dimension formula for the quotient algebra, which turns out to be useful when we consider degenerations of 0-Schur algebras in Section 5.
Let I i (n, r) be the ideal generated by the idempotents k λ with the number of nonzero entries in λ less than or equal to i. That is, I i (n, r) is generated by idempotents corresponding to (i − 1)-faces of the simplex. There are in general several (i − 1)-faces in (n, r), but as the following lemma shows, any one of them contains enough idempotents to generate the whole of I i (n, r).

Lemma 3.1 The ideal I i (n, r) is generated by all idempotents k λ lying in any chosen
Proof For any λ = (. . . , λ i−1 , 0, λ i+1 , . . . ) ∈ (n, r), with the other entries equal to those of λ. Therefore k λ is contained in the ideal generated by k μ . Similarly, k μ is contained in the ideal generated by k λ , and so these two ideals are equal. This shows that I i (n, r) is generated by all idempotents k λ lying in any chosen (i − 1)-face in (n, r).
Denote by I (n, r) the ideal of S 0 (n, r) generated by all the boundary idempotents. If r ≥ n, then I (n, r) = I n−1 (n, r). If r < n, then I (n, r) = S 0 (n, r).

Lemma 3.2 The ideal I (n, r) has a basis consisting of e A , where A ∈ (n, r) is a matrix with at least one diagonal entry equal to 0.
Proof Denote by S the subspace of S 0 (n, r) spanned by e A with at least one diagonal entry of A equal to 0. We first show that S is an ideal. It is enough to prove that for any generator x and any e D ∈ S, both xe D and e D x are contained in S. We only prove that xe D ∈ S for x = e j and D = (d ij ) ij with d ii = 0, as the other case can be done similarly. Let e X = e j e D and X = (x ij ) ij . By Lemma 2.3, multiplying e j with e D from the left only affects two entries in the (j + 1)th-row and j th-row in D, respectively. We have either The latter case occurs only if d i+1,i+1 = 0 (and j = i + 1), but then x i+1,i+1 = 0 as well. In either case, xe D = e X ∈ S, as required. So S is an ideal in S 0 (n, r).
As I (n, r) is generated by idempotents k λ with λ i = 0 for some i, we have k λ ∈ S and therefore I (n, r) ⊆ S, since S is an ideal.
It remains to show that S ⊆ I (n, r). Suppose that e A ∈ S has the diagonal entry a ii = 0, for some 1 ≤ i ≤ n. We will show that e A ∈ I (n, r). Let B be the matrix with To complete the proof we will construct elements x and y such that xye B = e A , which implies that e A ∈ I (n, r). In fact, y is a product of generators e j for j > i and x is a product of f j for j ≤ i. Explicitly, for instance, we have We give an example to illustrate the construction in the proof above. Lemma 3.2 shows that I (n, r) is a summand of S 0 (n, r) as Z-modules, and so S 0 (n, r)/I (n, r) is a free Z-module. We have the following formulas which will be used Section 5. (2) rank S 0 (n, r)/I (n, r) = n 2 + r − n − 1 r − n .
Note that in part (1), each term in the sum counts the matrices with exactly s zero diagonal entries.

Modified Algebras of S 0 (n, r)
In the remainder of this paper, we will work with algebras defined over a field F. Note that the 0-Schur algebra S 0 (n, r) is a free Z-module. When we extend the ground ring Z to a field F, S 0 (n, r) ⊗ Z F, the results in Sections 2 and 3 still hold, where replace rank by dimension in Corollary 3.4. By abuse of notation, we continue to denote the F-algebra by S 0 (n, r).
Modifying the generating relations in Section 2.4, we construct a family of modified algebras D t (n, r) of S 0 (n, r) and prove when two modified algebras are isomorphic. Further properties will be discussed in the next section, in relation to degenerations of 0-Schur algebras and their geometric interpretation.
Let B(n, r) ⊆ (n, r) be the set of elements corresponding to boundary idempotents. That is, this is the set of λ such that there is some with each entry of t in F. Denote by J (t) the ideal of F (n, r) generated by P ij,λ , N ij,λ and C ij,λ (t), where P ij,λ and N ij,λ are defined as in Section 2.4 and where Note that if λ i = λ i+1 = 0, then C ij,λ (t) is an empty relation.  When i = j and either λ i = 0 or λ i+1 = 0, we have the following three cases For the first case, we have the commutative relation at k λ as follows For the second and the third case, λ is boundary and the relations are not admissible if t i = 0. When all the t i = 1, we recover the 0-Schur algebra S 0 (n, r). When all the t i = 0, D t (n, r) is a basic algebra, which we call the degenerate 0-Schur algebra.
Define the map t,s : D t (n, r) −→ D s (n, r) given by if t i s i = 0 and e i,λ → e i,λ otherwise. For a = (a 1 , a 2 , . . . , a n−1 ) ∈ F n−1 , we write t = a s if t i = a i s i for all 1 ≤ i ≤ n − 1. We have the following proposition.

