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Journal of Pure and Applied Algebra

ELSEVTER www.elsevier.com/locate/jpaa

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

Bernt Tore Jensen a, Xiuping Su b'*

a Gj0vik University College, Postboks 191, 2802 Gj0vik, Norway

b Department of Mathematical Sciences, University of Bath, Bath BA2 7JY, United Kingdom

article info abstract

We define a new product on orbits of pairs of flags in a vector space over a field k, using open orbits in certain varieties of pairs of flags. This new product defines an associative Z-algebra, denoted by G(n,r). We show that G(n,r) is a geometric realisation of the 0-Schur algebra So(n,r) over Z, which is the q-Schur algebra Sq (n, r) at q = 0. A pair of flags naturally determines a pair of projective resolutions for a quiver of type A with linear orientation, and we study q-Schur algebras from this point of view. This allows us to understand the relation between q-Schur algebras and Hall algebras and to construct bases of q-Schur algebras. 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 ground ring, where q is not invertible. Crown Copyright © 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

CrossMark

Article history: Received 1 November 2012 Received in revised form 18 March 2014

Available online 12 June 2014 Communicated by S. Donkin

0. 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 FxF, Beilinson, Lusztig and MacPherson gave a geometric construction of some finite dimensional quotients of the quantised enveloping algebra Uq(gln). In [11], Du remarked that the quotients are isomorphic to the g-Schur algebras defined by Dipper and James in [7]. So in this paper the g-Schur algebras Sq (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 FxF. Denote by [/, g] the GL(V )-orbit of (/, g) GFx F and by F x F/GL(V) the set of GL(V )-orbits on FxF. Let A and n be the maps

* Corresponding author.

E-mail addresses: bernt.jensen@hig.no (B.T. Jensen), xs214@bath.ac.uk (X. Su).

http://dx.doi.Org/10.1016/j.jpaa.2014.04.022

0022-4049/Crown Copyright © 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

Fx Fx F —^^ (Fx F) x (Fx F)

given by A(f, ff") = ((f, f'), (f/")) and n(f, ff") = (f, f").

Let Z[q] be the ring of polynomials in q over Z, where q is an indeterminate. The q-Schur algebra Sq(n, r) is a free Z[q]-module with basis F x F/GL(V) and with multiplication

[f,g][h,l]= £ FfMf'i [f',l'], [f',i']

where Ff,g,h,i,f',i' is the polynomial in Z[q] such that Ffgh,i,f',i'(|k|) is the cardinality of the set

n-1(f',l') n A-1([f,g] x [h,l])

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 S0(n,r), which is the q-Schur algebra Sq(n,r) at q = 0. We define a new Z-algebra G(n,r) with basis F x F/GL(V) by defining the product of [f,f'] and [f/,f"] to be the unique open orbit (see Section 6) in

nA-1([f,f'] x [f'J'']).

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 So(n,r).

Remark 2. Throughout, Sq(n,r) is a Z[q]-algebra, G(n,r) and S0(n,r) are Z-algebras and Sv(n,r) = Sq(n,r) ®z[q] Q(v) is a Q(v)-algebra, where v2 = 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

Sq(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 An. In this way, a pair of flags (f,g) naturally determines a pair of projective resolutions f n g C f and f n g C 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 Sq(n,r),

{[f,f + g][f + g,f] I [f,g] €FxF/GL(V)}

{[/,/n g][f n g,f] | [/,g] GFx F/GL(V)}.

Using the new bases and the main result, we give new presentations of g-Schur algebras over a commutative Z[g]-algebra Q, where g is not invertible. Using open orbits, we construct families of idempotents and an ideal M(n,r) C 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 H0(n).

The remainder of the paper is organised as follows. In Section 1, we explain the construction of Sq (n, r) by Beilinson, Lusztig and MacPherson in more detail. In particular, we recall the description of GL(V)-orbits in Fx F using matrices and the fundamental multiplication rules. In Section 2, we give a new description of the GL(V)-orbits in F x F using representations of linear quivers of type An. In Section 3, we recall the definition of the positive and negative parts of the g-Schur algebras and their relationship to the Hall algebras. In Section 4, we construct new bases of Sq(n,r). In Section 5, we describe g-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 Fx 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.

1. Flag varieties and g-Schur algebras

In this section, we fix notation and recall some definitions and results of Beilinson, Lusztig and MacPher-son on g-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 / and /' be flags in F with

/ : {0} = Vo C V1 C ... C Vn = V

/' : {0} = Uo C U1 C ... C Un = V. We say that /' is a subflag of /, denoted by /' C /, if for all i,

Ui C Vi.

Denote the intersection of / and /' by / n /', which is the flag

{0}C V1 n U1 C ... C Vn n Un = V, and the sum of / and /' by / + /', which is the flag

{0}C V1 + U1 C ... C Vn + Un = v. Let ai = dim Vi — dim Vi-1 for i = 1, ••• ,n. Then

a = (a1, •••, an)

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 ^(n, r) denote the set of all decompositions of r in n parts, and let Fa C F denote the orbit corresponding to a € A(n, r).

The natural GL(V)-action on F induces a diagonal GL(V)-action on F x F. Associate a matrix A = A(f, f') = (Aij) to the pair of flags (f,f) with

Aij = dim(Vi-i + Vi n Vj) — dim(Vi-i + Vi n V—) = dim Vi n V! — dim(Vi n V/-X + Vi_i n V/).

This defines a bijection between the GL(V)-orbits in F x F and n x 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 ca 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

iff) ^ {g,g').

Let eA,eA' ,eA" €F x F/GL(V) and (f1,f2) € eA". Let

S(A,A',A") = {f € F | (fi,f) € eA, (f,f2) € eA'}. Following Proposition 1.1 in [1], there exists a polynomial gA,A',A" € Z[q], such that

gA,A',A'' (|k|) = |S(A,A',A")|

for any finite field k. The projection

Fx Fx F^F

onto the middle factor maps A-1(eA x eA') n n-1(f1,f2) bijectively onto S(A,A',A"), and so these two sets have the same cardinality. Define

caCA' = gA , A', A'' eA''.

eA" eFxF/GL(V)

This gives an associative algebra over Z[q] with basis F x F/GL(V). Denote this algebra by Sq(n,r). Du proved in [11] that Sq (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 gA, 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 gA,A',A'' using Hall polynomials for any A and AA [12]. Let

qm — 1

[m] = q-- = qm—1 + ••• + q +1

q — 1

for m € N and let Ej,, j be the (i,j) elementary matrix.

Lemma 1.1. (See [1].) Assume that 1 < h < n. Let eA C Fp x Fy . Assume that eg C Fa x Fp and ec CF<5 x Fp such that B — Ehhh+1, C — Eh+1hh are diagonal matrices. Then the following multiplication formulas hold in Sq(n,r),

egeA = ^ Ah,p + 1 ex,

{p\Ah+1,p>0}

ec eA = qE3<P Ah+1'j Ah+ip + 1]ey,

{p\Ah,p>0}

where X = A + Ehp — Eh+ip and Y = A — Ehp + Eh+ip.

Note that the classical Schur algebra [13] can be obtained by evaluating q =1, i.e.,

S(n, r) = Sq (n, r) <g>z[q] Z[q]/(q — 1), the 0-Schur algebra S0(n,r), obtained by evaluating q = 0, is

S0(n,r) = Sq(n,r) <z[q] Z[q]/(q)

Sv (n,r) = Sq (n,r) <Z[q] Q(v),

where v2 = q.

2. Representations of linear quivers

In this section, we describe orbits of pairs of flags using representations of the linear quiver A of type An

A : 1 -^ 2-^ • • • -^ n.

