Prime ideals in the quantum grassmannian
aa r X i v : . [ m a t h . QA ] A ug Prime ideals in the quantum grassmannian
S Launois, T H Lenagan and L Rigal ∗ Abstract
We consider quantum Schubert cells in the quantum grassmannian and give a celldecomposition of the prime spectrum via the Schubert cells. As a consequence, weshow that all primes are completely prime in the generic case where the deformationparameter q is not a root of unity. There is a natural torus action of H = ( k ∗ ) n on G q ( m, n ) and the cell decomposition of the set of H -primes leads to a parameterisationof the H -spectrum via certain diagrams on partitions associated to the Schubertcells. Interestingly, the same parameterisation occurs for the non-negative cells inrecent studies concerning the totally non-negative grassmannian. Finally, we usethe cell decomposition to establish that the quantum grassmannian satisfies normalseparation and catenarity. Key words:
Quantum matrices, quantum grassmannian, quantum Schubert variety, quan-tum Schubert cell, prime spectrum, total positivity.
Introduction
Let m ≤ n be positive integers and let O q ( M m,n ( k )) denote the quantum deformationof the affine coordinate ring on m × n matrices, with nonzero deformation parameter q in the base field. The quantum deformation of the homogeneous coordinate ring of thegrassmannian, denoted O q ( G m,n ( k )), is defined as the subalgebra of O q ( M m,n ( k )) generatedby the maximal quantum minors of the generic matrix of O q ( M m,n ( k )). To simplify, thesealgebras will be referred to in the sequel as the algebra of quantum matrices and thequantum grassmannian, respectively.The main goal of this work is the study of the prime spectrum of the quantum grass-mannian. This algebra is naturally endowed with the action of a torus H . Thus, according ∗ This research was supported by a Marie Curie Intra-European Fellowship within the 6th EuropeanCommunity Framework Programme and by Leverhulme Research Interchange Grant F/00158/X
1o the philosophy of the stratification theory as developed by Goodearl and Letzter (see[1]), our main concern is the set of H -prime ideals (namely, the prime ideals invariant underthe action of H ). Recall that if A is an algebra and H a torus which acts on A by algebraautomorphisms then the stratification theory suggests a study of the prime spectrum of A by means of a partition into strata, each stratum being indexed by an H -prime ideal.For many algebras arising from the theory of quantum groups, general results have beenproved about such a stratification. For example, when such an algebra is a certain kindof iterated skew polynomial extension, general results show that it has only finitely many H -primes and that each stratum is homeomorphic to the spectrum of a suitable commu-tative Laurent polynomial ring. However, the algebra which interests us here is far frombeing such an extension and it is not even clear at the outset that it has finitely many H -primes. For this reason, these general results do not apply and we are led to use a verydifferent approach which has a geometric flavour. Recall that a classical approach to thestudy of the grassmanian variety G m,n ( k ) is to use its partition into Schubert cells and theirclosures which are the so-called Schubert subvarieties of the grassmannian. Notice that, inthis decomposition, Schubert cells are indexed by Young diagrams fitting in a rectangular m × ( n − m ) Young diagram. Our method is inspired by this classical geometric setting.Quantum analogues of Schubert varieties (or rather of their coordinate rings) werestudied in [14] in order to show that the quantum grassmannian has a certain combinato-rial structure, namely the stucture of a quantum graded algebra with a straightening law .Subsequently, some of their properties have been established in [15]. In this paper, wedefine quantum Schubert cells as noncommutative dehomogenisations of quantum Schu-bert varieties. Using the structure of a quantum graded algebra with a straightening lawenjoyed by the quantum grassmannian, we are then in position to define a partition of itsprime spectrum. This partition is called a cell decomposition since it turns out that the setof H -primes of a given component is in natural one-to-one correspondence with the set of H -primes of an associated quantum Schubert cell. Hence, the description of the H -primesof the quantum grassmannian reduces to that of the H -primes of each of its associatedquantum Schubert cells. (Here, the actions of H on the quantum Schubert varieties andcells are naturally induced by its action on the quantum grassmannian.)On the other hand, we can show that a quantum Schubert cell can be identified asa subalgebra of a quantum matrix algebra, with the variables that are included sittingnaturally in the Young diagram associated to that cell. As a consequence, we can establishproperties for quantum Schubert cells akin to known properties of quantum matrix alge-bras. For example, we are able to parameterise the H -prime ideals of a quantum Schubertcell by Cauchon diagrams on the corresponding Young diagram, in the same way thatCauchon was able to parameterise the H -prime ideals in quantum matrices, see [3]. This isachieved by using the theory of deleting derivations as developed by Cauchon in [2]. This2heory utilizes certain changes of variables in the field of fractions of the algebra underconsideration. In the case of quantum matrices, these changes of variable can be reinter-preted using quasi-determinants, see [5]. Recently, Cauchon diagrams in Young diagramshave appeared in the literature under the name Le-diagrams see, for example, [16] and[17].By using this approach, we are able to show that there are only finitely many H -primeideals in O q ( G m,n ( k )). More precisely, we show that such H -primes are in natural one-to-one correspondence with Cauchon diagrams defined on Young diagrams fitting into arectangular m × ( n − m ) Young diagram. Following on from this description, we are ableto calculate the number of H -prime ideals in the quantum grassmannian.In addition, we are able to show that prime ideals in the quantum grassmannian arecompletely prime, and that this algebra satisfies normal separation and, hence, is catenary.Again, the method is to establish these properties for each quantum Schubert cell and thentransfer them to the quantum grassmannian.To conclude this introduction, it should be stressed that there are very interestingconnections between our results in the present paper and recent results in the theory oftotal positivity. More details on this are given in Section 5. Throughout the paper, k is a field and q is a nonzero element of k that is not a root ofunity. Occasionally, we will remind the reader of this restriction in the statement of results.In this section, we collect some basic definitions and properties about the objects we in-tend to study. Most proofs will be omitted since these results already appear in [10, 14, 15].Appropriate references will be given in the text.Let m, n be positive integers.The quantisation of the coordinate ring of the affine variety M m,n ( k ) of m × n ma-trices with entries in k is denoted O q ( M m,n ( k )). It is the k -algebra generated by mn indeterminates x ij , with 1 ≤ i ≤ m and 1 ≤ j ≤ n , subject to the relations: x ij x il = qx il x ij , for 1 ≤ i ≤ m, and 1 ≤ j < l ≤ n ; x ij x kj = qx kj x ij , for 1 ≤ i < k ≤ m, and 1 ≤ j ≤ n ; x ij x kl = x kl x ij , for 1 ≤ k < i ≤ m, and 1 ≤ j < l ≤ n ; x ij x kl − x kl x ij = ( q − q − ) x il x kj , for 1 ≤ i < k ≤ m, and 1 ≤ j < l ≤ n. To simplify, we write M n ( k ) for M n,n ( k ) and O q ( M n ( k )) for O q ( M n,n ( k )). The m × n matrix X = ( x ij ) is called the generic matrix associated with O q ( M m,n ( k )).3s is well known, there exists a k -algebra transpose isomorphism between O q ( M m,n ( k ))and O q ( M n,m ( k )), see [14, Remark 3.1.3]. Hence, from now on, we assume that m ≤ n ,without loss of generality.An index pair is a pair ( I, J ) such that I ⊆ { , . . . , m } and J ⊆ { , . . . , n } are subsetswith the same cardinality. Hence, an index pair is given by an integer t such that 1 ≤ t ≤ m and ordered sets I = { i < · · · < i t } ⊆ { , . . . , m } and J = { j < · · · < j t } ⊆ { , . . . , n } .To any such index pair we associate the quantum minor[ I | J ] = X σ ∈ S t ( − q ) ℓ ( σ ) x i σ (1) j . . . x i σ ( t ) j t . Definition 1.1 – The quantisation of the coordinate ring of the grassmannian of m -dimensional subspaces of k n , denoted by O q ( G m,n ( k )) and informally referred to as the( m × n ) quantum grassmannian is the subalgebra of O q ( M m,n ( k )) generated by the m × m quantum minors. An index set is a subset I = { i < · · · < i m } ⊆ { , . . . , n } . To any index set weassociate the maximal quantum minor [ I ] := [ { , . . . , m }| I ] of O q ( M m,n ( k )) which is, thus,an element of O q ( G m,n ( k )). The set of all index sets is denoted by Π m,n . Since Π m,n is inone-to-one correspondence with the set of all maximal quantum minors of O q ( M m,n ( k )),we will often identify these two sets. We equip Π m,n with a partial order ≤ st defined in thefollowing way. Let I = { i < · · · < i m } and J = { j < · · · < j m } be two index sets, then I ≤ st J ⇐⇒ i s ≤ j s for 1 ≤ s ≤ m. For example, Figure 1 shows the partial ordering on generators of O q ( G , ( k )).Let A be a noetherian k -algebra, and assume that the torus H := ( k ∗ ) r acts rationallyon A by k -algebra automorphisms. (For details concerning rational actions of tori, see [1,Chapter II.2].) A two-sided ideal I of A is said H -invariant if h · I = I for all h ∈ H . An H -prime ideal of A is a proper H -invariant ideal J of A such that whenever J contains theproduct of two H -invariant ideals of A then J contains at least one of them. We denoteby H -Spec( A ) the H -spectrum of A ; that is, the set of all H -prime ideals of A . It followsfrom [1, Proposition II.2.9] that every H -prime ideal is prime when q is not a root of unity;so that in this case H -Spec( A ) coincides with the set of all H -invariant prime ideals of A .There are natural torus actions on the classes of algebras that we study here, includingquantum matrices, partition subalgebras of quantum matrices and quantum grassmanni-ans. These actions are rational; and so the remarks above apply.4irst, there is an action of a torus H := ( k ∗ ) m + n on O q ( M m,n ( k )) given by( α , . . . , α m , β , . . . , β n ) ◦ x ij := α i β j x ij . In other words, one is able to multiply through rows and columns by nonzero scalars.Next, there is an action of the torus H := ( k ∗ ) n on O q ( G m,n ( k )) which comes from thecolumn action on quantum matrices. Thus, ( α , . . . , α n ) ◦ [ i , . . . , i m ] := α i . . . α i m [ i , . . . , i m ].We shall be interested in prime ideals left invariant under the action of this torus. The setof such prime ideals is the H -spectrum of O q ( G m,n ( k )). [456][356] [346] [256] [345] GGGGG [246] [156] [245]
GGGGG [236] [146] [235] [145] [136] [234]
GGGGG [135] [126] [134]
GGGGG [125] [124][123]
Figure 1: The partial ordering ≤ st on O q ( G , ( k )).We recall the definition of quantum Schubert varieties given in [15]. Definition 1.2 – Let γ ∈ Π m,n and put Π γm,n = { α ∈ Π m,n | α st γ } . The quantumSchubert variety S ( γ ) associated to γ is S ( γ ) := O q ( G m,n ( k )) / h Π γm,n i . (Note that S ( γ ) was denoted by O q ( G m,n ( k )) γ in [15].) This definition is inspired by the classical description of the coordinate rings of Schu-bert varieties in the grassmannian. For more details about this matter, see [6, Section5.3.4].Note that each of the maximal quantum minors that generate O q ( G m,n ( k )) is an H -eigenvector. Thus, the H -action on O q ( G m,n ( k )) transfers to the quantum Schubert vari-eties S ( γ ).In order to study properties of the quantum grassmannian, the notion of a quantumgraded algebra with a straightening law (on a partially ordered set Π) was introduced in[14]. We now recall the definition of these algebras and mention various properties thatwe will use later.Let A be an algebra and Π a finite subset of elements of A with a partial order < st . A standard monomial on Π is an element of A which is either 1 or of the form α . . . α s , forsome s ≥
1, where α , . . . , α s ∈ Π and α ≤ st · · · ≤ st α s . Definition 1.3 – Let A be an N -graded k -algebra and Π a finite subset of A equipped witha partial order < st . We say that A is a quantum graded algebra with a straightening law ( quantum graded A.S.L. for short) on the poset (Π , < st ) if the following conditions aresatisfied.(1) The elements of Π are homogeneous with positive degree.(2) The elements of Π generate A as a k -algebra.(3) The set of standard monomials on Π is a linearly independent set.(4) If α, β ∈ Π are not comparable for < st , then αβ is a linear combination of terms λ or λµ , where λ, µ ∈ Π , λ ≤ st µ and λ < st α, β .(5) For all α, β ∈ Π , there exists c αβ ∈ k ∗ such that αβ − c αβ βα is a linear combination ofterms λ or λµ , where λ, µ ∈ Π , λ ≤ st µ and λ < st α, β . By [14, Proposition 1.1.4], if A is a quantum graded A.S.L. on the partially orderedset (Π , < st ), then the set of standard monomials on Π forms a k -basis of A . Hence, in thepresence of a standard monomial basis, the structure of a quantum graded A.S.L. may beseen as providing more detailed information on the way standard monomials multiply andcommute. Example 1.4 – It is shown, in [14, Theorem 3.4.4], that O q ( G m,n ( k )) is a quantum gradedalgebra with a straightening law on (Π m,n , ≤ st ).From our point of view, one important feature of quantum graded A.S.L. is the follow-ing. Let A be a k -algebra which is a quantum graded A.S.L. on the set (Π , ≤ st ). A subsetΩ of Π will be called a Π-ideal if it is an ideal of the partially ordered set (Π , ≤ st ) in the6ense of lattice theory; that is, if it satisfies the following property: if α ∈ Ω and if β ∈ Π,with β ≤ st α , then β ∈ Ω. We can consider the quotient A/ h Ω i of A by the ideal generatedby Ω. Clearly, it is still a graded algebra and it is generated by the images in A/ h Ω i ofthe elements of Π \ Ω. The important point here is that A/ h Ω i inherits from A a naturalquantum graded A.S.L. structure on Π \ Ω (or, more precisely, on the canonical image ofΠ \ Ω in A/ h Ω i ). In particular, the set of homomorphic images in A/ h Ω i of the standardmonomials of A which either equal 1 or are of the form α . . . α t ( t ∈ N ∗ ) and α / ∈ Ω forma k -basis for A/ h Ω i . The reader will find all the necessary details in § Example 1.5 – Let γ ∈ Π m,n . It is clear that the set Π γm,n introduced in Definition 1.2 isa Π m,n -ideal. Hence, the discussion above shows that the quantum Schubert variety S ( γ )is a quantum graded A.S.L. on the canonical image in S ( γ ) of Π m,n \ Π γm,n . In particular,the canonical images in S ( γ ) of the standard monomials of O q ( G m,n ( k )) which either equalto 1 or are of the form [ I ] . . . [ I t ], for some t ≥ γ ≤ st [ I ], form a k -basis of S ( γ ). Remark 1.6 – Let γ ∈ Π m,n . As mentioned in Example 1.5, the quantum Schubertvariety S ( γ ) is a quantum graded A.S.L. on the canonical image in S ( γ ) of Π m,n \ Π γm,n . Atthis point, it is worth noting that the set Π m,n \ Π γm,n has a single minimal element, namely γ , and that the image of γ is a normal nonzerodivisor in S ( γ ), by [14, Lemma 1.2.1]. Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m ≥
