A geometric realisation of 0-Schur and 0-Hecke algebras

We define a new product on orbits of pairs of flags in a vector space, using open orbits in certain varieties of pairs of flags. This new product defines an associative $\mathbb{Z}$-algebra, denoted by $G(n,r)$. We show that $G(n,r)$ is a geometric realisation of the 0-Schur algebra $S_0(n, r)$ over $\mathbb{Z}$, which is the $q$-Schur algebra $S_q(n,r)$ at q=0. We view a pair of flags as a pair of projective resolutions for a quiver of type $\mathbb{A}$ with linear orientation, and study $q$-Schur algebras from this point of view. This allows us to understand the relation between $q$-Schur algebras and Hall algebras and construct bases of $q$-Schur algebras, which are used in the proof of the main results. Using the geometric realisation, we construct idempotents and multiplicative bases for 0-Schur algebras. We also give a geometric realisation of 0-Hecke algebras and a presentation of the $q$-Schur algebra over a base ring where $q$ is not invertible.


Introduction
Let k be a finite or an algebraically closed field and F the variety of partial n-step flags in an r-dimensional vector space V over k. Denote by |k| the cardinality of k. In [1], using the double flag variety F ×F, Beilinson, Lusztig and MacPherson gave a geometric construction of some finite dimensional quotients of the quantised enveloping algebra U q (gl n ). In [11], Du remarked that the quotients are isomorphic to the q-Schur algebras defined by Dipper and James in [7]. So in this paper the q-Schur algebras S q (n, r) are defined as the quotients constructed in [1], which we recall below.
Note that the natural GL(V )-action on V induces a GL(V )-action on the flag variety F and a diagonal GL(V )-action on the double flag variety F × F. Denote by [f, g] the GL(V )-orbit of (f, g) ∈ F × F and by F × F/GL(V ) the set of GL(V )-orbits on F × F. Let Δ and π be the maps given by Δ(f, f , f ) = ((f, f ), (f , f )) and π(f, f , f ) = (f, f ).
Let Z[q] be the ring of polynomials in q over Z, where q is an indeterminate. The q-Schur algebra S q (n, r) is a free Z[q]-module with basis F × F/GL(V ) and with multiplication where F f,g,h,l,f ,l is the polynomial in Z [q] such that F f,g,h,l,f ,l (|k|) is the cardinality of the set for any finite field k. The multiplication implies in particular that if g and h are not in the same GL(V )-orbit, then [f, g][h, l] = 0.
The main goal of this paper is to give a geometric realisation of the 0-Schur algebra S 0 (n, r), which is the q-Schur algebra S q (n, r) at q = 0. We define a new Z-algebra G(n, r) with basis F × F/GL(V ) by defining the product of [f, f ] and [f , f ] to be the unique open orbit (see Section 6) in The definition of the new product is similar to the one defined by Reineke [20] for Hall algebras and the following main result generalises Theorem 2.3 in [23]. Theorem 1. As Z-algebras, G(n, r) is isomorphic to S 0 (n, r). Remark 2. Throughout, S q (n, r) is a Z[q]-algebra, G(n, r) and S 0 (n, r) are Z-algebras and S v (n, r) = S q (n, r) ⊗ Z[q] Q(v) is a Q(v)-algebra, where v 2 = q. In these cases, we don't always emphasise the ground rings Z[q], Z and Q(v). We will occasionally consider q-Schur algebras over other ground rings, which will then be specified. For instance, if R is a commutative Z[q]-algebra, then by applying the functor − ⊗ Z[q] R we obtain a q-Schur algebra S q (n, r) ⊗ Z[q] R with the ground ring R. We also say that this q-Schur algebra is an R-algebra to emphasise the ground ring R.
We intend to understand the algebras from the viewpoint of representation theory of quivers. Where it is possible, we give explanations using representations. In particular, we view a flag as a projective representation of the linear quiver of type A n . In this way, a pair of flags (f, g) naturally determines a pair of projective resolutions f ∩ g ⊆ f and f ∩ g ⊆ g. We will show that a pair of flags and its corresponding pair of projective resolutions uniquely determine each other and give a criterion, using representations, for when two pairs of flags are in the same orbit. Applying the criterion, we construct two new bases for S q (n, r), Using the new bases and the main result, we give new presentations of q-Schur algebras over a commutative Z[q]-algebra Q, where q is not invertible. Using open orbits, we construct families of idempotents and an ideal M (n, r) ⊆ G(n, r), which splits off as a direct factor of the algebra G(n, r). In the case n = r, we obtain a geometric realisation of the 0-Hecke algebra H 0 (n).
The remainder of the paper is organised as follows. In Section 1, we explain the construction of S q (n, r) by Beilinson, Lusztig and MacPherson in more detail. In particular, we recall the description of GL(V )-orbits in F × F using matrices and the fundamental multiplication rules. In Section 2, we give a new description of the GL(V )-orbits in F × F using representations of linear quivers of type A n . In Section 3, we recall the definition of the positive and negative parts of the q-Schur algebras and their relationship to the Hall algebras. In Section 4, we construct new bases of S q (n, r). In Section 5, we describe q-Schur algebras over Q using quivers and relations and obtain presentations of the algebras, modified from the presentations given in [9] by Doty and Giaquinto. We define the generic algebra in Section 6 and show that it is isomorphic to the 0-Schur algebra in Section 7. In Section 8, we consider the degeneration order of orbits in F × F, and use open orbits to construct idempotents for the 0-Schur algebra in Section 9. Finally, we discuss 0-Hecke algebras in Section 10.