We first recall a few definitions regarding representations of quivers. Given a quiver Q with the set of vertices Q0 = {1,...,n} and the set of arrows Q1, we denote a representation of Q by

X = {{Xi}ieQo, {Xi,j}i^jeQi ),

where each Xi is a vector space and each Xi j is a linear map from Xi to Xj. A homomorphism h : X ^ Y between two representations X and Y is a collection of linear maps {hi : Xi ^ Y:}"=1, satisfying

hj Xi,j Yi,j hi

for all arrows i ^ j G Q1. A homomorphism h is an isomorphism if all the hi are bijective. We write

X ~ Y,

if X and Y are isomorphic. The direct sum X © Y of representations X and Y is the representation with

(X © Y)i = Xi © Yi and (X © Y)i j = Xi j © Yi j.

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

dim X = (dim X1,..., dimXn).

Given a vector d € Z" 0, define the representation variety of Q, parameterising representations of Q of dimension vector d, to be

Rep(Q,d)= 0 Hom(kd* ,kd).

i^jeQi

The group G = n i GL(di) acts on Rep(Q,d) by conjugation, i.e. for g = (g1,...,gn) € G and X = (X1,2, . ..,Xn — 1,n ) € Rep(Q,d),

(g •X= gjXi,jg-1.

Then the G-orbits in Rep(Q, d) are in one-to-one correspondence to the isomorphism classes of representations of Q with dimension vector d. In this paper, we only consider quivers of type A, so there are a unique closed orbit and a unique open orbit in Rep(Q, d).

Now let Mij, for j > i, be the indecomposable representation of A supported on the interval of vertices [i,j] = {i, •••, j}, with vector spaces in the support equal to k and all non-zero maps equal to the identity. Any representation M is isomorphic to the direct sum of some Mij, i.e.

m ~0 (Mij a,

where each aij is a non-negative integer. For each vertex i, let Si = Mii and Pi = Min be the simple and indecomposable projective representation associated to i, respectively. A representation P is projective if and only if each map Pi,i+1 is injective.

For any vector a € Z"0, let P(a) be the projective representation defined by

P (a) = 0(Pi)ai.

P(a)1 C P(a)2 C---C P(a)n

is a flag in P(a)n. Any projective representation is isomorphic to P(a) for some a and so we can view a projective representation as a flag. Conversely, an n-step flag

{0} = Vo C V1 C ••• C Vn = V

can be naturally viewed as a projective representation of A

V1 -^ V2 -------- Vn,

with the natural inclusion Vi ^ Vi+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 Ui © U[.

A pair of flags (g, f) with g C f can be viewed as a projective resolution

0 ^ g ^ f ^ f/g ^ 0.

If (fi, f2), (f1 ,f2) € F x F, with / C f2 and J C f2, then (f1,f2) — (J,f2), i.e. they are in the same GL(V)-orbit, if and only if f2/f1 — f2/f1 and f2 — f2, as representations of A. This fact generalises to arbitrary pairs.

Lemma 2.1. Let (fi, f2), (fi, f2) €F x F. T^e following are equivalent. i) (fl,f2) - (fl ,f2).

ii) (fi + f2)/fi - (fl + f2)/fl for i = 1,2, and fi + f2 - fl + f2.

iii) (fi,fi + f2) - (fl,fl + f2) for i = 1,2.

Proof. The implication from i) to ii) is trivial.

We prove that ii) implies i). By ii), fi/(fi n f2) - fl/(fi n f2), and fi n f2 - fl n f2. Let gi : fi/fi n f2 ^ fl/fl n f2 be isomorphisms. Consider the following diagram,

fi + f2

(fi/fi n f2) © (f2/fi n f2)

0 N 92 ,

fi + f2

(fi/fi n f2 ) © (f2/fl n f2 ),

where n and n' are natural projections. Since Jl + f2 and fi + f2 are isomorphic projective representations, there is an isomorphism h such that the above diagram commutes. Thus h(f1) C (n')-1(f1 /fi fl /2) = fi. Therefore h(fL) = fi. Similarly h(/2) = /2. Hence (/, f2) and (fi, J2) are in the same orbit. This proves i). iii) is a reformulation of ii). So the proof is done. □

Note that the isomorphism of two pairs of flags can also be characterised using the inclusions / f f2 C fi and the isomorphisms (/ + /2)// — fj/(fi f /2) for i = j € {1,2}.

The lemma shows that a pair of flags in F x F and a triple (a, [M], [N]), where a € A(n,r) and [M], [N] are isomorphism classes of representations M and N of A with a surjection P(a) ^ M © N, mutually determine each other. In fact, given such a triple (a, [M], [N]), we can construct a corresponding pair (g1,g2) as follows. Recall that a surjective homomorphism ^ : P ^ M from a projective representation P is a projective cover if ker ^ C rad P where rad P denotes the Jacobson radical of P. As inner direct sum, we have

P(a) = fi © f2 © c

such that is a projective cover of M and is a projective cover of N, and then

ker ^ = fl © f2 © c

with fl = ker . Now let

(gi,g2) = (fi © f2 © c,fi © f2 © c).

i) gi + 92 = fi © /2 © C and gi n g2 = fi © /2 © c.

ii) f 1 Ç rad f i and /2 Ç rad f2.

iii) 9i + 92/92 - gi/gi n 92 - M and gi + 92/91 - 92/91 n 92 - N.

Given a pair of flags (f,f') GFxF, we have the following description of the matrix corresponding to the orbit [f,f1].

Lemma 2.2. Let eA = [f, f'] = [fi © f2 © c,f[ © f2 © c] as above. Then

i) if i < j, then Aij is the multiplicity of Mij—i as a direct summand in fi/fi,

ii) if i > j, then Aij is the multiplicity of Mjii—i as a direct summand in f2/f2.

iii) Aii is the multiplicity of Min as a direct summand in c.

Proof. By definition, Aij is equal to the dimension of the space

_(fi,i © f2,i © Ci) n (f1,j © f2,j © Cj)_

(fi,i © f2,i © Ci) n (f1,j-i © f2,j—i © Cj-i) + (fi,i-i © f2,i-i © Ci-i) n (f1,j © f2,j © Cj) •

For i < j,

fi,i n flj © f2i © Ci

Aij = dim

fii n f1,j—i + f2,i + Ci + fii—i n fi,j

fi,i n f1,j

¡1,i n /1,j_1 + f1,i-1 n /1,j-

which is the multiplicity of Mitj-1 as a direct summand in /1//1. This proves i). Similarly, ii) holds. For i = j,

, f1,i © f2,i © ci Aii = dim ■

fi,i—i n fi i + fli n f2,i—i + f2,i + fl,i + Ci

As /1 C /1 and /2 C /2 are minimal projective resolutions, we have inclusion of vector spaces /1 i C /1i-1 and /2 i C /2,i-1. Therefore

A,-,- = dim - i

which is the multiplicity of Min as a direct summand in c. □

3. The non-negative g-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 A, 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 orbit [/,/'] gFx F/GL(V) with /' C / decomposes as

[f,f] = [c © g,C © g']

where f/f - g/g' and g' C radg. That is, such an orbit is determined by the minimal projective resolution

of f/f

0 ^ g' ^ g ^ f/f ^ 0 and a projective representation c such that

f - g ® c.

The non-negative Z[g]-subalgebra S+(n,r) is the subalgebra of Sq (n,r) with basis consisting of all orbits iff] with f' C f. Similarly, the non-positive g-Schur algebra S-(n,r) has the corresponding basis of all orbits [ff with f' C f.

Let M, N and L be representations of A. Recall that the Hall polynomial hMN € Z[g] defined by Ringel [21] is the polynomial such that hMN(|k|) is equal to the number of subrepresentations X C L such that

X - N and L/X - M

for any finite field k.

Lemma 3.1. Let f D f2 D f3 be flags and let = [fi,f2], ¿A' = [/2,fs], ¿A" = [f ,f ] with f D fS, f - fi and f3 - fs, M = fi/f2, N = f2/fs and L = f /fS. Then

gA,A',A" = hMN.