0. The partitionsubalgebra A λ of O q ( M m,n ( k )) is defined to be the subalgebra of O q ( M m,n ( k )) generated bythe variables x ij with j ≤ λ i . By looking at the defining relations for quantum matrices,it is easy to see that A λ can be presented as an iterated Ore extension with the variables x ij added in lexicographic order. As a consequence, partition subalgebras are noetheriandomains. Recall that there is an action of a torus H := ( k ∗ ) m + n on O q ( M m,n ( k )) given by( α , . . . , α m , β , . . . , β n ) ◦ x ij := α i β j x ij . This action induces an action on A λ , by restriction.Our main aim in this section is to observe that the Goodearl-Letzter stratification theoryand the Cauchon theory of deleting derivations apply to partition subalgebras of quantummatrices. As a consequence, we can then exploit these theories to obtain information aboutthe prime and H -prime spectra of partition subalgebras.The conditions needed to use the theories have been brought together in the notionof a (torsion-free) CGL-extension introduced in [12, Definition 3.1]; the definition is givenbelow, for convenience. 7 efinition 2.1 An iterated skew polynomial extension A = k [ x ][ x ; σ , δ ] . . . [ x n ; σ n , δ n ]is said to be a CGL extension (after Cauchon, Goodearl and Letzter) provided that thefollowing list of conditions is satisfied: • With A j := k [ x ][ x ; σ , δ ] . . . [ x j ; σ j , δ j ] for each 1 ≤ j ≤ n , each σ j is a k -algebraautomorphism of A j − , each δ j is a locally nilpotent k -linear σ j -derivation of A j − ,and there exist nonroots of unity q j ∈ k ∗ with σ j δ j = q j δ j σ j ; • For each i < j there exists a λ ji ∈ k ∗ such that σ j ( x i ) = λ ji x i ; • There is a torus H = ( k ∗ ) r acting rationally on A by k -algebra automorphisms; • The x i for 1 ≤ i ≤ n are H -eigenvectors; • There exist elements h , . . . , h n ∈ H such that h j ( x i ) = σ j ( x i ) for j > i and suchthat the h j -eigenvalue of x j is not a root of unity.If, in addition, the subgroup of k ∗ generated by the λ ji is torsionfree then we will saythat A is a torsionfree CGL extension .For a discussion of rational actions of tori, see [1, Chapter II.2].It is easy to check that all of these conditions are satisfied for partition subalgebras (forexactly the same reasons that quantum matrices are CGL-extensions). Proposition 2.2
Partition subalgebras of quantum matrix algebras are CGL-extensionsand are torsion-free CGL extensions when the parameter q is not a root of unity.Proof: It is only necessary to show that we can introduce the variables x ij that define thepartition subalgebra in such a way that the resulting iterated skew polynomial extensionsatisfies the list of conditions above. Lexicographic ordering is suitable. Corollary 2.3
Let A λ be a partition subalgebra of quantum matrices and suppose that A λ is equipped with the induced action of H . Then A λ has only finitely many H -prime idealsand all prime ideals of A λ are completely prime when the parameter q is not a root ofunity. roof: This follows immediately from the previous result and [1, Theorem II.5.12 andTheorem II.6.9].In fact, we can be much more precise about the number of H -primes. We will provebelow that there exists a natural bijection between the H -prime spectrum of A λ andCauchon diagrams defined on the Young diagram corresponding to the partition λ .Suppose that Y λ is the Young diagram corresponding to the partition λ . Then a Cau-chon diagram on Y λ is an assignment of a colour, either white or black, to each square of thediagram Y λ in such a way that if a square is coloured black then either each square aboveis coloured black, or each square to the left is coloured black. These diagrams were firstintroduced by Cauchon, [3], in his study of the H -prime spectrum of quantum matrices.Recently, they have occurred with the name Le-diagrams in work of Postnikov, [16], andWilliams, [17]. Lemma 2.4
Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m > .The number of H -prime ideals in A λ is equal to the number of Cauchon diagrams definedon the Young diagram corresponding to the partition λ .Proof: Let n λ denote the number of H -prime ideals in A λ . First, we obtain a recurrencerelation for n λ .The H -prime spectrum of A λ can be written as a disjoint union: H -Spec( A λ ) = { J ∈ H -Spec( A λ ) | x m,λ m ∈ J } ⊔ { J ∈ H -Spec( A λ ) | x m,λ m / ∈ J } . It follows from the complete primeness of every H -prime ideal of A λ that an H -primeideal J of A λ that contains x m,λ m must also contain either x i,λ m for each i ∈ { , . . . , m } or x m,α for each α ∈ { , . . . , λ m } . Let I be the ideal generated by x i,λ m for i ∈ { , . . . , m } ,and let I be the ideal generated by x m,α for α ∈ { , . . . , λ m } . Set I := I + I . As A λ I ≃ A ( λ − ,λ − ,...,λ m − , A λ I ≃ A ( λ ,λ ,...,λ m − ) and A λ I ≃ A ( λ − ,λ − ,...,λ m − − , we obtain n λ = n ( λ − ,λ − ,...,λ m − + n ( λ ,λ ,...,λ m − ) − n ( λ − ,λ − ,...,λ m − − + |{ J ∈ H -Spec( A λ ) | x m,λ m / ∈ J }| . (Even though the above isomorphisms are not always H -equivariant, they preserve theproperty of being an H -prime.) 9s A λ is a CGL extension, one can apply the theory of deleting derivations to thisalgebra. In particular, it follows from [2, Th´eor`eme 3.2.1] that the multiplicative systemof A λ generated by x m,λ m is an Ore set in A λ , and A λ [ x − m,λ m ] ≃ A ( λ ,λ ,...,λ m − ,λ m − [ y ± ; σ ] , where σ is the automorphism of A ( λ ,λ ,...,λ m − ,λ m − defined by σ ( x iα ) = q − x iα if i = m or α = λ m , and σ ( x iα ) = x iα otherwise. Denote this isomorphism by ψ , and note that ψ ( x m,λ m ) = y . As x m,λ m is an H -eigenvector, the action of H on A λ extends to an actionof H on A λ [ x − m,λ m ], and so on A ( λ ,λ ,...,λ m − ,λ m − [ y ± ; σ ]. It is easy to show that thisaction restricts to an action on A ( λ ,λ ,...,λ m − ,λ m − which coincides with the “natural”action of H on this algebra. Hence the isomorphism ψ induces a bijection from { J ∈H -Spec( A λ ) | x m,λ m / ∈ J } to H -Spec( A ( λ ,λ ,...,λ m − ,λ m − [ y ± ; σ ]); and so it follows from [12,Theorem 2.3] that there exists a bijection between { J ∈ H -Spec( A λ ) | x m,λ m / ∈ J } and H -Spec( A ( λ ,λ ,...,λ m − ,λ m − ). Hence |{ J ∈ H -Spec( A λ ) | x m,λ m / ∈ J }| = n ( λ ,λ ,...,λ m − ,λ m − ;so that n λ = n ( λ − ,λ − ,...,λ m − + n ( λ ,λ ,...,λ m − ) − n ( λ − ,λ − ,...,λ m − − + n ( λ ,λ ,...,λ m − ,λ m − . On the other hand, it follows from [17, Remark 4.2] that the number of Cauchon di-agrams (equivalently, Le-diagrams) defined on the Young diagram corresponding to thepartition λ satisfies the same recurrence. As the number of H -prime ideals in A (1) is equalto 2 which is also the number of Cauchon diagrams defined on the Young diagram corre-sponding to the partition λ = (1), the proof is complete.Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m > A λ be the corresponding partition subalgebra of generic quantum matrices. Let C λ denote theset of Cauchon diagrams on the Young diagram Y λ corresponding to the partition λ . Wehave just seen that the sets H -Spec( A λ ) and C λ have the same cardinality. In fact, thereis a natural bijection between these two sets which carries over important algebraic andgeometric information. This natural bijection arises by using Cauchon’s theory of deletingderivations developed in [2] and [3].As A λ is a CGL extension, the theory of deleting derivations can be applied to theiterated Ore extension A λ = k [ x , ] . . . [ x m,λ m ; σ m,λ m , δ m,λ m ] (where the indices are increas-ing for the lexicographic order). Before describing the deleting derivations algorithm, weintroduce some notation. Denote by ≤ lex the lexicographic ordering on N and set E :=( F mi =1 { i } × { , . . . , λ i } ∪ { ( m, λ m + 1) } ) \ { (1 , } . If ( j, β ) ∈ E with ( j, β ) = ( m, λ m + 1),then ( j, β ) + denotes the least element (relative to ≤ lex ) of the set { ( i, α ) ∈ E | ( j, β ) < ( i, α ) } .10he deleting derivations algorithm constructs, for each r ∈ E , a family of elements x ( r ) i,α for α ≤ λ i of F := Frac( A λ ), defined as follows.1. If r = ( m, λ m + 1), then x ( m,λ m +1) i,α = x i,α for all ( i, α ) with α ≤ λ i .2. Assume that r = ( j, β ) < ( m, λ m + 1) and that the x ( r + ) i,α are already constructed.Then, it follows from [2, Th´eor`eme 3.2.1] that x ( r + ) j,β = 0 and, for all ( i, α ), we have: x ( r ) i,α = x ( r + ) i,α − x ( r + ) i,β (cid:16) x ( r + ) j,β (cid:17) − x ( r + ) j,α if i < j and α < βx ( r + ) i,α otherwise.As in [2], we denote by A λ the subalgebra of Frac( A λ ) generated by the indeterminatesobtained at the end of this algorithm; that is, we denote by A λ the subalgebra of Frac( A λ )generated by the t i,α := x (1 , i,α for each ( i, α ) such that α ≤ λ i . Cauchon has shown that A λ can be viewed as the quantum affine space A λ generated by indeterminates t ij for j ≤ λ i with relations t ij t il = qt il t ij for j < l , while t ij t kj = qt kj t ij for i < k , and allother pairs commute. Observe that the torus H still acts by automorphisms on A λ via( a , . . . , a m , b , . . . , b n ) .t ij = a i b j t ij . The theory of deleting derivations allows the explicit(but technical) construction of an embedding ϕ , called the canonical embedding, from H -Spec( A λ ) into the H -prime spectrum of A λ . The H -prime ideals of A λ are well-known:they are generated by the subsets of { t ij } . If C is a Cauchon diagram defined on theYoung tableau corresponding to λ , then we denote by K C the (completely) prime ideal of A λ generated by the subset of indeterminates t ij such that the square in position ( i, j ) isa black square of C . Theorem 2.5
Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m > and let A λ be the corresponding partition subalgebra of generic quantum matrices. Let C λ denote the set of Cauchon diagrams defined on the Young tableau corresponding to λ .For every Cauchon diagram C ∈ C λ , there exists a unique H -invariant (completely) primeideal J C of A λ such that ϕ ( J C ) = K C . Moreover there is no other H -prime in A λ ; so that H - Spec( A λ ) = { J C | C ∈ C λ } . Proof:
As the sets H -Spec( A λ ) and { J C | C ∈ C λ } have the same cardinality by theprevious lemma, it is sufficient to show that H -Spec( A λ ) ⊆ { J C | C ∈ C λ } . This inclusioncan be obtained by following the arguments of [3, Lemmes 3.1.6 and 3.1.7]. The detailsare left to the interested reader. 11 emark 2.6 Theorem 2.5 provides more than just an explicit bijection between the H -spectrum of A λ and C λ . This natural bijection carries algebraic and geometric data. Forexample, it can be shown that the height of J C is given by the number of black boxes of theCauchon diagram C . Also, the dimension of the H -stratum (in the sense of [1, Definition2.2.1]) associated to J C can be read off from the Cauchon diagram C .An algebra A is said to be catenary if for each pair of prime ideals Q ⊆ P of A allsaturated chains of prime ideals between Q and P have the same length. Our next aimis to show that partition subalgebras of quantum matrix algebras are catenary. The keyproperty that we need to establish in order to prove catenarity is the property of normalseparation. Two prime ideals Q $ P are said to be normally separated if there is anelement c ∈ P \ Q such that c is normal modulo Q . The algebra is normally separated ifeach such pair of prime ideals is normally separated. In our case, a result of Goodearl,see [7, Section 5], shows that it is enough to concentrate on the H -prime ideals. Supposethat A is a k -algebra with a torus H acting rationally. If Q is any H -invariant ideal of A then an element c is said to be H -normal modulo Q provided that there exists h ∈ H suchthat ca − h ( a ) c ∈ Q for all a ∈ A . Goodearl observes that in this case one may choosethe element c to be an H -eigenvector. The algebra A has H -normal separation providedthat for each pair of H -prime ideals Q $ P there exists an element c ∈ P \ Q such that c is H -normal modulo Q .A slightly weaker notion, also introduced by Goodearl, is that of normal H -separation .The algebra A has normal H -separation provided that for each pair of H -primes Q $ P there is an H -eigenvector c ∈ P \ Q which is normal modulo Q . Goodearl shows that inthe situation that we are considering, normal H -separation implies normal separation, see[7, Theorem 5.3].Notice that, as explained in paragraph 5.1 of [7], the action of H induces a grading on A by a suitable free abelian group. Using this grading, it is easy to see that A has normal H -separation if and only if for each pair of H -primes Q $ P there is an element c ∈ P \ Q whose image in A/Q is normal and an H -eigenvector. This fact will be freely used in thesequel.Recall, from [12, Definition 2.5], the definition of a Cauchon extension. Let A bea domain that is a noetherian k -algebra and let R = A [ X ; σ, δ ] be a skew polynomialextension of A . We say that R = A [ X ; σ, δ ] is a Cauchon Extension provided that • σ is a k -algebra automorphism of A and δ is a k -linear locally nilpotent σ -derivationof A . Moreover we assume that there exists q ∈ k ∗ which is not a root of unity suchthat σ ◦ δ = qδ ◦ σ . 12 There exists an abelian group H which acts on R by k -algebra automorphisms suchthat X is an H -eigenvector and A is H -stable. • σ coincides with the action on A of an element h ∈ H . • Since X is an H -eigenvector and since h ∈ H , there exists λ ∈ k ∗ such that h .X = λ X . We assume that λ is not a root of unity. • Every H -prime ideal of A is completely prime. Lemma 2.7