Proposition 4.3 The map t,s defined above is an isomorphism of algebras if and only if the sets {i | t i = 0} and {i | s i = 0} coincide.
Proof If the sets {i | t i = 0} and {i | s i = 0} coincide, it can be easily verified that the maps s,t and t,s are well-defined and that they are mutually inverse. Therefore t,s is an isomorphism.
On the other hand, if the two sets do not coincide, then either t,s has a non-trivial kernel or is not a homomorphism of algebras. So it is not an isomorphism. In particular, if t i = 0 for all 1 ≤ i ≤ n − 1, then D t (n, r) ∼ = S 0 (n, r).

A New Algebra Defined on Double Flag Varieties
Let D t S 0 (n, r) be the F-space with basis {e A | A ∈ (n, r)}. Define a multiplication on D t S 0 (n, r) as follows, where t E(e X ) is the product of t E i (e X ) i and we use the convention that t 0 i = 1 when t i = 0. Following Lemma 3.2, let I (n, r) ⊆ D t S 0 (n, r) be the subspace with basis e A , where A has a zero on the diagonal. Lemma 5.1 D t S 0 (n, r) with the product t is an associative F-algebra. Moreover, I (n, r) is a two-sided ideal.
Proof We need only show that the product t is associative. We have

Similarly, e A t (e B t e C ) = t E(e A )+E(e B )+E(e C )−E(e A e B e C ) e A e B e C
and so the product is associative. As e A t e B and the multiplication e A e B in S 0 (n, r) only differ by a scalar, that I (n, r) is a two sided ideal in D t S 0 (n, r) follows from Lemma 3.2.
For the special case where t i = 0 for all i, we write for the product t , and we write DS 0 (n, r) for D 0 S 0 (n, r). By Lemma 3.2, the e C in Example 5.2 is contained in the ideal I (2,5). This indicate that there should be a link between DS 0 (n, r) and the quotient algebra S 0 (n, r)/I (n, r). We will explore the link and their relation to D 0 (n, r) in the remaining part of this section.

Lemma 5.3
The algebra D t S 0 (n, r) is generated by e i,λ , f i,λ and k λ , where λ ∈ (n, r) and 1 ≤ i ≤ n − 1.
Proof Recall that Lemma 2.7 expresses each basis element e A in S 0 (n, r) as reduced paths. Observe that the corresponding products in D t S 0 (n, r) determined by the reduced paths are not zero, as the products satisfy

E(e A ) + E(e B ) = E(e A · e B ).
So the lemma follows.

Recall the algebra D t (n, r) = F (n, r)/J (t)
defined in Section 4. Up to isomorphism (see Corollary 4.4), the difference between D t (n, r) and S 0 (n, r) is that some of the defining relations of S 0 (n, r) involving idempotents are replaced by zero relations. Recall also for j ≥ i, e(i, j ) = e i · e i+1 · ... · e j .