Proof. Denote the set {X C L | X - N,L/X - M} by U. Note that

gA,A' ,A'' (|k|) = \S(A,A',A")\ and hLMN (|k|) = |U | for any finite field k. We need only to show that

\S (A, A' A" )\ = |U |.

We will define two mutually inverse maps between U and S(A,A',A"). Given f2 € S(A,A',A"), we have the following commutative diagram of short exact sequences

0-- fS-- fS-- 0-^ 0

0-- f2-- fi —- fi/f2 —- 0

0 —- f2/fs-- l-- fi/f2 —- 0,

with fi /f2 - fi /f2 - M and f2 /fs - f2/fS - N. Define maps

S(A,A',A") ^ U, f2 ^ n(f2)

U ^ S(A,A',A"), X ^ n-1(X).

It is easy to check that these two maps are mutually inverse, and so the equality follows. □

Denote the (non-twisted) Ringel-Hall algebra [21] by Hq(A). That is, Hq(A) is the free Z[q]-module with basis isomorphism classes [M] of representations of A and multiplication

[M ][W ]—£ hLMN [L].

Mapping representations to choices of projective resolutions induces an algebra homomorphism 0+ : Hq (A) ^ S+(n,r), [M] ^ £ [f,f]

{[f,f']\f 'QfJ/f'-M }

with kernel spanned by those [M] with the number of indecomposable direct summands bigger than r [14]. There is a similar map 0- : Hq(A) ^ S—(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.

Corollary 3.2. We have

9A+D,A' + D,A'' + D — hM,N;

for any diagonal matrix D = diag(^1, • • • , an) with ai > 0.

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 Hq(A) is isomorphic to the algebra with basis consisting of all formal sums

[/, g] M is a representation of A

[f/g]=[M ]

and with multiplication induced by the multiplication in q-Schur algebras.

As before, let Mij be the indecomposable representation of A supported on the interval [i,j]. Let Mij < Mitji if j < j1 or i < i' when j = j'. This is a total order on the indecomposable representations of A. Observe that if Mij < M^jt then

Ext1(Mj/j/,Mij) = 0,

i.e., any extension of Mitjt by Mij splits. The next lemma follows easily from Lemma 3.1 and the corresponding computations in the Ringel-Hall algebra Hq(A).

Lemma 3.4. Suppose that g is a subflag of / and //g = (J)ij M™j. Then there exists a filtration

f — fn D fn-i D • • • D fi — g D 0

with indecomposable factors fi/fi-i < fi+i/fi for all i, such that

[fn,fn-i] • • • [f2 ,fi] = [f,g]Y\[mij ]! and mij is the multiplicity of Mj^j as a subfactor in the filtration.

4. Bases of Sq (n, r)

In this section, we describe a basis for Sq(n,r) using the non-negative and non-positive subalgebras defined in the previous section. Let

B = {[fi,fi + f2][fi + f2,f2] | [fi,f2] €FxF/GL(V)}. Lemma 2.1 shows that the map F x F/GL(V) ^ B given by

[fi,f2] ^ [fi ,fi + f2][fi + f2,f2]

is well-defined. We will prove that B is a Z[q]-basis of Sq(n,r). Note that there is a similar basis B' of Sq(n,r) consisting of elements of the form [fi,fi H f2][fi H f2,f2].

Lemma 4.1. Let (fi,f2) €F x F and let ea = [fi,fi + f2] and eA' = [fi + f2,f2]. Then [fi,fi + f2 ][f i + f2, f2 ] = [fi,f2] + E gA,A',A'' [fi ,f2 ] .

{eA" = [/! ,/2 ]i/' +/2 C/1+/2}

Proof. Suppose that [f', f2 ] is one of the terms with a non-zero coefficient in the sum

[fi,fi + f2][f i + f2,f2] = ^ gA,A',A'' eA''.

Then there exists an f € Fsuch that (fi,f) - (fi,fi + f2) and (f,f2) - (fi + f2,f2). Thus f,f2 C f and so fi + f2 C f. Note that if fi + f2 = f, then (fi,f) = (fi,f + f2) - (fi,fi + f2) and (f, f2) = (fi + f2,f2) -(fi + f2,f2). Therefore (fi + f2)/f - (fi + f2)/fi for i = 1, 2. By Lemma 2.1, (fi,f2) - (fi J2). Moreover gA,A',A'' = 1 for ¿a = [fi,f2]. So the lemma follows. □

There is a similar formula for the product [fi, fi H f2][fi H f2, f2]. Theorem 4.2. The set B is a Z[q]-basis of Sq(n,r). Proof. By Lemma 4.1,

[fi,f2] = [fi,fi + f2 ][fi + f2 , f2 ] - E gA,A',A'' [fi ,f2 ].

{eA'' = [/1 ,/2 ]i/1 +/2 C/1+/2 }

We prove by induction on the dimension of fi + f2, considered as a representation of A, that [fi,f2] is a Z[q]-linear combination of elements in B. If fi = f2, then fi + f2 = fi and

[fi , f2] = [fi, fi][f2 ,f2] = [fi ,fi + f2][fi + f2 , f2],

and so it is done. In general, for each term [fi ,f2] in the sum, fi + f22 is a proper subrepresentation of fi + f2. By induction, [fi ,f2] can be written as a Z[q]-linear combination of elements in B and therefore so

can [fi, f2]. This proves that B spans Sq(n,r) as a Z[q]-module. On the other hand, note that the map from F x F/GL(V) to B, sending [fi,f2] to [fi,fi + f2][fi + f2,f2], is surjective. Therefore B is a Z[q]-basis of Sq (n,r). □

By Lemma 2.2, each basis element in B has a decomposition

[c © fi © f2,c © fi © f2] [c © fi © f2,c © fi © f2] where fi C rad fi and f2 C rad f2.

Lemma 4.3. Lei fi, g and f2 be flags in F with fi C g and f2 C g. Then (fi, g) — (hi, hi + h.2) and (g, f2) — (hi + h-2, h2) /or a pair of flags (hi, h.2) €FxF if and only if there is a surjective map g ^ g/fi © g/f2.

Proof. Assume that ^ : g ^ g/fi © g/f2 is surjective. Let hi = ^-i(g/f2) and h2 = ^-i(g/fi). Then (hi,g) - (fi,g) and (g,h2) — (g,f2) and g = 4>-1(g/fi © g/f2) = hi + h2.

The converse holds, since the map n : hi + h2 ^ (hi + h2)/hi © (hi + h2)/h2 is surjective. □

The lemma shows that a surjective map g ^ g/fi © g/f2 implies [fi,g][g,f2] € B, but the converse is not true. An example is given below.

Example 4.4. Let n = 3, r = 2, V = span{xi,x2|, fi : 0 C kxi C V for i = 1, 2, and g : V C V C V. Then

[fi,g][g,f2] = [fi,f2] + [fi,fi] = [fi,fi + f2][fi + f2,f2] € B, with no surjective map g ^ g/fi © g/f2. In this case g — fi + f2.

5. Quiver and relations for g-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 g-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.

5.1. The quiver S(n,r)

Let ei = (0,...,0,1,0,...,0) be the ith unit vector in Z". Let S(n,r) be the quiver with vertices Ka and arrows Eia and Fia,

Ka+ei-ei+i

where a, a + ei — ei+i € ^(n, r). The vertices can be drawn on a simplex, where the vertices Ka with ai = 0 for some i = 0 are on the boundary, and vertices Ka with ai = 0 for all i are in the interior of the simplex.