Suppose that R = A [ X ; σ, δ ] is a Cauchon extension. Moreover, assume that H is a torus and that the action of H on R is rational. If R has H -normal separation then A has H -normal separation.Proof: First, note that { X n } is an Ore set in R , by [2, Lemme 2.1]; and so we can form theOre localization b R := RS − = S − R . As X is an H -eigenvector, the rational action of H on R extends to a rational action on b R . We claim that b R has H -normal separation. Supposethat Q $ P are H -prime ideals of b R . Then Q ∩ R $ P ∩ R are distinct H -prime idealsof R . Thus, there exist an element c ∈ ( P ∩ R ) \ ( Q ∩ R ) and an element h ∈ H such that cr − h ( r ) c ∈ Q ∩ R for all r ∈ R . In particular, cX − λXc = cX − h ( X ) c ∈ Q ∩ R for some λ ∈ k ∗ , as X is an H -eigenvector. From this it is easy to calculate that ( λX ) − k c − cX − k ∈ Q .Now, let y = rX − k be an element of b R . Then, working modulo Q , we calculate cy = crX − k = h ( r )( λX ) − k c = h ( r ) h ( X − k ) c = h ( rX − k ) c = h ( y ) c ;so that b R has H -normal separation, as claimed.For each a ∈ A , set θ ( a ) = + ∞ X n =0 (1 − q ) − n [ n ]! q δ n ◦ σ − n ( a ) X − n ∈ b R (Note that θ ( a ) is a well-defined element of b R , since δ is locally nilpotent, q is not aroot of unity, and 0 = 1 − q ∈ k .)The following facts are established in [2, Section 2]. The map θ : A −→ b R is a k -algebra monomorphism. Let A [ Y ; σ ] be a skew polynomial extension. Then θ extends to amonomorphism θ : A [ Y ; σ ] −→ b R with θ ( Y ) = X . Set B = θ ( A ) and T = θ ( A [ Y ; σ ]) ⊆ b R .Then T = B [ X ; α ], where α is the automorphism of B defined by α ( θ ( a )) = θ ( σ ( a )).The element X is a normal element in T , and so the set S is an Ore set in T andCauchon shows that T S − = S − T = b R . Thus, b R = B [ X, X − ; α ]. Also, the H -actiontransfers to B via θ , by [12, Lemma 2.6]. Note, in particular, that α coincides with theaction of an element of H on B . 13hus, it is enough to show that B has H -normal separation, given that B [ X, X − ; α ]has H -normal separation.Let Q $ P be H -prime ideals of B . Set b Q = ⊕ i ∈ Z QX i and b P = ⊕ i ∈ Z P X i . Then b Q ∩ B = Q and b P ∩ B = P , and it follows that b Q $ b P are H -prime ideals in B [ X, X − ; α ],see [12, Theorem 2.3]. As B [ X, X − ; α ] has H -normal separation, there is an element c ∈ b P \ b Q and an element h ∈ H such that cs − h ( s ) c ∈ b Q , for each s ∈ B [ X, X − ; α ]. Now,write c = P i ∈ Z c i X i . Note that each c i ∈ P and at least one c i Q , say c i Q . Let b ∈ B . Then, cb − h ( b ) c ∈ b Q . Therefore, P i c i X i b − h ( b ) c i X i ∈ b Q ; and so X i ( c i α i ( b ) − h ( b ) c i ) X i ∈ b Q As b Q = ⊕ i ∈ Z QX i , this forces c i α i ( b ) − h ( b ) c i ∈ Q for each i , and, in particular, c i α i ( b ) − h ( b ) c i ∈ Q . As b was an arbitrary element of B , we may replace b by α − i ( b ) to obtain c i b − hα − i ( b ) c i ∈ Q As α coincides with the action of an element of H on B , this produces an element h i ∈ H such that c i b − h i ( b ) c i ∈ Q, as required to show that B has H -normal separation. Theorem 2.8
Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m ≥ and let A λ be the corresponding partition subalgebra of generic quantum matrices. Then A λ has H -normal separation.Proof: Let µ = ( n, . . . , n ) ( m times); so that Y µ is an m × n rectangle. Then A µ = O q ( M m,n ( k )); and so A µ has H -normal separation, by [3, Th´eor`eme 6.3.1]. We can con-struct A µ from A λ by adding the missing variables x ij in lexicographic order. At each stage,the extension is a Cauchon extension. Thus, A λ has H -normal separation, by repeatedapplication of the previous lemma. Corollary 2.9
Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m ≥ and let A λ be the corresponding partition subalgebra of generic quantum matrices. Then A λ has normal H -separation and normal separation.Proof: We have seen earlier that H -normal separation implies normal H -separation.Normal separation now follows from [7, Theorem 5.3]. Corollary 2.10
Let λ = ( λ , λ , . . . , λ m ) be a partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m ≥ and let A λ be the corresponding partition subalgebra of generic quantum matrices. Then A λ is catenary. roof: This follows from the previous results and [18, Theorem 0.1] which states thatif A is a normally separated filtered k -algebra such that gr( A ) is a noetherian connectedgraded k -algebra with enough normal elements then Spec( A ) is catenary. (For the notionof an algebra with enough normal elements see [19].)Note that it is also possible to deduce this result from [8, Theorem 1.6] Quantum Schubert cells in the quantum grassmannian are obtained from quantum Schu-bert varieties via the process of noncommutative dehomogenisation introduced in [10].Recall that if R = ⊕ R i is an N -graded k -algebra and x is a regular homogeneous normalelement of R of degree one, then the dehomogenisation of R at x , written Dhom( R, x ), isdefined to be the zero degree subalgebra S of the Z -graded algebra S := R [ x − ]. If R isgenerated as a k -algebra by a , a , . . . , a s and each a i has degree one, then it is easy tocheck that Dhom( R, x ) = k [ a x − , . . . , a s x − ].If σ denotes the automorphism of S given by σ ( s ) = xsx − for s ∈ S then σ inducesan automorphism of S , also denoted by σ , and there is an isomorphism θ : Dhom( R, x )[ y, y − ; σ ] −→ R [ x − ]which is the identity on Dhom( R, x ) and sends y to x .Let γ ∈ Π m,n . Recall from Remark 1.6 that S ( γ ) = O q ( G m,n ( k )) / h Π γm,n i and that γ isa homogeneous regular normal element of degree one in S ( γ ). It follows that we can formthe localisation S ( γ )[ γ − ] and that S ( γ ) ⊆ S ( γ )[ γ − ]. Definition 3.1
The quantum Schubert cell associated to the quantum minor γ is denotedby S o ( γ ) and is defined to be Dhom( S ( γ ) , γ ) . Remark 3.2
In the classical case when q = 1, it can be seen that this definition coincideswith the usual definition of Schubert cells, as discussed, for example, in [4, Section 9.4]It follows from the definition that S o ( γ ) is generated by the elements x γ − , for x ∈ Π m,n \ (Π γm,n ∪ { γ } ). However, these elements are not independent; so we will pick out abetter generating set for the quantum Schubert cell.This is achieved by using the quantum ladder matrix algebras introduced in [15, Section3.1]. Let us recall the definition. To each γ = ( γ , . . . , γ m ) ∈ Π m,n , with 1 ≤ γ < · · · < m ≤ n , we associate the substet L γ of { , . . . , m } × { , . . . , n } defined by L γ = { ( i, j ) ∈ { , . . . , m } × { , . . . , n } | j > γ m +1 − i and j = γ ℓ for 1 ≤ ℓ ≤ m } , which we call the ladder associated with γ .Consider the quantum minors m ij defined by m ij := [ { γ , . . . , γ m } \ { γ m +1 − i } ∪ { j } ],for each ( i, j ) ∈ L γ . These are the quantum minors that are above γ in the standard orderand differ from γ in exactly one position. Denote the set of these quantum minors by M γ . Proposition 3.3 S o ( γ ) = k [ m ij γ − | m ij ∈ M γ ] Proof:
In the proof of [15, Theorem 3.1.6] it is shown that S ( γ )[ γ − ] is generated bythe elements γ, γ − and the m ij . The Schubert cell S o ( γ ) is the degree zero part of thisalgebra. As γ and m ij commute up to scalars, see [15, Lemma 3.1.4(v)], it is easy to checkthat S o ( γ ) is generated by m ij γ − , as required.Set g m ij := m ij γ − . Lemma 3.4
There is an induced action of H = ( k ∗ ) n on S o ( γ ) given by ( α , α , . . . , α n ) ◦ g m ij := α − γ m +1 − i α j g m ij . Proof:
This follows immediately from the fact that g m ij = [ { γ , . . . , γ m }\{ γ m +1 − i } ∪ { j } ] [ γ , . . . , γ m ] − . We now need to establish the commutation relations between the g m ij . Definition 3.5 – Let γ = ( γ , . . . , γ m ) ∈ Π m,n , with ≤ γ < · · · < γ m ≤ n . The quantumladder matrix algebra associated with γ , denoted O q ( M m,n,γ ( k )) , is the k -subalgebra of O q ( M m,n ( k )) generated by the elements x ij ∈ O q ( M m,n ( k )) such that ( i, j ) ∈ L γ . The following example, taken from [15] will help clarify this definition.