Flag varieties and q-Schur algebras
In this section, we fix notation and recall some definitions and results of Beilinson, Lusztig and MacPherson on q-Schur algebras in [1].
Let n, r ≥ 1 be integers and V an r-dimensional vector space over a field k. Denote by F the set of all n-steps flags in V . Let f and f be flags in F with We say that f is a subflag of f , denoted by f ⊆ f , if for all i, Denote the intersection of f and f by f ∩ f , which is the flag and the sum of f and f by f + f , which is the flag is a decomposition of r into n parts. Two flags are in the same GL(V )-orbit if and only if they have the same decomposition, in this case, we write f g.
Let Λ(n, r) denote the set of all decompositions of r in n parts, and let F α ⊆ F denote the orbit corresponding to α ∈ Λ(n, r).
This defines a bijection between the GL(V )-orbits in F × F and n × n matrices of non-negative integers with the sum of entries equal to r. We denote the GL(V )-orbit of (f, f ) by [f, f ] and by e A if we want to emphasise the matrix A = A(f, f ). If two pairs of flags (f, f ) and (g, g ) belong to the same GL(V )-orbit, we write Following Proposition 1.
This gives an associative algebra over Z[q] with basis F × F/GL(V ). Denote this algebra by S q (n, r). Du proved in [11] that S q (n, r) is isomorphic to the q-Schur algebras defined by Dipper and James in [7]. Although, in general it is difficult to compute the polynomial g A,A ,A , the following lemma from [1], dealing with special A and A , gives clear multiplication rules. Also, Deng and Yang give a recursive formula of g A,A ,A using Hall polynomials for any A and A [12]. Let for m ∈ N and let E i,j be the (i, j) elementary matrix.
Note that the classical Schur algebra [13] can be obtained by evaluating q = 1, i.e., the 0-Schur algebra S 0 (n, r), obtained by evaluating q = 0, is where v 2 = q.