For a commutative ring R, denote by RS(n, r) the path R-algebra of S(n, r), which is the free R-module with basis all paths in E(n,r), and multiplication given by composition of paths. The vertices Kd form an orthogonal set of idempotents in RS(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,

Eia = 0 if a £ A(n,r), i = n, or ai+x = 0, Fia = 0 if a £ A(n,r), i = n, or ai = 0, Ka = 0 if a £ A(n,r),

Ei = ^2 Ei,a and Fi = ^2 Fi,a-

Recall that a relation in RS(n, r) is an R-linear combination of paths with common starting and ending vertex

P = ^ riPi,

where ri £ R and pi is a path. Let I(n,r) Ç Z[q]S(n, r) be the ideal generated by the relations

Pij,a Ka+pij Pij Ka,

Nij,a = Ka-Pij Nij Ka, and

Cij,a Ka+€i + €j + i — €i+i — €j Cij Ka,

2ei + tj - 2ei+x - ej+i, if |i - j| = 1, ti + tj - ti+i - tj+i, if |i - j| > 1;

( E2Ej - (q + 1)EiEjEi + qEjE2, if i = j - 1,

i qE2Ej - (q + 1)EiEjEi + EjE2, if i = j + 1, ( EiEj - EjEi, otherwise;

qFfFj - (q + 1)FiFjFi + FjF2, if i = j - 1, F?Fj - (q + 1)FiFjFi + qFjF2, if i = j + 1,

FF F F,

i J j i >

otherwise;

Cij = EiFj - Fj Ei - 5ij

qai - qai+i

ei,a = [f,f/] , fi,a+ei-ei+i = [f',f] and ka = [h, h],

where (f, f') € FxF with f C f, f/f - Si, and f, h € Fa. Lemma 5.1.1. There is a homomorphism of Z[q]-algebras

0 : Z[q]^(n,r)/I(n,r) ^ Sq(n,r)

defined by

0(Fi,a) = ei,a,0(Fi,a) = fi,a and 0(Ka) = ka.

Proof. By Lemma 1.1, the relations Pij, Nj and Cj hold in Sq(n,r), and so 0 is an algebra homomor-phism.

We remark that the relations Pij and Nij hold in Sq(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 0 is not surjective in general, since for instance [to] is not invertible in Z[q]. So 0 does not give a presentation of the q-Schur algebra over Z[q].

5.2. 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 ^ : R 4 S be a homomorphism of commutative rings, which gives S the structure of an R-algebra. Let E be a quiver, and let I C RE be an ideal. There are induced maps of R-algebras ^ : RE 4 SE and RE/I 4 SE/S^(I), where S^(I) C SE is the ideal generated by ^(I).

Lemma 5.2.1. The induced map (RE/I) <r S 4 SE/S^(I) is an isomorphism of S-algebras. Proof. The natural isomorphism R <r S 4 S of S-algebras induces an S-algebra isomorphism

to : RE <R S 4 SE. Applying the functor — <r S to the short exact sequence

0 4 I 4 RE 4 RE/I 4 0

gives us the exact sequence

I <R S -4 SE 4 (RE/I) <R S 4 0

where j = to o (i < IdS), which shows that

(RE/I) <R S — SE/ im(TO o (i < IdS)). As im(TO o (i < IdS)) = S^(I), the proof is complete. □

5.3. q-Schur algebras over Q(v)

Let v be an indeterminate with v2 = q and

Sv (n,r) = Sq (n,r) <Z[q] Q(v). Lemma 5.3.1. There is an isomorphism of Q(v)-algebras Q(v)E(n,r)/Q(v)I(n,r) 4 Sv (n,r) with Eia 4

ei,a: Fi,a ' ^ fi,a and Ka ' ^ ka.

Proof. Let Ei = a v-ai+ieita, Fi = a v-ai+1+i fia and Ka = ka, and by abuse of notation, in this proof we let Ei = ^a ei,a, Fi = a fia and Ka = ka. Then both {Ei,Fj,Ka} and {Ei,Fj,Ka} generate Sv (n, r). Moreover, by a straightforward computation, Ei, Fj, Ka satisfy the defining relations in Theorem 4' in [9] by Doty and Giaquinto if and only if Ei, Fj, Ka satisfy the relations Pij, Nij and Cj. Therefore we have the isomorphism as required.

Proposition 5.3.2. The induced map 0 < IdQ(v) : Z[q]E(n,r)/I(n,r) <z[q] Q(v) ^ Sq(n,r) <z[q] Q(v) is a Q(v)-algebra isomorphism, where 0 is as in Lemma 5.1.1.

Proof. By Lemma 5.2.1, the natural inclusion Z[q] ^ Q(v) induces an isomorphism

Z[q]r(n,r)/I(n,r) ®Z[q] Q(v) ~ Q(v)r(n,r)/Q(v)I(n,r),

which composed with the isomorphism in Lemma 5.3.1 is 0 < IdQ(v). Thus the proposition follows. □

Since q is invertible in Q(v) and thus in Sv(n,r), we cannot evaluate q = 0 in Sv(n,r). We will modify the ground ring in the next subsection, so that q can be evaluated at 0.

5.4. 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

Z[q] CQC

and q is not invertible in Q. So we can evaluate q = 0.

Proposition 5.4.1. The induced map 0 <g) Idg : Z[q]S(n,r)/I(n,r) ®z[q] Q ^ Sq(n,r) ®z[q] Q is a surjective Q-algebra homomorphism.

Proof. The image of 0® Idg is the subalgebra of Sq(n,r) ®z[q] Q generated by the set of all eia, fia and ka. Lemma 3.4 shows that the Z[q]-subalgebra of S+(n,r) generated by all eia and ka contains all

[f,g]l[[mij ]!

where g C f and mj is the multiplicity of Mj as a direct summand in f/g. Since [m] is invertible in Q for any m, the image contains S+(n,r) ®z[q] Q. Similarly, the image contains S-(n,r) ®z[q] Q. By Theorem 4.2,

B = {[fi,fi + f2][fi + f2,f2] | [fi,f2] € FxF/GL(V)}

is a Z[q]-basis of Sq(n,r), and thus a Q-basis of S+(n,r) ®z[q] Q. Thus the map is surjective. □

By Lemma 5.2.1, QS(n,r)/QI(n,r) - Z[q]^(n,r)/I(n,r) ®z[q] Q, and so the following theorem gives a presentation of q-Schur algebras over Q and will be proven in Section 7.

Theorem 5.4.2. The induced map 0 <g) Idg : Z[q]S(n,r)/I(n,r) ®z[q] Q ^ Sq(n,r) ®z[q] Q is a Q-algebra isomorphism.

6. The generic algebras

In this section let k be algebraically closed. We define a generic multiplication of orbits in F x 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.

Let A : FxFxF4 (FxF) x (FxF) be the morphism given by

A(Pi,P2,P3) = ((P1,P2), (P2,P3)) .

n : FxFxF4FxF

be the projection onto the left and right components. The map n is open, and A is a closed embedding. Given two orbits ca and ca' , define

S(A,A') = nA-i(eA x eA'). That is, S(A, A') is the union of the orbits with non-zero coefficients in the product ca • ca' in Sq(n,r). Lemma 6.1. The closure of S(A, A') in FxF is irreducible.

Proof. Let [fi,f2] = eA, [f3,f4] = ca' , and S = A-i(eA x ca'). We first show that S is irreducible. If f2 — f3 then S is empty, and we are done. So we may assume that f2 = f3. Let (p1,p2,p3) € S then there exists g € GL(V) such that (p2,p3) = g(f3,f4) and g(g-ipi,f3,f4) = (Pi,P2,P3), where GL(V) acts diagonally. Since (g-ipi,f3) — (fi,f3), there is an a € Aut(f3) such that g-ipi = afi. Hence S is the image of the morphism

Aut(f3) x GL(V) 4FxFxF

given by

(a,g) 4 (gafi,gf3,gf4)

and is therefore irreducible. Now S(A, A') = n(S), and so its closure is irreducible. □

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

eA * eA' = eA''

if S(A, A') is non-empty and ca" is the open orbit in S(A, A'), and

eA *ca' = 0

if S(A, A') is empty. Denote by G(n,r) the free Z-module with basis F x F/GL(V). Proposition 6.3. The free Z-module 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 [fi,f2], [f3,f4], [f5,f6] € F, we have

([fi,f2] * [f3,f4]) * [f5,fe] = [fi,f2] * ([f3,f4] * [f5,fe]).