Lemma 5.4 Let
in D t (n, r).
Proof It suffices to prove that e(i, j ) · e l = e l · e(i, j ) for all l = i, · · · , j. In fact, due to the commutativity relation e m e m = e m e m when |m − m | > 1, we need only prove e(i, j ) · e l = e l · e(i, j ) for (i, j ) = (l − 1, l), (l − 1, l + 1), (l, l + 1). All the three cases follow from the relations P mm .
Proof Observe that in both algebras, a complete list of representatives of nonzero paths is a basis. In particular, the paths of the form in Lemma 2.7 (2), which are all reduced paths, form a basis for S 0 (n, r).
As the relations P ij and N ij are the same in both algebra, if two nonzero paths are equal in D t (n, r), they are equal in S 0 (n, r). Further, any reduced path that is nonzero in S 0 (n, r) is also nonzero in D t (n, r). Indeed, take a nonzero reduced path ρ = ...e i,λ ..f j,μ ... in (n, r). Now suppose that 0 = ρ ∈ D t (n, r). This implies that using relations P ij,λ , N ij,λ , with λ at the boundary. So the new expression ( †) also holds in S 0 (n, r). By the relations which contradicts the minimality of the number of arrows in ρ. So ρ is a non-zero path in D t (n, r). Therefore dim F D t (n, r) ≥ dim F S 0 (n, r).
Next we claim that any reduced path ρ in D t (n, r) is equal to a path of the form (2) in Lemma 2.7 and thus by the inequality (7), the dimensions of the two algebras are the same. We proceed with the proof of the claim by induction on the length of ρ. When the length of ρ is at most 1, then it already has the required form. Assume that the length is larger than 1, and that ρ = ρ · k λ . We have ρ = ρ e i k λ or ρ f i k λ where ρ is a path of length one less than ρ. By the induction hypothesis, we may write ρ of the form (2) in Lemma 2.7, ρ = EF k λ where E = e(i s , j s ) · .... · e(i 1 , j 1 ) and F = f (i 1 , j 1 ) · ... · f (i t , j t ). We first consider the case ρ = ρ e i k λ . Since ρ is reduced, we have ρ = EF e i k λ = Ee i F k λ .
If j m = i − 1 for all m, the relations P ij give us the required form where l is the smallest integer such that j l > i. Let m be the smallest integer with j m = i − 1. We have Ee i = e(i s , j s ) · .... · e(i m , j m + 1) · · · e(i 1 , j 1 ).
By repeatedly applying Lemma 5.4 and relations P ij , we can move e(i m , j m + 1) to the appropriate position. The case ρ = ρ f i k λ is similar, and so we skip the details. In either case, the path ρ is equal to a path of the form (2) in Lemma 2.7, as claimed.
We can now prove the first main result of this paper.

Theorem 5.6
There is an isomorphism of F-algebras Proof As the defining relations of D t (n, r) are satisfied in D t S 0 (n, r), the map is welldefined. By Lemma 5.3, the map is an epimorphism. Note that by definition, dim D t S 0 (n, r) = dim S 0 (n, r) and thus by Proposition 5.5, dim D t S 0 (n, r) = dim D t (n, r).
Thus the map is an isomorphism, as claimed.
Our second main result realises the algebra DS 0 (n, r) as a quotient of the algebra D t S 0 (n, r + n).
Further, following the isomorphism in Theorem 5.6, which maps the ideal I (n + r) of D t (n, r + n) into the ideal I (n, n + r) of D t S 0 (n, n + r), we have a surjective map D t (n, r + n)/I (n, r + n) D t S 0 (n, r + n)/I (n, r + n).
Applying the isomorphism in Theorem 5.6 for t = 0, we have surjective maps DS 0 (n, r) D t (n, r + n)/I (n, r + n) D t S 0 (n, r + n)/I (n, r + n).
with the dimensions of two end terms the same. Therefore we have the isomorphism as claimed.
The construction of the isomorphism in the previous theorem gives us the following inverse system of algebra surjections (similar to the analogous inverse system for q-Schur algebras when q is invertible [1]).