Representations of linear quivers
In this section, we describe orbits of pairs of flags using representations of the linear quiver Λ of type A n , We first recall a few definitions regarding representations of quivers. Given a quiver Q with the set of vertices Q 0 = {1, . . . , n} and the set of arrows Q 1 , we denote a representation of Q by where each X i is a vector space and each X i,j is a linear map from X i to X j . A homomorphism h : X → Y between two representations X and Y is a collection of linear maps if X and Y are isomorphic. The direct sum X ⊕ Y of representations X and Y is the representation with A non-zero representation is indecomposable if it is not isomorphic to a direct sum of non-zero representations. We denote the dimension vector of X by Given a vector d ∈ Z n ≥0 , define the representation variety of Q, parameterising representations of Q of dimension vector d, to be The group G = i GL(d i ) acts on Rep(Q, d) by conjugation, i.e. for g = (g 1 , . . . , g n ) ∈ G and X = (X 1,2 , . . . , X n−1,n ) ∈ Rep(Q, d), where each α ij is a non-negative integer. For each vertex i, let S i = M ii and P i = M in be the simple and indecomposable projective representation associated to i, respectively. A representation P is projective if and only if each map P i,i+1 is injective.
For any vector α ∈ Z n ≥0 , let P (α) be the projective representation defined by Then is a flag in P (α) n . Any projective representation is isomorphic to P (α) for some α and so we can view a projective representation as a flag. Conversely, an n-step flag can be naturally viewed as a projective representation of Λ with the natural inclusion V i → V i+1 the linear map on the arrow i → i + 1. Two flags are in the same GL(V )-orbit if and only if they are isomorphic as representations. So if two flags are in the same GL(V )-orbit, we also say that they are isomorphic. If f is a flag in U and f is a flag in U , then f ⊕ f denotes the flag in U ⊕ U with vector space at ith step equal to U i ⊕ U i . A pair of flags (g, f ) with g ⊆ f can be viewed as a projective resolution (f 1 , f 2 ), i.e. they are in the same GL(V )-orbit, if and only if f 2 /f 1 f 2 /f 1 and f 2 f 2 , as representations of Λ. This fact generalises to arbitrary pairs.
Proof. The implication from i) to ii) is trivial.
We prove that ii) implies i). By ii), Consider the following diagram, where π and π are natural projections. Since f 1 + f 2 and f 1 + f 2 are isomorphic projective representations, there is an isomorphism h such that the above diagram commutes. Thus Similarly h(f 2 ) = f 2 . Hence (f 1 , f 2 ) and (f 1 , f 2 ) are in the same orbit. This proves i). iii) is a reformulation of ii). So the proof is done. 2 Note that the isomorphism of two pairs of flags can also be characterised using the inclusions f 1 ∩ f 2 ⊆ f i and the isomorphisms ( The , we can construct a corresponding pair (g 1 , g 2 ) as follows. Recall that a surjective homomorphism ψ : P → M from a projective representation P is a projective cover if ker ψ ⊆ rad P where rad P denotes the Jacobson radical of P . As inner direct sum, we have such that ψ |f 1 is a projective cover of M and ψ |f 2 is a projective cover of N , and then iii) g 1 + g 2 /g 2 g 1 /g 1 ∩ g 2 M and g 1 + g 2 /g 1 g 2 /g 1 ∩ g 2 N .
Given a pair of flags (f, f ) ∈ F × F, we have the following description of the matrix corresponding to the orbit [f, f ].
Proof. By definition, A ij is equal to the dimension of the space .
For i < j, which is the multiplicity of M i,j−1 as a direct summand in f 1 /f 1 . This proves i). Similarly, ii) holds. For i = j, As which is the multiplicity of M in as a direct summand in c. 2

The non-negative q-Schur algebras
In this section, we describe the non-negative part of a q-Schur algebra as a Hall algebra of projective resolutions of representations of the linear quiver Λ, defined in Section 2. We also include some easy lemmas on the computation of Hall numbers for the linear quiver which are needed in subsequent sections. An where f/f g/g and g ⊆ rad g. That is, such an orbit is determined by the minimal projective resolution of f/f The non-negative Z[q]-subalgebra S + q (n, r) is the subalgebra of S q (n, r) with basis consisting of all orbits Let M , N and L be representations of Λ. Recall that the Hall polynomial h L MN ∈ Z[q] defined by Ringel [21] is the polynomial such that h L MN (|k|) is equal to the number of subrepresentations X ⊆ L such that X N and L/X M for any finite field k.
for any finite field k. We need only to show that We will define two mutually inverse maps between U and S(A, A , A ). Given f 2 ∈ S(A, A , A ), we have the following commutative diagram of short exact sequences It is easy to check that these two maps are mutually inverse, and so the equality follows. 2 Denote the (non-twisted) Ringel-Hall algebra [21] by H q (Λ). That is, H q (Λ) is the free Z[q]-module with basis isomorphism classes [M ] of representations of Λ and multiplication Mapping representations to choices of projective resolutions induces an algebra homomorphism with kernel spanned by those [M ] with the number of indecomposable direct summands bigger than r [14].
There is a similar map Θ − : H q (Λ) → S − q (n, r). As a consequence, we have the following special case of Corollary 4.5 in [1] (see also Proposition 14.1 in [10]). The assumptions are as in Lemma 3.1.
This yields a different proof of Theorem 14.27 in [10], which we restate as follows.

Theorem 3.3. As Z[q]-algebras, the Hall algebra H q (Λ) is isomorphic to the algebra with basis consisting of all formal sums
and with multiplication induced by the multiplication in q-Schur algebras.
i.e., any extension of M i j by M ij splits. The next lemma follows easily from Lemma 3.1 and the corresponding computations in the Ringel-Hall algebra H q (Λ).
and m ij is the multiplicity of M ij as a subfactor in the filtration.