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, f2 — f3 and f4 — f5. By the GL(V)-action on F, we may assume that f2 = f3 and f4 = f5. Denote the sets

{(Pi,P2,P3,P4) | (Pi,P2) — (fi,f2), (P2,P3) — (f3,f4), (P3,P4) — (f5,f6)}, {(Pi,P3,P4) 1 3p such that (pi,p) — (fi ,f2), (P,P3) — (f3,f4), (P3,P4) — (f5,f6)}, {(Pi,P2,P4) 1 3p such that (Pi,P2) — (fi ,f2), (P2,P) — (f3,f4), (P,P4) — (f5 ,f6^ ,

{(Pi,P4) | 3p,p' such that (pi,p) — (fi,f2), (p,p') — (f3,f4), (p',p4) — (f5,f6)} by Ti, T2, T3 and T4, respectively. We have natural surjections

nij : Ti ^ Tj

for (i,j) = (1,2), (1,3), (2,4), (3,4). Similar to the proof of Lemma 6.1, we see that Ti is irreducible, and so the closures of all the Ti are irreducible. In particular, there is a unique open orbit O in T4. Then n24i(O) intersects with the open subset of T2, consisting of triples (pi,p3,p4) with [pi,p3] open in S(A, A'). That is, ([fi,f2] * [f3,f4]) * [f5,f6] is the open orbit O in T4. Similarly, [fif * ([f3,f4] * [f5,f6]) is also the open orbit O. Therefore the equality holds and so * is associative.

The following is a direct consequence of the definition of the product in G(n,r).

Corollary 6.4. The set F x F/GL(V) is a multiplicative basis of G(n,r).

In addition to the basis of G(n,r) consisting of orbits [fi,f2] we can also consider bases analogous to the bases B and B1 defined in Section 4 for the q-Schur algebras. We show that these three bases of G(n,r) coincide.

Lemma 6.5. Let (/i,/2) GFxF. Then

[fi,fi + f2] * [fi + f2, f2 ] = [fi,f2] = [fi ,fi PI f2] * [fi PI f2,f2].

Proof. We prove the first equality. Let ca = [fi,fi + f2] and ca' = [fi + f2,f2]. We prove that the orbit [fi,f2] is open in S(A, A'). For any (fi,f2) € S(A,A'), fi + f2 is isomorphic to a subflag of fi + f2. By Lemma 4.1, for (fi,f2) € S(A, A'), we have (fi,f2) — (fif if and only if fi + f1 — fi + f2. That the dimension of fi + f2 is maximal is an open condition. Therefore ca *ca' = [fi,f2]. Similarly, [fi, fi n f2] * [fi n f2, f2] = [fi,f2]. □

We now prove that the Z-algebra G(n,r) is generated by the orbits ei, a, fi , a and ka. 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 D g D 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 a € A(n,r) with a — ej+i non-negative, let

e(i, j, a) ei,a+6i+i — £j+i * • •• * ej,a .

Similarly, let f (i,j,a) = fja—ei+ej * • •• * fi,a for a — ei non-negative.

Lemma 6.7. Let f D h be flags with h € Fa and f/h - Mij. Then [f,h] = e(i,j,a) and [h, f] = f (i,j,a + ei — £j+i).

Proof. If i = j, then [f, h] = eia. Now assume j > i. Then there is f D g D h with f/g - Mi,j —i and g/h - Mjj. 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. □

Using the order < on representations defined in Section 3, we can write each orbit [f, g] with f D g as a product over indecomposable summands of f/g.

Lemma 6.8. Let f D g be flags with f/g - (J)Mi and Mi < Mi+i. Then there is a filtration f = ft D ft—i D • •• D fo = g D 0 with indecomposable factors Mi = fi/fi—i and [f, g] = [ft, ft—i] * •••* [fi,fo].

Proof. The lemma follows from the vanishing of extension groups along the filtration and Lemma 6.6.

Lemma 6.9. The Z-algebra G(n,r) is generated by the orbits eia, fia and ka.

Proof. Lemma 6.7 and Lemma 6.8 imply that any orbit [f, g] with f D g is in the subalgebra of G(n,r) generated by ej,, a and ka. Similarly, any orbit [f, g] with f C g is generated by ^ , a and ka. The lemma now follows from Lemma 6.5.

Following Lemmas 6.5, 6.7 and 6.8, we obtain the following basis of G(n,r) in terms the generators eia and fia.

Lemma 6.10. The Z-algebra G(n,r) has a basis consisting of all ka and all non-zero monomials

e(i s , j s , as ) * • ••*e(ii,ji,ai) *f(ii ,ji ,ai) * • • • * f(i't,j/t ,a't) ,

where Midl < Mii+Ui+i, Milj' < Mil+1j'+1 and ai > Ei eji+i ^ Ei j'+i.

Proof. First observe that any basis element [f, g] = [f,f Hg][f Hg, g] can be written as a monomial described in the statement. So we need only show that for any such monomial

e(is,js,as) *• • • *e(ii ,ji,ai) *f(i/i,j/i ,a'i) *• • • *f(i/t,j'i,at)

there is a unique orbit [f, g] such that

[f,f H g] = e(is,js,as) *• • • *e(ii,ji,ai) and [f H g, g] = f(ii,jii,ai) *• • • *f(i't,j/t,a't).

Write ai = ¡3 + a' + a", where a' = El eji+i and a" = El £ji'+i. Consider P(ai) as a flag in V, and decompose as P(ai) = P(3) © P(a') © P(a"). Let Q(a') and Q(a") be minimal flags containing P(a') and P(a"), respectively, such that

Q(a>) /P(a') ~ 0 Mkjl and Q(a")/P(a") ~ 0 Mj.

Let f = P(ft) © Q(a') © P(a") and g = P(ft) © P(a') © Q(a"), then [f,g] is an orbit as required. The uniqueness is determined by the two quotients f/f fi g and g/f fi g and ai. □

We compute the multiplication in G(n, r) of an arbitrary element with a generator.

Lemma 6.11. Let ca C Fa x Fp.

i) If ai+i > 0, then ei,a * ca = ex where X = A + Eip — Ei+ip and p = maxjj | Ai+i j > 0}.

ii) If ai > 0, then fi, a * ca = ey where Y = A — Ej,, p + Ei+i ,p and p = min{j | Aj,, j > 0}.

Proof. We prove i). By Lemma 1.1, the orbit ex has a non-zero coefficient in the product ej,, a -ca in Sq (n, r). Now, by Lemma 2.2 in [1], among all terms A + Eitj — Ei+ij with Ai+i,j > 0, the elements in the orbit ex has the smallest stabiliser, and so eia *ca = ex. The proof of ii) is similar. □

7. 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 S0(n,r) and G(n,r) are isomorphic as Z-algebras by an isomorphism which is the identity on the closed orbits eia, fia and ka. Finally, we prove Theorem 5.4.2.

7.1. A presentation of G(n,r)

Let E(n,r), Ei and Fi be as in Section 5. Let

( EfEj — EiEjEi, if i = j — 1,

Pij(0) = ^ —EiEjEi + EjE2, if i = j + 1,

[ EiEj — EjEi, otherwise;

( — FiFj Fi + Fj F2, if i = j — 1,

Nij (0) = ^ FfFj — FiFj Fi, if i = j + 1,

[ FiFj — FjFi, otherwise;

Cij(0) = EiFj — FiEj — SijJ2 Aij(a) • Ka,

(1, if ai > ai+i = 0, — 1, if ai+i > ai = 0, 0, otherwise.

That is, Pij(0), Nij(0) and Cj(0) are obtained by evaluating Pij, Nij and Cij at q = 0. Let I0(n,r) C ZE(n,r) be the ideal generated by Pij,a(0), Nij,a(0), and Cij,a(0), which are obtained by evaluating Pij,a, Nij,a and Cija at q = 0.