Example 3.6 – We put ( m, n ) = (3 ,
7) and γ = ( γ , γ , γ ) = (1 , , ∈ Π , . In the 3 × X = ( x ij ) associated with O q ( M , ( k )), put a bullet on each row as follows:on the first row, the bullet is in column 6 because γ is 6, on the second row, the bullet isin column 3 because γ is 3 and on the third row, the bullet is in column 1 because γ = 1.16ow, in each position which is to the left of a bullet, or which is below a bullet, put a star.To finish, place x ij in any position ( i, j ) that has not yet been filled. We obtain ∗ ∗ ∗ ∗ ∗ • x ∗ ∗ • x x ∗ x • x ∗ x x ∗ x . By definition, the quantum ladder matrix algebra associated with γ = (1 , ,
6) is thesubalgebra of O q ( M , ( k )) generated by the elements x , x , x , x , x , x , x , x .Notice that if we rotate the matrix above through 180 ◦ then the x ij involved in thedefinition of O q ( M , ,γ ( k )) sit naturally in the Young Diagram of the partition λ = (4 , , Lemma 3.7
The quantum Schubert cell S o ( γ ) is isomorphic to the quantum ladder matrixalgebra O q ( M m,n,γ ( k )) .Proof: Lemma 3.1.4 of [15] shows that the commutation relations for the m ij are thesame as the commutation relations for corresponding variables x ij in the quantum laddermatrix algebra O q ( M m,n,γ ( k )). As γm ij = qm ij γ , for each i, j , by [15, Lemma 3.1.4(v)],it follows that the g m ij satisfy the same relations. Thus there is an epimorphism from O q ( M m,n,γ ( k )) onto S o ( γ ). A comparison of Gelfand-Kirillov dimensions similar to thatused in [15, Theorem 3.1.6] now shows that this epimorphism is in fact an isomorphism. Theorem 3.8
The quantum Schubert cell S o ( γ ) is (isomorphic to) a partition subalgebraof O q − ( M m,n − m ( k )) .Proof: For any n , there is an isomorphism δ : O q ( M n ( k )) −→ O q − ( M n ( k )) defined by δ ( x ij ) = x n +1 − i,n +1 − j , see the proof of [9, Corollary 5.9]. The isomorphism δ can be usedto convert quantum ladder matrix algebras into partition subalgebras. As Schubert cellsare isomorphic to quantum ladder matrix algebras, the result follows.The isomorphism described in the previous result carries over the H -action on S o ( γ ) tothe partition subalgebra, and this induced action acts via row and column multiplications.After suitable re-numbering of the components of H , this action coincides with the actiondiscussed at the beginning of Section 2. As a consequence of Theorem 3.8, the resultsobtained in Section 2 apply to quantum Schubert cells. In particular, the following resultshold. 17 heorem 3.9 Let λ = ( λ , λ , . . . , λ m ) be the partition with n ≥ λ ≥ λ ≥ · · · ≥ λ m ≥ defined by λ i + γ i = n − m + i and let Y λ be the corresponding Young diagram. Then the H -prime spectrum of S o ( γ ) is in bijection with the set of Cauchon diagrams on the Youngdiagram, Y λ , as described in Theorem 2.5. Theorem 3.10
The quantum Schubert cell S o ( γ ) has H -normal separation, normal H -separation and normal separation. Corollary 3.11
The quantum Schubert cell S o ( γ ) is catenary. In this section, we use the quantum Schubert cells to obtain information concerning theprime spectrum of the quantum grassmannian. We show that, in the generic case, where q is not a root of unity, all primes are completely prime and that there are only finitely manyprimes that are invariant under the natural torus action on the quantum grassmannian.By using a result of Lauren Williams, we are able to count the number of H -primes. Also,we are able to show that the quantum grassmannian is catenary.Note that the following result is valid for any q = 0. Theorem 4.1
Let P be a prime ideal of O q ( G m,n ( k )) with P = h Π i ; that is, P is not theirrelevant ideal. Then there is a unique γ in Π with the property that γ P but π ∈ P forall π st γ .Proof: If Π ⊆ P then P is the irrelevant ideal. Otherwise, there exists γ ∈ Π with γ P .Choose such a γ that is minimal in Π with this property. Then λ ∈ P for all λ < st γ .Note that h{ λ | λ < st γ }i ⊆ P and that γ is normal modulo h{ λ | λ < st γ }i , by [14,Lemma 1.2.1]; so that γ is normal modulo P .Suppose that π st γ . If π < st γ then π ∈ P by the choice of γ . If not, then π and γ are not comparable. Thus, we can write πγ = X k λµ λµ with k λµ ∈ k while λ, µ ∈ Π with λ < st γ , by [14, Theorem 3.3.8].It follows that πγ ∈ P . Thus, π ∈ P , since γ P and γ is normal modulo P .This shows that there is a γ with the required properties. It is easy to check that therecan only be one such γ . 18his enables us to give a decomposition of the prime spectrum, Spec( O q ( G m,n ( k ))).Set Spec γ ( O q ( G m,n ( k ))) to be the set of prime ideals P such that γ P while π ∈ P forall π st γ . The previous result shows thatSpec( O q ( G m,n ( k ))) = G γ ∈ Π Spec γ ( O q ( G m,n ( k ))) G h Π i . We now re-instate our convention that q is not a root of unity. Theorem 4.2
Let q be a non root of unity. Then all prime ideals of the quantum grass-mannian O q ( G m,n ( k )) are completely prime.Proof: Let P be a prime ideal of O q ( G m,n ( k )). If P = h Π i then O q ( G m,n ( k )) /P ∼ = k ; so P is completely prime.Otherwise, suppose that P ∈ Spec γ ( O q ( G m,n ( k ))). In this case, P = P/ h Π γm,n i is aprime ideal in S ( γ ) = O q ( G m,n ( k )) / h Π γm,n i and it is enough to show that P is completelyprime. Set T := S ( γ )[ γ − ]. Then P T is a prime ideal of T and P T ∩ S ( γ ) = P . Thus S ( γ ) /P ⊆ T /P T and so it is enough to show that
P T is completely prime.Now, the dehomogenisation isomorphism shows that T ∼ = S o ( γ )[ y, y − ; σ ], where σ is theautomorphism determined by the conjugation action of γ , see the beginning of Section 3.We know that S o ( γ ) is a torsionfree CGL-extension by Proposition 2.2 and Theorem 3.8.It is then easy to check that S o ( γ )[ y ; σ ] is also a torsionfree CGL-extension. Thus, all primeideals of S o ( γ )[ y ; σ ] are completely prime, by [1, Theorem II.6.9], and it follows that allprime ideals of T ∼ = S o ( γ )[ y, y − ; σ ] are completely prime, as required.Of course, the decomposition of Spec( O q ( G m,n ( k ))) above induces a similar decompo-sition of H -Spec( O q ( G m,n ( k ))): H -Spec( O q ( G m,n ( k ))) = G γ ∈ Π H -Spec γ ( O q ( G m,n ( k ))) G h Π i , where H -Spec γ ( O q ( G m,n ( k ))) is the set of H -prime ideals P such that γ P while π ∈ P for all π st γ .Our next task is to show that H -Spec γ ( O q ( G m,n ( k ))) is in natural bijection with H -Spec( S o ( γ )) and hence in bijection with Cauchon diagrams on the associated Youngdiagram Y λ . As a consequence, we are able to calculate the size of H -Spec( O q ( G m,n ( k ))). Remark 4.3