Bases of S q (n, r)
In this section, we describe a basis for S q (n, r) using the non-negative and non-positive subalgebras defined in the previous section. Let Proof. Suppose that [f 1 , f 2 ] is one of the terms with a non-zero coefficient in the sum Then there exists an f ∈ F such that ( There is a similar formula for the product [ We prove by induction on the dimension of f 1 + f 2 , considered as a representation of Λ, can be written as a Z[q]-linear combination of elements in B and therefore so can [f 1 , f 2 ]. This proves that B spans S q (n, r) as a Z[q]-module. On the other hand, note that the map from The converse holds, since the map π : The lemma shows that a surjective map , but the converse is not true. An example is given below.
with no surjective map g → g/f 1 ⊕ g/f 2 . In this case g f 1 + f 2 .

Quiver and relations for q-Schur algebras
In this section, we present an algebra using quivers and binomial relations, which will be shown to be the 0-Schur algebra in Section 7. This will lead to presentations of the q-Schur algebras over a ground ring, where q is not invertible. Also, following from the relations, the 0-Schur algebra has a multiplicative basis of paths, which will be constructed geometrically in Sections 6 and 7.

The quiver Σ(n, r)
Let i = (0, . . . , 0, 1, 0, . . . , 0) be the ith unit vector in Z n . Let Σ(n, r) be the quiver with vertices K α and arrows E i,α and F i,α , r). The vertices can be drawn on a simplex, where the vertices K α with α i = 0 for some i = 0 are on the boundary, and vertices K α with α i = 0 for all i are in the interior of the simplex.
For a commutative ring R, denote by RΣ(n, r) the path R-algebra of Σ(n, r), which is the free R-module with basis all paths in Σ(n, r), and multiplication given by composition of paths. The vertices K d form an orthogonal set of idempotents in RΣ(n, r) and the composition of two paths p and q is pq, if q ends where p starts, and zero otherwise. To simplify our formulas we define and finally, Recall that a relation in RΣ(n, r) is an R-linear combination of paths with common starting and ending vertex where r i ∈ R and p i is a path. Let I(n, r) ⊆ Z[q]Σ(n, r) be the ideal generated by the relations and Proof. By Lemma 1.1, the relations P ij , N ij and C ij hold in S q (n, r), and so φ is an algebra homomorphism. 2 We remark that the relations P ij and N ij hold in S q (n, r) also follows from Lemma 3.1 and the proposition in Section 2 of [22], and that the lemma can also be deduced from Lemma 5.6 in [1].
The homomorphism φ is not surjective in general, since for instance [m] is not invertible in Z[q]. So φ does not give a presentation of the q-Schur algebra over Z[q].

Change of rings
We need the following change of rings lemma for presentations of algebras using quivers with relations. The proof is similar to an argument at the end of Chapter 5 in [15]. Let ψ : R → S be a homomorphism of commutative rings, which gives S the structure of an R-algebra. Let Σ be a quiver, and let I ⊆ RΣ be an ideal. There are induced maps of R-algebras ψ : RΣ → SΣ and RΣ/I → SΣ/Sψ(I), where Sψ(I) ⊆ SΣ is the ideal generated by ψ(I).

q-Schur algebras over Q(v)
Let v be an indeterminate with v 2 = q and and by abuse of notation, in this proof we let r). Moreover, by a straightforward computation,Ẽ i ,F j ,K α satisfy the defining relations in Theorem 4' in [9] by Doty and Giaquinto if and only if E i , F j , K α satisfy the relations P ij , N ij and C ij . Therefore we have the isomorphism as required. 2 Since q is invertible in Q(v) and thus in S v (n, r), we cannot evaluate q = 0 in S v (n, r). We will modify the ground ring in the next subsection, so that q can be evaluated at 0.

A presentation of q-Schur algebra over Q
Let Q be the ring obtained from Z[q] by inverting all polynomials of the form 1 + qf (q). In particular, all [m] for m ∈ N are invertible. We have and q is not invertible in Q. So we can evaluate q = 0.
Proof. The image of φ ⊗Id Q is the subalgebra of S q (n, r) ⊗ Z[q] Q generated by the set of all e i,α , f i,α and k α .  Q, and so the following theorem gives a presentation of q-Schur algebras over Q and will be proven in Section 7.

The generic algebras
In this section let k be algebraically closed. We define a generic multiplication of orbits in F × F and obtain an associative Z-algebra G(n, r), which we call a generic algebra. This multiplication generalises the one for positive 0-Schur algebras in [23] and is similar to the product defined by Reineke [20] for Hall algebras. We also give generators for G(n, r) and find a standard decomposition of each basis element [f, g] into a product of the generators. Given two orbits e A and e A , define That is, S(A, A ) is the union of the orbits with non-zero coefficients in the product e A · e A in S q (n, r).
. We first show that S is irreducible. If f 2 f 3 then S is empty, and we are done. So we may assume that and is therefore irreducible. Now S(A, A ) = π(S), and so its closure is irreducible. 2 Since there are only finitely many orbits in S(A, A ), as a consequence of Lemma 6.1, we have the following corollary.