Lemma 7.1.1. ZE(n,r)/Io(n,r) has a multiplicative basis of paths in E(n,r).

Proof. The lemma holds since each relation Pij,a(0), Nj,a (0), and Cij,a(0) is a binomial in Ei,a, Fia and Ka. This is obvious for Pij,a (0), Nij,a (0). For Cija (0), if the coefficient of Ka is non-zero then either Cij,a (0) = KaEiFjKa — Ka or Cy,a(0) = KaFiEjKa — Ka . □

For an interval [i,j] in {1, • • • ,n} and a € A(n,r) with a — Cj+i non-negative, let

E(i,j, a) Ei,a+ei+1—ej+1 • • • Ej,a

and F(i,j,a) = Fjta—e.+ej • • • Fi a for a — ei non-negative. The E(i,j,a) and F(i,j,a) are analogous to e(i,j,a) and f (i,j,a), respectively, defined in Section 6.

Theorem 7.1.2. The map n : ZS(n,r)/Io(n,r) ^ G(n,r) given by n(Ei,a) = eia, n(Fi,a) = fi,a and n(Ka) = ka is an isomorphism of Z-algebras.

Proof. By Lemma 6.11, it is straightforward to check that ei a, fia, and ka satisfy the relations Pij,d(0), Nij,d(0), and Cij,d(0). Thus n is well-defined. Also, Lemma 6.9 implies that the map is surjective. It remains to prove that n is injective.

We claim that, modulo the relations in I0(n, r), any path p in S(n,r) is either equal to ka or a path of the form

E (i s , j s , as ) • • • E(ii ,ji,ai)F(ii,j^ ,ai) • • • F(i't,ft ,a't) ,

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 n 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 ka or one of the arrows Fia and Eia, and so the claim follows. Assume that p has length greater than one. Then we have

p = p'Fifi or p = p'Eifi where p' is a non-trivial path of smaller length, and so by induction has the required form

p' = EF = E(is,js,as) • • • E(ii,ji,ai)F(ii,j/i,ai) • • • F(i't,j/t,at),

where E and F are products of the E(ia,ja,aa) and F(ib,jb,a'b), respectively.

We first consider p = p'Eitp. If p' contains no Fja, then the claim follows using the relations Paba(0). Otherwise, by the relations Cab a(0), either p = EF' with the length of F' smaller than that of F or p = EEia1— ei+ei+1 F' with each factor F(i'l,j[,a'l) in F replaced with a factor F(i'l,j'l,3'i). In the first case, the claim follows by induction. Otherwise, by the relations Pab,a(0), there are two possibilities. First, there exists a minimal m with jm = i — 1. Then EEia1—Ci+ei+1 F' is equal to

E(is,js,as) • • • E(im+i,jm+i,am+i)E( im i Jm + 1,3 m )E( im—i , jm—i , ¡¡m— i ) • • • E (ii,ji,3i)F'. We have

¡i = ai — 6i + Ci+i >^2 ejl + i — ei + £i+i + ej[ + i = ^2 €jl + i + Cjm+2 +^2 j' + i.

l l l=m l

Moreover, again using the relations Pab,a(0), the factors can be reordered (up to change of al, ¡m) to obtain a path of the required form.

Second, there is no such m with jm = i — 1. Then EEiai—ei+ei+1 F' is equal to

E(is, js,as) • • • E( im, jm, am )E(i,i,ftm)E ( ftm— i ) • • • E (ii,ji,fti)F',

with jm— i < i and jm > 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 jm = i — 1, the sum£; £ji+i contain no ei. Since FEip = Eip1 F' with the length of F' equal to that of F, we must have (ai — ei)i > (^; eji'+i)i and so the inequality follows.

Finally, we consider p = p'Fitp, where p' is a path of the required form p' = EF as above. If there are no factor Eja in p', then the claim follows from the relations Naba (0). Otherwise p = E'Ejai FFitp, which following Caba (0) is either p = E'F' with F' not longer than F, or p = E'F'Ej$. In the first case, the length of the path E'F' is smaller than p in E(n, r) and so the claim follows by induction; in the second case, the claim is proved above. In either case, the claim holds. □

7.2. A geometric realisation of So(n,r)

We now prove the main result.

Theorem 7.2.1. The map

^ : G(n,r) 4 So(n,r)

defined by ei, a) = ei, a, ^(fi,a) = fi,, a and ^(ka) = ka is an isomorphism of Z-algebras.

Proof. From Proposition 5.4.1, we have the surjective Q-algebra homomorphism

QE(n, r)/QI(n,r) 4 Sq (n,r) ®z[,j Q,

which, since Q/qQ ~ Z, induces a surjective Z-algebra homomorphism

{QE(n,r)/QI(n,r)) ®Q Q/qQ 4 Sq(n,r) ®Z[g] q®q Q/qQ.

Following the definition of S0(n,r) and the isomorphisms

Q/qQ~Z[q]/qZ[q] ~ Z,

we have

Sq (n,r) ®Z[q] Q®Q Q/qQ = So(n,r)

and by Lemma 5.2.1

(QE(n,r)/QI (n,r)) ®q Q/qQ ~ (ZE(n,r)/Io(n,r)).

So there is a surjective Z-algebra homomorphism

ZS(n,r)/Io(n,r) ^ So(n,r)

given by

Ei,a 1 ^ ei,a, Fi,a 1 ^ fi,a, Ki,a 1 ^ ki,a.

The theorem now follows from Theorem 7.1.2, since G(n,r) = So(n,r) as Z-modules. □

Via the isomorphism in Theorem 7.2.1, the presentation of G(n,r) in Section 7.1 becomes a presentation of So(n,r). We remark that Deng and Yang [6] have independently given a similar presentation for So(n,r), using a different approach.

Corollary 7.2.2. Let ^ be the map in Theorem 7.2.1. The set ^(F x F/GL(V)) is a multiplicative basis for So(n,r).

7.3. Proof of Theorem 5.4.2

By Proposition 5.4.1, the map 0 < Ida induces a short exact sequence

0 ^ K ^ QS(n, r)/QI(n, r) ^ Sq(n, r) ®z[g] Q^ 0. Since Sq(n,r) <z[q] Q is a free Q-module, applying — <q Q/qQ gives the exact sequence

0 ^ K ®Q Q/qQ ^ QS(n, r)/QI(n, r) ®Q Q/qQ ^ Sq (n, r) <z[q] Q < Q/qQ ^ 0. As in the proof of Theorem 7.2.1, we have isomorphisms

Sq (n,r) <z[q] Q< Q/qQ = So (n,r)

(QS(n,r)/QI(n,r)) ®Q Q/qQ ~ (ZS(n,r)/Io(n,r)). Furthermore, via these two isomorphisms the map 0 < Ida < Ida/qQ is the composition of the isomorphism

ZS(n,r)/Io(n,r) ^ G(n,r)

in Theorem 7.1.2 and the isomorphism

G(n,r) ~ So(n,r)

in Theorem 7.2.1. Therefore

K <g)Q Q/qQ = K/qK = 0.

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 0 < Ida is an isomorphism.

8. 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 FxF using quivers and the symmetric group Sr. Let r = r(n) be the quiver of type A2n-1,

r : 1l ->■ 2l -^ • • • -^ n ^- • • • ^-2R ^- 1R

constructed by joining two linear quivers AL = A and Ar = A at the vertex n. Often it will be clear from the context which side of r we are considering, and then we drop the subscripts on the vertices.

A pair (/,/') G F x F is a representation of r, where / is supported on AL, /' is supported on Ar. Conversely, any representation M of r that is projective when restricted to both AL and Ar and has dimMn = r determines uniquely an orbit of pair of flags [/,/'] G F x F/GL(V). Moreover, two pairs of flags are isomorphic if and only if the corresponding representations are isomorphic.