Recall from the beginning of Section 3 that, for any γ ∈ Π m,n , there is thedehomogenisation isomorphism θ : S o ( γ )[ y, y − ; σ ] −→ S ( γ )[ γ − ] , σ is conjugation by γ . Hence, the action of H on S ( γ )[ γ − ] transfers, via θ , toan action on S o ( γ )[ y, y − ; σ ]. By Lemma 3.4, S o ( γ ) is stable under this action and itis clear that y is an H -eigenvector. Further, let h = ( α , . . . , α n ) ∈ H be such that α i = q if i / ∈ { γ , . . . , γ m } and α i = q otherwise. Then, by using [15, Lemma 3.1.4(v)] andLemma 3.4, it is easily verified that the action of h on S o ( γ ) coincides with σ . In addition, h ( y ) = q m y , since h ( γ ) = q m γ . It follows that S o ( γ )[ y, y − ; σ ] satisfies Hypothesis 2.1 in[12]. Theorem 4.4
Let P ∈ H - Spec γ ( O q ( G m,n ( k ))) ; so that P is an H -prime ideal of O q ( G m,n ( k )) such that γ P , while π ∈ P for all π st γ . Set T = S ( γ )[ γ − ] ∼ = S o ( γ )[ y, y − ; σ ] .Then the assignment P P T ∩ S o ( γ ) defines an inclusion-preserving bijection from H - Spec γ ( O q ( G m,n ( k ))) to H - Spec( S o ( γ )) , with inverse obtained by sending Q to the in-verse image in O q ( G m,n ( k )) of QT ∩ S ( γ ) . (Note, we are treating the isomorphism aboveas an id entification in these assignments.)Proof: This follows from the conjunction of two bijections. First, standard localisationtheory shows that P = P T ∩ S ( γ ); and this gives a bijection between H -Spec γ ( O q ( G m,n ( k )))and H -Spec( T ). For the second bijection, note that T ∼ = S o ( γ )[ y, y − ; σ ] and that the au-tomorphism σ is given by the action of an element of H , see Remark 4.3. Thus, it followsfrom [12, Theorem 2.3] that there is a bijection between H -Spec( T ) and H -Spec( S o ( γ ))given by intersecting an H -prime of T with S o ( γ ). The composition of these two bijectionsproduces the required bijection. Corollary 4.5 H - Spec γ ( O q ( G m,n ( k ))) is in bijection with the Cauchon diagrams on Y λ ,where λ is the partition associated with γ .Proof: This follows from the previous theorem and Theorem 3.9.It follows from this corollary and the partition of the H -spectrum of the quantumgrassmannian that the H -spectrum of the quantum grassmannian is finite. This finiteness isa crucial condition needed to investigate normal separation, Dixmier-Moeglin equivalence,etc. in the quantum case because of the stratification theory, see, for example, [7, Theorem5.3], [1, Theorem II.8.4 ]. However, in this situation, we can say much more: we can sayexactly how many H -primes there are in the quantum grassmannian O q ( G m,n ( k )). Thisis one more (the irrelevant ideal h Π i ) than the total number of Cauchon diagrams on theYoung diagrams Y λ corresponding to the partitions λ that fit into the partition ( n − m ) m .This combinatorial problem has been solved by Lauren Williams, in [17]. The followingresult is obtained by setting q = 1 in the formula for A k,n ( q ) in [17, Theorem 4.1].20 heorem 4.6 |H - Spec( O q ( G m,n ( k ))) | = 1 + m − X i =0 (cid:18) ni (cid:19) (cid:0) ( i − m ) i ( m − i + 1) n − i − ( i − m + 1) i ( m − i ) n − i (cid:1) Proof:
By using the results above, we see that, except for the irrelevant ideal, each H -prime corresponds to a unique Cauchon diagram drawn on the Young diagram Y λ thatcorresponds to the partition λ associated to the quantum minor γ which determines thecell that P is in.In [17, Theorem 4.1], Lauren Williams has counted the number of Cauchon diagramson the Young diagrams Y λ that fit into the partition ( n − m ) m ; and this count, plus one,gives the number of H -prime ideals of O q ( G m,n ( k )).For example, |H -Spec( O q ( G , )) | = 34 and |H -Spec( O q ( G , )) | = 884. (These numberscan be seen from the table in [16, Figure 23.1].)We turn now to the questions of normal separation and catenarity. In order to establishthese properties for the quantum grassmannian, we need to use the dehomogenisationisomorphism. Recall that the methods of [12] are available because of Remark 4.3. Lemma 4.7
Let Q $ P be H -prime ideals in S ( γ ) that do not contain γ . Then, there isan H -eigenvector in P \ Q that is normal modulo Q .Proof: Let Q $ P be H -prime ideals in S ( γ ) that do not contain γ . Set T := S ( γ )[ γ − ]and observe that there is an induced action of the torus H on T , because γ is an H -eigenvector. Note that P = P T ∩ S ( γ ) and Q = QT ∩ S ( γ ); so QT $ P T are H -primeideals in T . Now, set P := P T ∩ S o ( γ ) and Q := QT ∩ S o ( γ ) (here, we are treatingthe isomorphism T ∼ = S o ( γ )[ y, y − ; σ ] as an identification) and note that P T = ⊕ n ∈ Z P y n and QT = ⊕ n ∈ Z Q y n ; so Q $ P are H -prime ideals of S o ( γ ), see Remark 4.3 and [12,Theorem 2.3]. These observations make it clear that S o ( γ ) Q [ y, y − ; σ ] ∼ = TQT ∼ = S ( γ ) Q [ γ − ] . As usual, S o ( γ ) will denote S o ( γ ) /Q , etc.The quantum Schubert cell S o ( γ ) has H -normal separation, by Theorem 3.10. Thus,there exists an H -eigenvector c ∈ P \ Q and an element h ∈ H such that ca − h ( a ) c ∈ Q for all a ∈ S o ( γ ). Recall that the action of σ coincides with the action of an element h y of H ; so that yc = h y ( c ) y = λcy for some λ ∈ k ∗ . It follows that c is normal in T /QT .Define σ c : T /QT −→ T /QT by ct = σ c ( t ) c for all t ∈ T . Note that σ c | S o ( γ ) = h | S o ( γ ) andthat σ c ( y ) = λ − y . 