Corollary 6.2. There is a unique open GL(V )-orbit in S(A, A ).
We define a new multiplication  G(n, r) with the product is an associative Z-algebra.

Proof. We need only to show that is associative. That is, for any GL(V )-orbits [f
Following the definition, we see that if one side of the equality is zero, then so is the other side. We now suppose that both sides are not zero, that is, f 2 f 3 and f 4 f 5 . By the GL(V )-action on F, we may assume that f 2 = f 3 and f 4 = f 5 . Denote the sets by T 1 , T 2 , T 3 and T 4 , respectively. We have natural surjections The following is a direct consequence of the definition of the product in G(n, r).

Corollary 6.4. The set F × F/GL(V ) is a multiplicative basis of G(n, r).
In addition to the basis of G(n, r) consisting of orbits [f 1 , f 2 ] we can also consider bases analogous to the bases B and B defined in Section 4 for the q-Schur algebras. We show that these three bases of G(n, r) coincide. Similarly, We now prove that the Z-algebra G(n, r) is generated by the orbits e i,α , f i,α and k α . Recall that a representation X is said to be a generic extension of N by M , if the stabiliser of X is minimal among all representations that are extensions of N by M . Lemma 6.6. (See [23].

) Let f ⊇ g ⊇ h be flags. Then [f, h] = [f, g] [g, h] if and only if f/h is a generic extension of f/g by g/h.
For an interval [i, j] in {1, · · · , n} and α ∈ Λ(n, r) with α − j+1 non-negative, let M jj . Since f/h is a generic extension of f/g by the simple representation g/h, the lemma follows from Lemma 6.6 by induction. 2 Using the order ≤ on representations defined in Section 3, we can write each orbit [f, g] with f ⊇ g as a product over indecomposable summands of f/g.
Proof. The lemma follows from the vanishing of extension groups along the filtration and Lemma 6.6. 2 Lemma 6.9. The Z-algebra G(n, r) is generated by the orbits e i,α , f i,α and k α .
Proof. Lemma 6.7 and Lemma 6.8 imply that any orbit [f, g] with f ⊇ g is in the subalgebra of G(n, r) generated by e i,α and k α . Similarly, any orbit [f, g] with f ⊆ g is generated by f i,α and k α . The lemma now follows from Lemma 6.5. 2 Following Lemmas 6.5, 6.7 and 6.8, we obtain the following basis of G(n, r) in terms the generators e i,α and f i,α . Lemma 6.10. The Z-algebra G(n, r) has a basis consisting of all k α and all non-zero monomials Proof. First observe that any basis element [f, g] = [f, f ∩g][f ∩g, g] can be written as a monomial described in the statement. So we need only show that for any such monomial there is a unique orbit [f, g] such that [f, f ∩ g] = e(i s , j s , α s ) · · · e(i 1 , j 1 , α 1 ) and [f ∩ g, g] = f i 1 , j 1 , α 1 · · · f i t , j t , α t .
Let f = P (β) ⊕ Q(α ) ⊕ P (α ) and g = P (β) ⊕ P (α ) ⊕ Q(α ), then [f, g] is an orbit as required. The uniqueness is determined by the two quotients f/f ∩ g and g/f ∩ g and α 1 . 2 We compute the multiplication in G(n, r) of an arbitrary element with a generator.
Proof. We prove i). By Lemma 1.1, the orbit e X has a non-zero coefficient in the product e i,α ·e A in S q (n, r). Now, by Lemma 2.2 in [1], among all terms A + E i,j − E i+1,j with A i+1,j > 0, the elements in the orbit e X has the smallest stabiliser, and so e i,α e A = e X . The proof of ii) is similar. 2

A geometric realisation of the 0-Schur algebra
In this section we first give a presentation of G(n, r) using quivers and relations. Then we show that S 0 (n, r) and G(n, r) are isomorphic as Z-algebras by an isomorphism which is the identity on the closed orbits e i,α , f i,α and k α . Finally, we prove Theorem 5.4.2.