For integers i,j G {1,- • • , n}, let Nij be the indecomposable representation of r which is equal to the indecomposable projective representations Min and Mjn when restricted to AL and Ar, respectively. A representation N of r which is projective when restricted to AL and Ar, and dimNn = r, decomposes up to isomorphism as

N ^0 N

iiji ■ 1=1

We assume that ji < j2 < • • • < jr.

Let <deg denote the degeneration order on isomorphism classes of representations of r. That is, M <deg N for two representations M and N, if N is contained in the closure of the orbit of M in the space of all representations. The degeneration order on pairs of flags is also denoted by <deg, since there is a degeneration between two pair of flags if and only if there is a degeneration between the corresponding representations of r.

Since r 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

0-^ N'-^ M-^ N"-^ 0,

then M <ext N' © Nand more generally <ext is the transitive closure.

The symmetric group Sr of permutations of the set {1, • • • ,r} acts on representations with a decomposition N = 0r=1 NUjl by

ivljl l=1

for a G Sr.

The following facts are the key lemmas on degenerations in F x F. For the sake of completeness we include a brief sketch of the proofs.

Lemma 8.1. Let N = (J)r=i Niljl be a decomposition as above, and let (t,s) with t < s be a transposition. Then N <deg (t,s)N if and only if it > is.

Proof. Assume that it > is. There is a short exact sequence

ylt3t © 1 yls3s

Nitjt © Nt

<deg Ni

and thus

N<deg (t, s)N.

Conversely, assume that it < is. By comparing the dimensions of the stabilisers of N and (t, s)N we see that N £deg (t,s)N. □

We say that a degeneration M <deg N is minimal if M ^ N and M <deg X <deg N implies X ^ M or X ~ N.

Lemma 8.2. Let N = (J)Niljl and M <deg N be minimal. Then there exists a transposition (t,s) such that M ~ (t,s)N.

Proof. Since M <deg N is minimal, there is a non-split extension

0-^ N'-^ M-^ N"-^ 0,

where N — N' © N". We may choose summands Nisjs and Nitjt of N' and N", respectively, such that taking pushout along the projection N' 4 Nisjs and then pullback along the inclusion Nitjt 4 N" gives us a non-split extension

0-- N] -^ M'-- Nitjt -- 0.

This extension is of the form of the extension in the proof of Lemma 8.1. Hence

M' <deg Ni,], © Nitjt,

and so

M' © N /Nisjs ) © (N"/Nitjt ) <deg N /Nisjs ) © (N" /Nitjt ) © Nisjs © Nitjt - N.

By the construction of M',

M <deg M' © {N'/Nisjs) © (N"/Nidt), so by the minimality of the degeneration,

M ~ M' © (N' /Nisjs) © (N "/Nij),

and so the lemma follows.

There is a unique closed orbit in Fa x Fp. We describe a corresponding representation.

Lemma 8.3. The orbit of a pair of flags corresponding to a representation N is closed, if and only if N ~ @r=1 Niljl with ii < ii+1 for all l = 1, • • • ,r — 1.

Proof. A representation N = Or=1 Niljl with il < il+1 does not have any proper degenerations, according to Lemma 8.1. Hence N and thus the corresponding pair of flags have closed orbits.

Conversely, if il > il+1 for some l, then N has a degeneration again by Lemma 8.1, and so the orbit of N is not closed.

Alternatively, we may prove the lemma by observing that among all representations of the form N = @r=1 Niljl the representation with il < il+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(U).

There is a unique open orbit in Fa x Fp with a corresponding representation given as follows. The proof is similar to the proof of the previous lemma.

Lemma 8.4. The orbit of a pair of flags corresponding to a representation N is open, if and only if N ~ ®r=1 Niljl with il > il+1 for all l = 1, • • • ,r — 1.

Similar to the closed orbit, a representation of the form N ~ (J)r=1 Niljl with il > il+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(U). 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(U).

Let oap denote the unique open orbit and kap the unique closed orbit in Fa xFp. Then ka = kaa and we let oa = oa a. For t G Sr, denote by Toap the orbit of pairs of flags corresponding to the representation tN, where N = (J)r=1 Nil jl with il+1 < il is the representation corresponding to oap. Similarly, denote by Tkap the orbit corresponding to tN, where N = (J)r=1 Niljl with il+1 > il is the representation corresponding

to ka , p .

9. Idempotents from open orbits

Let M(n, r) be the Z-submodule of G(n, r) with basis the open orbits in FxF. 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 |A(n,r)| x |A(n, r)|-matrices with integer entries, where |A(n,r)| is the cardinality of A(n,r).

We start with two lemmas relating degeneration and multiplication in G(n,r). Let <deg be the degeneration order on orbits in (FxF) x (F x F) with the action of GL(U) x GL(U).

Lemma 9.1. If es x es' <deg eA x ca>, then es * es> <deg eA *&a>

Proof. Since es x es> Q eA x eA we have S(B,B') C S(A, A'). By Corollary 6.2, we have that S(A,A') is the orbit closure of eA * ea and S(B, B') is the orbit closure of eB * eB', the lemma follows. □

We have the following key lemma on degeneration and multiplication in G(n,r).

Lemma 9.2. Let a G Sr, es' C Fa x Fp and es C FY x Fs. Then es' * (aopr/) * es <deg aoa,s.

Proof. By Lemma 9.1, it suffices to consider the case where es and es' are closed orbits. By Lemma 8.3, we may choose the representation

QNjlkl, l=1

where kl+i > kl and ji+i > jl for the orbit eB. Similarly, Op Y is the orbit corresponding to the representation

(D Niiji, i=i

where il > il+i by Lemma 8.4. Then the coefficient of aOp s in the product (aOp Y) ■ eB in Sq(n, r) is non-zero, and so

(&Op,j) *es <deg &Op,s.

Similarly,

eB' *aOp,s <deg OOa,b.

By Lemma 9.1,

eB' *aOp a *eB <deg eB' *aOp,s <deg aOa,S,

as required.

Corollary 9.3. Let a G Sr, eB' C Fa x Fp and eB C FY x Fs. Then eB' * (akpr/) * eB <deg aka,s. Proof. The corollary follows from the previous lemma, since

aka,s = aiOa,p,

where i(i) = n — i + 1. □

Corollary 9.4. Let eB' C Fa x Fp and eB C FY x Fs. Then eB' *OpY *eB = Oas. In particular, Oa,p *OpY =

Proof. By the lemma we know that eB' * OpY *eB <deg Oaj. Since Oa^ is the unique dense open orbit in Fa xFs, the equality follows. □

Corollary 9.5. M(n,r) is an ideal in G(n,r).

Proof. The previous corollary shows that the Z-submodule M(n, r) C G(n, r) is closed under multiplication from both sides with elements from G(n,r), and so it is an ideal. □

Lemma 9.6. {Oa}a U {ka — Oa}a is a set of pairwise orthogonal idempotents in G(n,r).

Proof. By Corollary 9.4, (Oa)2 = Oa, (ka — Oa)Oa = Oa — Oa = 0, Oa(ka — Oa) = Oa — Oa = 0, and (ka — Oa)2 = (ka — Oa — Oa + Oa) = ka — Oa. All other orthogonality relations follow from the definition of multiplication in Sq(n,r). □

Let M(A(n,r)) be the Z-algebra of |A(n,r)| x \A(n, r)|-matrices with integer entries. Let

w0 : M(n,r) 4 M(A(n,r))

be the Z-linear map where u0(Oap) = Eap is the (a, ft)-elementary matrix in M(A(n,r)).

Lemma 9.7. The map wo : M(n,r) ^ M(A(n,r)) is a Z-algebra isomorphism. Proof. The result is an immediate consequence of Corollary 9.4. □ Lemma 9.8. The map w : G(n,r) ^ M(n,r) defined by

w(eA) = oap

for all eA C Fa x Fp is a surjective Z-algebra homomorphism.