21e claim that σ c ( S ( γ ) /Q ) = S ( γ ) /Q ; so that σ c induces an isomorphism on thisalgebra. In order to see this, note that S ( γ ) /Q is generated as an algebra by the imagesof the quantum minors [ α , . . . , α m ] for [ α , . . . , α m ] ≥ γ . Now, [ α , . . . , α m ] γ − ∈ S o ( γ ),because [ α , . . . , α m ] γ − has degree zero in T so that [ α , . . . , α m ] γ − ∈ S o ( γ ). Thus,recalling that γ is identified with y under the isomorphisms above, σ c ([ α , . . . , α m ]) = σ c ([ α , . . . , α m ] γ − ) σ c ( γ ) = h ([ α , . . . , α m ] γ − )( λ − y )= µ [ α , . . . , α m ] γ − ( λ − y ) = ( µλ − )[ α , . . . , α m ] γ − y = ( µλ − )[ α , . . . , α m ] , where the existence of µ ∈ k ∗ is guaranteed because h is acting as a scalar on the element[ α , . . . , α m ] γ − ∈ S o ( γ ) /Q . The claim follows.There exists d ≥ c γ d ∈ S ( γ ) /Q . It is obvious that cγ d is an H -eigenvector,because each of c and γ is an H -eigenvector. Also, cγ d ∈ P \ Q . Finally, c γ d is normal in S ( γ ) /Q , because S ( γ ) /Q is invariant under conjugation by each of c and γ . Theorem 4.8
The quantum grassmannian O q ( G m,n ( k )) has normal H -separation andhence normal separation.Proof: Suppose that Q $ P are H -prime ideals of O q ( G m,n ( k )). Suppose that Q ∈ Spec γ ( O q ( G m,n ( k ))). If γ ∈ P , then P contains the H -eigenvector γ .Otherwise, γ P and P ∈ Spec γ ( O q ( G m,n ( k ))). In this case, it is enough to show thatthere is a H -eigenvector in P \ Q which is normal modulo Q , where P = P/ h Π γm,n i and Q = Q/ h Π γm,n i are H -prime ideals in S ( γ ). However, this has been done in the previouslemma. Theorem 4.9
The quantum grassmannian O q ( G m,n ( k )) is catenary.Proof: As in Corollary 2.10, this follows from the previous results and [18, Theorem 0.1].
Remark 4.10
It is obvious from the style of proof of the preceding results that there isnow a good strategy for producing results concerning the quantum grassmannian: first,establish the corresponding results for partition subalgebras of quantum matrices, andthen use the theory of quantum Schubert cells and noncommutative dehomogenisation toobtain the result in the quantum grassmannian. We leave any further developments forinterested readers. 22
Concluding remark.
We end this work by stressing some important connections between the results establishedin Section 4 above, and recent work of Postnikov in total positivity, see [16].Let M + m,n ( R ) denote the space of m × n real matrices of rank m and whose m × m minors are nonnegative. The group GL + m ( R ) of m × m real matrices of positive determi-nant act naturally on M + m,n ( R ) by left multiplication. The corresponding quotient space G + m,n ( R ) = M + m,n ( R ) /GL + m ( R ) is the totally nonnegative grassmannian of m dimensionalsubspaces in R n . One can define a cellular decomposition of G + m,n ( R ) by specifying, foreach element of G + m,n ( R ), which m × m minors are zero and which are strictly positive. Thecorresponding cells are called the totally nonnegative cells of G + m,n ( R ). In [16], Postnikovshows that totally nonnegative cells in G + m,n ( R ) are in bijection with the Cauchon diagramson partitions fitting into the partition ( n − m ) m . For further details, see Sections 3 and 6in [16].Hence, by the results in Section 4 above, the set of totally nonnegative cells of G + m,n ( R )is in one-to-one correspondance with the set of H -prime ideals of O q ( G m,n ( k )) distinct fromthe augmentation ideal. We believe it would be interesting to understand this coincidenceand we intend to pursue this theme in a subsequent paper. References [1] K A Brown and K R Goodearl, Lectures on algebraic quantum groups. AdvancedCourses in Mathematics-CRM Barcelona. Birkh¨auser Verlag, Basel, 2002[2] G Cauchon,
Effacement des d´erivations et spectres premiers des alg`ebres quantiques ,J Algebra 260 (2003), 476-518.[3] G Cauchon,
Spectre premier de O q ( M n ( k )) : image canonique et s´eparation normale JAlgebra 260 (2003), no. 2, 519–569.[4] W Fulton, Young tableaux. With applications to representation theory and geome-try. London Mathematical Society Student Texts, 35. Cambridge University Press,Cambridge, 1997.[5] I M Gelfand and V S Retakh,
Determinants of matrices over noncommutative rings .(Russian) Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 13–25, 96; translation inFunct. Anal. Appl. 25 (1991), no. 2, 91–102236] N Gonciulea and V Lakshmibai, Flag Varieties, Actualit´es Math´ematiques, Hermann,Paris, 2001[7] K R Goodearl,
Prime spectra of quantized coordinate rings , Interactions between ringtheory and representations of algebras (Murcia), 205–237, Lecture Notes in Pure andAppl Math, 210, Dekker, New York, 2000[8] K R Goodearl and T H Lenagan,
Catenarity in quantum algebras , J Pure Appl Algebra111 (1996), 123–142[9] K R Goodearl and T H Lenagan,
Winding invariant prime ideals in quantum 3x3matrices , Journal of Algebra 260 (2003), 657-687[10] A Kelly, T H Lenagan and L Rigal,
Ring theoretic properties of quantum grassman-nians , J Algebra Appl 3 (2004), no. 1, 9–30[11] G R Krause and T H Lenagan, Growth of algebras and Gelfand-Kirillov dimension,Revised edition, Graduate Studies in Mathematics, 22, American Mathematical Soci-ety, Providence, RI, 2000[12] S Launois, T H Lenagan and L Rigal,
Quantum unique factorisation domains
J LondonMath Soc 74 (2006), 321-340[13] T H Lenagan and L Rigal,
The maximal order property for quantum determinantalrings , Proc Edinb Math Soc (2) 46 (2003), no. 3, 513–529[14] T H Lenagan and L Rigal,
Quantum graded algebras with a straightening law and theAS-Cohen-Macaulay property for quantum determinantal rings and quantum grass-mannians , J Algebra 301 (2006), no. 2, 670–702[15] T H Lenagan and L Rigal,
Quantum analogues of Schubert varieties in the grassman-nian , to appear in Glasgow Mathematical Journal[16] A Postnikov,
Total positivity, grassmannians, and networks , arxiv:math.CO/0609764[17] L K Williams,
Enumeration of totally positive Grassmann cells , Adv. Math. 190(2005), no. 2, 319–342[18] A Yekutieli and J J Zhang
Rings with Auslander Dualizing Complexes , J Algebra 213(1999), 1-51[19] J J Zhang,