A presentation of G(n, r)
Let Σ(n, r), E i and F i be as in Section 5. Let and That is, P ij (0), N ij (0) and C ij (0) are obtained by evaluating P ij , N ij and C ij at q = 0. Let I 0 (n, r) ⊆ ZΣ(n, r) be the ideal generated by P ij,α (0), N ij,α (0), and C ij,α (0), which are obtained by evaluating P ij,α , N ij,α and C ij,α at q = 0.  Σ(n, r).
Theorem 7.1.2. The map η : ZΣ(n, r)/I 0 (n, r) → G(n, r) given by η( Proof. By Lemma 6.11, it is straightforward to check that e i,α , f i,α , and k α satisfy the relations P ij,d (0), N ij,d (0), and C ij,d (0). Thus η is well-defined. Also, Lemma 6.9 implies that the map is surjective. It remains to prove that η is injective. We claim that, modulo the relations in I 0 (n, r), any path p in Σ(n, r) is either equal to k α or a path of the form satisfying the conditions in Lemma 6.10. Note that such a path is mapped onto one of monomial basis elements in Lemma 6.10, and so the injectivity of η follows.
We prove the claim by induction on the length of p. If p has length less than or equal to one, it is equal to k α or one of the arrows F i,α and E i,α , and so the claim follows. Assume that p has length greater than one. Then we have where p is a non-trivial path of smaller length, and so by induction has the required form where E and F are products of the E(i a , j a , α a ) and F (i b , j b , α b ), respectively.
We first consider p = p E i,β . If p contains no F j,α , then the claim follows using the relations P ab,α (0). Otherwise, by the relations C ab,α (0), either p = EF with the length of F smaller than that of F or p = EE i,α 1 − i + i+1 F with each factor F (i l , j l , α l ) in F replaced with a factor F (i l , j l , β l ). In the first case, the claim follows by induction. Otherwise, by the relations P ab,α (0), there are two possibilities. First, there exists a minimal m with j m = i − 1.
We have Moreover, again using the relations P ab,α (0), the factors can be reordered (up to change of α l , β m ) to obtain a path of the required form.
with j m−1 ≤ i and j m > i. In order to show that this path is of the required form, we need only to prove the inequality Clearly, the inequality holds for each component different from i. Since there are no m with j m = i − 1, the sum l j l +1 contain no i . Since FE i,β = E i,β 1 F with the length of F equal to that of F , we must have (α 1 − i ) i ≥ ( l j l +1 ) i and so the inequality follows.
Finally, we consider p = p F i,β , where p is a path of the required form p = EF as above. If there are no factor E j,α in p , then the claim follows from the relations N ab,α (0). Otherwise p = E E j,α 1 FF i,β , which following C ab,α (0) is either p = E F with F not longer than F , or p = E F E j,β . In the first case, the length of the path E F is smaller than p in Σ(n, r) and so the claim follows by induction; in the second case, the claim is proved above. In either case, the claim holds. 2

A geometric realisation of S 0 (n, r)
We now prove the main result. So there is a surjective Z-algebra homomorphism ZΣ(n, r)/I 0 (n, r) → S 0 (n, r) given by The theorem now follows from Theorem 7.1.2, since G(n, r) = S 0 (n, r) as Z-modules. 2 Via the isomorphism in Theorem 7.2.1, the presentation of G(n, r) in Section 7.1 becomes a presentation of S 0 (n, r). We remark that Deng and Yang [6] have independently given a similar presentation for S 0 (n, r), using a different approach.

Proof of Theorem 5.4.2
By Proposition 5.4.1, the map φ ⊗ Id Q induces a short exact sequence As in the proof of Theorem 7.2.1, we have isomorphisms S q (n, r) ⊗ Z[q] Q ⊗ Q Q/qQ = S 0 (n, r) and QΣ(n, r)/QI(n, r) ⊗ Q Q/qQ ZΣ(n, r)/I 0 (n, r) .
Furthermore, via these two isomorphisms the map φ ⊗ Id Q ⊗ Id Q/qQ is the composition of the isomorphism ZΣ(n, r)/I 0 (n, r) G(n, r) in Theorem 7.1.2 and the isomorphism G(n, r) S 0 (n, r) in Theorem 7.2.1. Therefore Now by Nakayama's lemma (see Theorem 2.2 in [18]), there is an element r = 1 + qf (q) ∈ Q such that rK = 0. Since r is invertible in Q, we have K = 0. Thus φ ⊗ Id Q is an isomorphism.