Proof. The map is clearly a surjective Z-module homomorphism. Let eA C Fa x Fp and es C Fp xFY. Then w(eA *es) is the unique open orbit in Fa x FY, which is equal to w(eA) *w(es), by Corollary 9.4. Moreover, w(1G(nr)) = w(Ea ka) = Ea oa = (n,r). This completes the proof of the lemma. □

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) x (G(n,r)/M(n,r)) given by eA ^ (w(eA),eA).

Proof. By Corollary 9.4, we have

M(n,r)=^2 oa} G(n,r)(j2 oa) ■

Now, 1G(n,r) = Ea ka, and again by Corollary 9.4, Ea oa 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. □

Let An denote the preprojective algebra of type An. See [4] for the definition and properties of prepro-jective algebras.

Corollary 9.10. S0(2,r) — M(2,r) x Ar-1 Proof. We need to show that

ka — oa G(n, r) ka — oa

oa ] — 1.

First observe that (Ea ka — oa)G(n,r)( J2a ka — oa) is generated by e^ — oa_£2+£lia, f1^ — oa+£2_£l!a and ka — oa .A direct computation shows that the generators satisfy the preprojective relations. By comparing dimensions we get the required isomorphism.

Let ¡3 = (n1, • • • ,nl) and 7 = (r1, • • • ,rl) be decompositions of n and r, respectively, into l parts, where ni > 0. Let mi = Ej=1 nj, where m1 = 0.

There is a map of flags of length nj to flags of length n given by (f )l = 0 for l < mj, (f )l = fl-mj for mj < l < mj+1 and (f )l = fnj for l > mj+1, where fi denotes the vector space at the ith-step of the

flag f. The corresponding map on orbits of pairs of flags

[f,f'] ^ [0j(f),0j(f')]

is also denoted by .

: G(ni,ri) x ••• x G(ni,ri) 4 G(n,r)

be the Z-linear map defined by

(N1, • • • ,Ni) 4 0i(N1) ©• • • © UN).

Lemma 9.11. The map

0ßr/ : G(ni,ri) x • • • x G(ni,ri) 4 G(n,r)

is an injective Z-algebra homomorphism. Moreover, 0ßY(Ni, • • • ,Ni) <deg 0ß,Y(N[, • • • ,N[) if and only if Ni <deg N for all i.

Proof. Since 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

0ß,Y (eAi, • • • ,eAi ) = eAi ®--®At .

Following Lemma 6.11, the map preserves multiplication and thus is an injective Z-algebra homomor-phism. Let

N = 0ß,7(Ni, • • • ,Ni) and N' = 0ß,7(N[r • • ,N/),

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 Ni, since the off-diagonal blocks of the matrices of both N and N' are zero, and so

(t, s)N = 0ß, 7(Ni, • • • ,Ni-i, (t', s')Ni,Ni+i, • • • ,Ni)

for a transposition (t',s'). This shows that Ni <deg N[ for all i. The converse also follows from Lemma 8.2.

Let ¡3, y and mi be as above. Let ai = (ami+i, ■ ■ ■ ,ami+1) be a decomposition of ri into ni parts. Then a = (ai, ■ ■ ■ , a„) is a decomposition in A(n, r). Let

O(a,p) = ^(Oa1 ■ ■ ■ ,Oal ).

By Lemma 6.11 and Corollary 9.4, we have the following.

Lemma 9.12. The orbit O(ap) is an idempotent.

We call Oapp an idempotent orbit. Note that ka = o^,^,...,^) and that Oa = O(a n), where n denotes the trivial decomposition of n into 1 part. For a given a, if ka is in the interior of the quiver S(n,r) viewed as an (n — 1)-simplex, there is exactly one idempotent orbit for each decomposition 3 and two different decompositions give two different idempotents, so there are 2n-i idempotent orbits in kaG(n,r)ka, If ka is on the boundary, but in the interior of a i-simplex, then there are 24 idempotent orbits. In particular, in the interior of a line, i.e. the 1-faces, there are the two idempotent orbits ka and oa, and for the vertices of the simplex, i.e. the 0-faces, there is a unique idempotent orbit ka = oa.

Lemma 9.13. If o(a,p) <deg N, then o(a,p) *N = N * o(ap) = o(ap). Proof. By Lemma 9.1,

o(a,p) = o(a,p) *o(a,p) <deg N * o(a,p) <deg ka * o(a,p) = o(a,p).

N * o(a,p) = o(a,p).

Similarly,

o(a,p) *N = o(a,p). □

10. Geometric realisation of 0-Hecke algebras

In this section, let n = r and a = (1,- • • , 1). In this case Fa is the complete flag variety and the idempotent ka is the unique interior vertex in the quiver E(n,n). Let

Ho(n) = kaSo(n,n)ka,

which is a Z-algebra. It is known that H0(n)®zC is isomorphic to the 0-Hecke algebra H0(n) (see [3,8,17,19]). We prove this fact below, using G(n, n) and thus give a geometric construction of 0-Hecke algebras.

From the previous section, we have 2n-1 distinct idempotents o(a p), one for each decomposition 3 = (n1, • • • ,nl) of n with ni > 0. Let

ti = (i,i + 1)ka.

We have

ti = o(a,p) ,

where 3 = (n1, • • • , nr-1) with ni = 2 and nj = 1 for j = i, and so ti is an idempotent. Also

ti fi,a+ei_ei+i * ei,a ei,a-£¿ + £¿+1 * fi,a.

Although it can be deduced from a bubble sort algorithm that the ti generate H0(n) as a Z-algebra, we will give an explicit construction of each of the basis element eA in H0(n), using the multiplication in H0 (n).

Lemma 10.1. Suppose i < j. Then ti *ti+1 * • • • *tj_1 = (i,i + 1, • • • ,j)ka, where (i,i + 1, • • • ,j) is a cycle in Sn.

Proof. It follows from the fact that ti = fia+£i_£i+1 * eia (or = eia_£i+£i+1 * fia) and the fundamental multiplication rules in Lemma 6.11. □

Let a be a permutation. Let

ta = ta>n *

be defined by

and then

, = ii * i2 *■ ■ ■ *iCT-i(i)-i

where Ti-i is given by

i<J,t = ii * ii+i * ■ ■ ■ *iTi_1a-1(i)-i,

Ti—i ka = iJ,i i *■ ■ ■ *iJ,i.

Theorem 10.2. With the notation above, iJ = aka. Consequently, the set of all iJ for a G Sn is a multiplicative basis of H](n).

Proof. By the previous lemma, iJ,i = (1, ■ ■ ■ , a-i(1))ka. As a representation Tika = iJ,i has the summand Ni,J-i(i), which is fixed by any ii for i > 1, and therefore by iJ,i for i > 1. By induction Tika has the summands Nj,J-i(j) for j = 1, ■ ■ ■ , i, which are fixed by iJ,j for j > i. Therefore iJ = aka. □

We construct the idempotents O(a,p) using the generators ii. Let [i, j] be an interval in [1, ■ ■ ■ ,n]. Define i[i,j] by induction as follows. Let i[i,i] = kd and

i[i,j] = i[i+i,j] *ii *■ ■ ■ *ij-i.

To each decomposition 3 = (ni, ■ ■ ■ ,nl), let

ip = i[m1 + i,m2] * ■ ■ ■ * i[ml + i,ml + l]

where mi = ^j=i nj and mi = 0.

Corollary 10.3. We have ip = O(ap).

Recall that the 0-Hecke algebra H0(n) is a C-algebra generated by % for i = 1, ■ ■ ■ ,n — 1 with generating relations

i) Ti2 = —Ti,

ii) TiTi+iTi = Ti+iTiTi+i and

iii) TiTj = TjTi for |i — j| > 1.

The algebra H0(n) is a specialisation of a Hecke algebra at q = 0 and has dimension n!. Theorem 10.4. As C-algebras, H0(n) <g>Z C ~ H0(n) Proof. Let

h : H0(n) 4 H0(n) ®z C

be given by h(Ti) = —ii. A direct computation in H0(n) shows that — ii 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.

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