Corollary 5.8
There is a sequence of surjective algebra homomorphisms DS 0 (n, r) ← DS 0 (n, r + n) ← · · · ← DS 0 (n, r + kn) ← · · · Proof If t i = 0 for all i, then the previous theorem gives us the surjective composition DS 0 (n, r + kn) → DS 0 (n, r + kn)/I (n, r + kn) ∼ = DS 0 (n, r + (k − 1)n) for any k ≥ 0. In particular, D t S 0 (n, r) is a degeneration of S 0 (n, r) and DS 0 (n, r) is a degeneration of D t S 0 (n, r) for all t. (2) These degenerations can be understood as passing from a filtered algebra to its associated graded algebra. In particular, DS 0 (n, r) is the graded algebra associated to the filtered algebra structure on S 0 (n, r) given in Corollary 2.6.
We end this section by giving a more geometric interpretation of the multiplication in DS 0 (n, r) using double flag varieties. Suppose that with e A · e B = e C in S 0 (n, r). Recall that V is an r-dimensional vector space defined over a field k. We view an n-step flag in V as a representation of a linear quiver A n , where vertex 1 is a source and vertex n is a sink. Associated to such a representation f , we denote the dimension vector by dim f.
Proof Note that and f ∼ = f as representations. Now the lemma follows from Lemma 2.5.
Therefore the product can be defined as below.

Centralizer Algebras of D t (n, r)
In this section, we discuss relations between the 0-Hecke algebra H 0 (r), the nil-Hecke algebra NH 0 (r), the nil-Temperley-Lieb algebra NT L(r) and the degenerate 0-Schur algebra DS 0 (r, r). Throughout this section, we let α = (1, . . . , 1) be the composition in (r, r) with all the entries equal to 1.

The Algebras H 0 (r), NH 0 (r) and DS 0 (r, r)
We first recall some definitions and key facts. We then show that the associated graded algebras of H 0 (r) and NH 0 (r), with the filtration induced by the degree function defined in Section 2.5, are isomorphic to a subalgebra of k α DS 0 (r, r)k α .
Definition 6.1 [2,8] The 0-Hecke algebra, denoted by H 0 (r), is the F-algebra generated by T 1 , T 2 , . . . , T r−1 with defining relations ⎧ ⎨ ⎩ Theorem 6.2 [11,Theorem 10.4] The algebras H 0 (r) and k α S 0 (r, r)k α are isomorphic via the the map T i → k α f i e i k α . Definition 6.3 [10,17] The nil-Hecke algebra, denoted by NH 0 (r), is the unital F-algebra generated by T 1 , T 2 , . . . , T r−1 with defining relations Assume that w = s i 1 · · · s i t = s j 1 · · · s j t are reduced expressions in the symmetric group S r on r letters, where s i is the transposition (i, i + 1). As reduced expressions of w can be obtained from each other by using the braid relations only (see [14,20]), we have T i 1 · · · T i t = T j 1 · · · T j t in both H 0 (r) and NH 0 (r), and thus the element T w = T i 1 · · · T i t is well-defined in both algebras. One can also deduce that {T w | w ∈ S r } is a basis for both algebras. Further, and where : W → N ∪ {0} is the length function of elements in S r . By Theorem 6.2, the filtration of the 0-Schur algebra S 0 (r, r) discussed in Section 2.5 induces a filtration on H 0 (r). When the length equations in Eqs. 10 or 11 hold, the multiplications of T w and T w in H 0 (r) and NH 0 (r) are the same, so we also have a filtration on NH 0 (r). Further, together with the fact that DS 0 (r, r) is the associated graded algebra of S 0 (r, r), this implies the following.

Proposition 6.4
The associated graded algebras of H 0 (r) and NH 0 (r) are isomorphic to the algebra k α DS 0 (r, r)k α .
Define a degree function d for a Z-grading on k α DS 0 (r, r)k α given by where A is the permutation matrix corresponding to w. Then via the isomorphism in Theorem 6.2, we can reformulate the multiplication in k α DS 0 (r, r)k α as follows Note that in k α DS 0 (r, r)k α , some T w can not be expressed as a product in the T i .
Example 6.5 Consider the product T 1 T 2 T 1 , which has length 3 and so is non-zero in the nil-Hecke algebra. The corresponding element T s 1 s 2 s 2 has degree 2, therefore the product T 1 T 2 T 1 is zero in k α DS 0 (r, r)k α .
By inspecting the entries above the diagonal in the matrix A, we can reinterpret d(e A ) in terms of permutations as It is shown in [15], that any permutation can be written as a product of transpositions, such that the degrees add up. In other words, the corresponding product in k α DS 0 (r, r)k α is non-zero. Hence the set of elements T (i,j ) corresponding to transpositions (i, j ) generates the algebra k α DS 0 (r, r)k α .