The degeneration order on pairs of flags
In this section, let k be algebraically closed. We describe the degeneration order on GL(V )-orbits in F × F using quivers and the symmetric group S r .
Let Γ = Γ (n) be the quiver of type A 2n−1 , constructed by joining two linear quivers Λ L = Λ and Λ R = Λ at the vertex n. Often it will be clear from the context which side of Γ we are considering, and then we drop the subscripts on the vertices. For integers i, j ∈ {1, · · · , n}, let N ij be the indecomposable representation of Γ which is equal to the indecomposable projective representations M in and M jn when restricted to Λ L and Λ R , respectively. A representation N of Γ which is projective when restricted to Λ L and Λ R , and dim N n = r, decomposes up to isomorphism as We assume that j 1 ≤ j 2 ≤ · · · ≤ j r .
Let Since Γ is a Dynkin quiver, by a result of Bongartz [2], the degeneration ≤ deg is the same as the degeneration ≤ ext given by a sequence of extensions. That is, if there is an extension then M ≤ ext N ⊕ N , and more generally ≤ ext is the transitive closure. The symmetric group S r of permutations of the set {1, · · · , r} acts on representations with a decompo- The following facts are the key lemmas on degenerations in F × F. For the sake of completeness we include a brief sketch of the proofs. Proof. Assume that i t > i s . There is a short exact sequence Conversely, assume that i t ≤ i s . By comparing the dimensions of the stabilisers of N and (t, s)N we see where N N ⊕ N . We may choose summands N i s j s and N i t j t of N and N , respectively, such that taking pushout along the projection N → N i s j s and then pullback along the inclusion N i t j t → N gives us a non-split extension This extension is of the form of the extension in the proof of Lemma 8.1. Hence and so By the construction of M , so by the minimality of the degeneration, and so the lemma follows. 2 There is a unique closed orbit in F α × F β . We describe a corresponding representation. Conversely, if i l > i l+1 for some l, then N has a degeneration again by Lemma 8.1, and so the orbit of N is not closed. 2 Alternatively, we may prove the lemma by observing that among all representations of the form N = r l=1 N i l j l the representation with i l ≤ i l+1 has a stabiliser of maximal dimension, and so this representation has a closed orbit. The stabiliser in this case is a parabolic in GL(V ).
There is a unique open orbit in F α × F β with a corresponding representation given as follows. The proof is similar to the proof of the previous lemma. Similar to the closed orbit, a representation of the form N r l=1 N i l j l with i l ≥ i l+1 has a stabiliser of minimal dimension, and so the orbit is open. The stabiliser in this case is the intersection of two opposite parabolics in GL(V ). Such stabilisers are called seaweeds [5] (see also [16]). The stabiliser of an arbitrary pair of flags is equal to the intersection of two parabolics in GL(V ).
Let o α,β denote the unique open orbit and k α,β the unique closed orbit in F α ×F β . Then k α = k α,α and we let o α = o α,α . For τ ∈ S r , denote by τ o α,β the orbit of pairs of flags corresponding to the representation τ N, where N = r l=1 N i l j l with i l+1 ≤ i l is the representation corresponding to o α,β . Similarly, denote by τ k α,β the orbit corresponding to τ N, where N = r l=1 N i l j l with i l+1 ≥ i l is the representation corresponding to k α,β .

Idempotents from open orbits
Let M (n, r) be the Z-submodule of G(n, r) with basis the open orbits in F × F. In this section we prove that M (n, r) is a subalgebra G(n, r) that is also a direct factor. We also show that M (n, r) is isomorphic to the Z-algebra of |Λ(n, r)| × |Λ(n, r)|-matrices with integer entries, where |Λ(n, r)| is the cardinality of Λ(n, r).
We start with two lemmas relating degeneration and multiplication in G(n, r). Let ≤ deg be the degeneration order on orbits in (F × F) × (F × F) with the action of GL(V ) × GL(V ).
where k l+1 ≥ k l and j l+1 ≥ j l for the orbit e B . Similarly, o β,γ is the orbit corresponding to the representation where i l ≥ i l+1 by Lemma 8.4. Then the coefficient of σo β,δ in the product (σo β,γ )·e B in S q (n, r) is non-zero, and so Similarly, Proof. The corollary follows from the previous lemma, since where ι(i) = n − i + 1. Proof. The previous corollary shows that the Z-submodule M (n, r) ⊆ G(n, r) is closed under multiplication from both sides with elements from G(n, r), and so it is an ideal. 2 a set of pairwise orthogonal idempotents in G(n, r).
All other orthogonality relations follow from the definition of multiplication in S q (n, r). 2 Let M (Λ(n, r)) be the Z-algebra of |Λ(n, r)| × |Λ(n, r)|-matrices with integer entries. Let (Λ(n, r)). for all e A ⊆ F α × F β is a surjective Z-algebra homomorphism.
Proof. The map is clearly a surjective Z-module homomorphism. Let e A ⊆ F α × F β and e B ⊆ F β × F γ .
Then ω(e A e B ) is the unique open orbit in F α × F γ , which is equal to ω(e A ) ω(e B ), by Corollary 9.4. Moreover, ω(1 G(n,r) ) = ω( α k α ) = α o α = 1 M (n,r) . This completes the proof of the lemma. 2 We can now prove the main result of this section, which implies that M (n, r) is a direct factor of the Z-algebra G(n, r). Theorem 9.9. We have an isomorphism of Z-algebras G(n, r) → M (n, r) × (G(n, r)/M (n, r)) given by e A → (ω(e A ), e A ).
Proof. By Corollary 9.4, we have Now, 1 G(n,r) = α k α , and again by Corollary 9.4, α o α is a central idempotent in G(n, r). This proves that M (n, r) is a direct factor of G(n, r), and so the theorem follows. 2 Let A n denote the preprojective algebra of type A n . See [4] for the definition and properties of preprojective algebras. Proof. We need to show that First observe that ( α k α − o α )G(n, r)( α k α − o α ) is generated by e 1,α − o α− 2 + 1 ,α , f 1,α − o α+ 2 − 1 ,α and k α − o α . A direct computation shows that the generators satisfy the preprojective relations. By comparing dimensions we get the required isomorphism. 2 Let β = (n 1 , · · · , n l ) and γ = (r 1 , · · · , r l ) be decompositions of n and r, respectively, into l parts, where n i > 0. Let m i = i−1 j=1 n j , where m 1 = 0. There is a map φ j of flags of length n j to flags of length n given by φ j (f ) l = 0 for l ≤ m j , φ j (f ) l = f l−m j for m j < l ≤ m j+1 and φ j (f ) l = f n j for l > m j+1 , where f i denotes the vector space at the ith-step of the flag f . The corresponding map on orbits of pairs of flags f, f → φ j (f ), φ j f is also denoted by φ j .
Proof. Since φ β,γ is injective on basis elements, it is an injective Z-linear map. By Lemma 2.2, in terms of matrices, the map is given by φ β,γ (e A 1 , · · · , e A l ) = e A 1 ⊕···⊕A l .
Following Lemma 6.11, the map φ β,γ preserves multiplication and thus is an injective Z-algebra homomorphism. Let N = φ β,γ (N 1 , · · · , N l ) and N = φ β,γ N 1 , · · · , N l , and N ≤ deg N . We may assume that the degeneration is minimal. By Lemma 8.2, N = (t, s)N for a transposition (t, s). Then the transposition (t, s) must act within one N i , since the off-diagonal blocks of the matrices of both N and N are zero, and so (t, s)N = φ β,γ N 1 , · · · , N i−1 , t , s N i , N i+1 , · · · , N l for a transposition (t , s ). This shows that N i ≤ deg N i for all i. The converse also follows from Lemma 8.2. 2 Let β, γ and m i be as above. Let α i = (α m i +1 , · · · , α m i+1 ) be a decomposition of r i into n i parts. Then α = (α 1 , · · · , α n ) is a decomposition in Λ(n, r). Let o (α,β) = φ(o α 1 · · · , o α l ). By Lemma 6.11 and Corollary 9.4, we have the following. We call o α,β an idempotent orbit. Note that k α = o (α,(1,···,1)) and that o α = o (α,n) , where n denotes the trivial decomposition of n into 1 part. For a given α, if k α is in the interior of the quiver Σ(n, r) viewed as an (n − 1)-simplex, there is exactly one idempotent orbit for each decomposition β and two different decompositions give two different idempotents, so there are 2 n−1 idempotent orbits in k α G(n, r)k α , If k α is on the boundary, but in the interior of a t-simplex, then there are 2 t idempotent orbits. In particular, in the interior of a line, i.e. the 1-faces, there are the two idempotent orbits k α and o α , and for the vertices of the simplex, i.e. the 0-faces, there is a unique idempotent orbit k α = o α . be defined by t σ,1 = t 1 t 2 · · · t σ −1 (1)−1 and then where τ i−1 is given by τ i−1 k α = t σ,i−1 · · · t σ,1 .
Theorem 10.2. With the notation above, t σ = σk α . Consequently, the set of all t σ for σ ∈ S n is a multiplicative basis of H 0 (n).

Proof. Let
h : H 0 (n) → H 0 (n) ⊗ Z C be given by h(T i ) = −t i . A direct computation in H 0 (n) shows that −t i satisfy the 0-Hecke relations i), ii) and iii) above, so the map is well defined. The two algebras have the same dimension over C, and so it suffices to have that the map is surjective, which is indeed true by Theorem 10.2. So the two algebras are isomorphic. 2