Nil-Temperley-Lieb Algebras
The nil-Temperley-Lieb algebra NT L(r) is the quotient algebra of NH 0 (r) modulo the ideal generated by T i T i+1 T i for 1 ≤ i ≤ r − 2 (see e.g. [9]).

Lemma 6.6
The algebra NT L(r) has a basis consisting of T w , where w does not contain a subword of the form s i s j s i in any of its reduced expressions, for any i and j with |i −j | = 1.
Proof The lemma follows from the fact that {T w |w ∈ S r } is a basis of NH 0 (r) and the definition of NT L(r).
Note that the dimension of NTL(r) is the Catalan number 1 r+1 2r r and there is a useful combinatorial parametrisation of the elements in a basis of NT L(r), using Dyck words. Pictorially, Dyck words can be described using peak pictures in a triangle with r dots on each edge (cf [18]

Let
x i = k α f i e i k α ∈ k α DS 0 (n, r)k α . Denote by DS(r, α) the subalgebra of k α DS 0 (r, r)k α generated by x i for 1 ≤ i ≤ r − 1. When r = 3, the algebra DS(r, α) is five dimensional, the orbit basis elements are determined by the following matrices. Similar to T w , x w = x i 1 . . . x i t is well-defined, where s i 1 . . . s i t is a reduced expression of w. By direct computation following the fundamental multiplication rules and the definition of DS 0 (n, r), we have the following lemma.

Lemma 6.7
The elements x i for 1 ≤ i ≤ r − 1 satisfy the generating relations of NT L(r).

Consequently, there is an epimorphism
Any matrix that determines a nonzero orbit in DS(r, α) is a permutation matrix. We call a nonzero entry that is below the diagonal a peak entry. For a peak entry (i, j ), which implies that j < i, we call the entries (j, j ) and (i, i) the feet of the peak. For the five matrices above, which determine the orbit basis elements in DS(r, α), we connect the peaks, feet and diagonal 1-entry, using zig-zag lines as follows. In this way, we obtain a well defined one-to-one correspondence between the basis elements of NT L(3) and DS (3, α). In fact, such a one-to-one correspondence exists for all r. Theorem 6.8 The two algebras NT L(r) and DS(r, α) are isomorphic.
Proof First note the nonzero orbits e A in DS(r, α) form a basis and each orbit e A is uniquely determined by the matrix A. By Lemma 6.7, there is a surjective homomorphism from NT L(r) to DS(r, α). So it suffices to show that there exists a nonzero element generated by the x i s such that the corresponding matrix produces the peak picture.
First identify the triangle, in which we draw the peak pictures, with the lower triangular part of an r × r matrix. The peaks give us peak entries, (i 1 , j 1 ), . . . , (i s , j s ) where for any l, j l < i l < i l+1 and j l < j l+1 .
By the fundamental multiplication rules, the multiplication of x i+1 with x i . . . x j s k α moves a 1 on or above the diagonal further up in the same column and a 1 on or below the diagonal further down in the same column and thus That is, the equality in the definition of holds. Therefore, Similarly, x (l) (x (l+1) . . . x (s) k α ) = 0 and when compared with x (l+1) . . . x (s) k α , it has a new peak at (i l , j l ). Thus we have found a nonzero element x = xk α such that the corresponding matrix gives us the required peak picture.
Example 6.9 In this example, we demonstrate the process of obtaining the matrix 0 0 1 1 0 0 0 1 0 which as indicated has two peaks, using the construction in the proof of the theorem. The two peak entries are (2, 1) and (3,2). By definition, x (2) = x 2 and x (1) = x 1 .
Then x (2) k α and x (1) x (2)  We have the following special version of Lemma 5.11. Lemma 6.10 Use the same notation Lemma 5.11. Assume that all the flags are isomorphic to ⊕ n i=1 P i and [f, g] = x i for some i. Then the following is true in DS(r, α) Proof Note that as a representation, g/(f ∩ g) is isomorphic to the simple representation of the linear quiver A n at vertex i, since [f, g] = x i . Now the lemma follows from Lemma 5.10 and 5.11.

Remark 6.11
In the light of Lemma 6.10, Theorem 6.8 gives a geometric realisation of nil-Temperley-Lieb algebras, via double flag varieties.