On Cohen-Macaulayness of S_n-invariant subspace arrangements
aa r X i v : . [ m a t h . A C ] J un ON COHEN–MACAULAYNESS OF S n -INVARIANTSUBSPACE ARRANGEMENTS AARON BROOKNER, DAVID CORWIN, PAVEL ETINGOF, AND STEVEN V SAM
Abstract.
Given a partition λ of n , consider the subspace E λ of C n where the first λ coordinates are equal, the next λ coordinates are equal, etc. In this paper, we studysubspace arrangements X λ consisting of the union of translates of E λ by the symmetricgroup. In particular, we focus on determining when X λ is Cohen–Macaulay. This is inspiredby previous work of the third author coming from the study of rational Cherednik algebrasand which answers the question positively when all parts of λ are equal. We show that X λ is not Cohen–Macaulay when λ has at least 4 distinct parts, and handle a large number ofcases when λ has 2 or 3 distinct parts. Along the way, we also settle a conjecture of Sergeevand Veselov about the Cohen–Macaulayness of algebras generated by deformed Newtonsums. Our techniques combine classical techniques from commutative algebra and invarianttheory; in many cases we can reduce an infinite family to a finite check which can sometimesbe handled by computer algebra. Contents
1. Introduction 12. Preliminaries and known results 33. The formal neighborhoods method 54. The algebra of deformed Newton sums 65. The case λ = ( a, b, c ). 96. The CM property of X b + c,b,c Introduction
Let λ = ( λ , . . . , λ r ) be a partition of n . Then we have a subspace E λ in C n defined bythe equations x = · · · = x λ , x λ +1 = · · · = x λ + λ , . . . , x n − λ r +1 = · · · = x n . Let S n be the group of permutations of n elements, and X λ = S n · E λ be the union of the S n -translates of E λ . Then X λ is an algebraic variety with an action of S n . Note that dim X λ = r , and X λ = X λ × C , where X λ = { x ∈ X λ | X x i = 0 } Date : June 13, 2015.2010
Mathematics Subject Classification. and dim X λ = r − Question 1.1. (i) For which λ is the variety X λ Cohen–Macaulay (CM)?(ii) For which λ is the variety X λ /S n CM?Clearly, in these questions we can replace X λ with X λ (this does not change the questions).Also, it is clear that if the answer to Question 1.1(i) is “yes” for some partition λ , then theanswer to Question 1.1(ii) for this partition is also “yes”.Questions 1.1(i),(ii) seem to be difficult to treat by standard methods of commutativealgebra (see [SS] for a survey of the commutative algebra of subspace arrangements; wealso point out [GW, Re, Y] for treatments of the Cohen–Macaulayness property of subspacearrangements). They are motivated by the paper [EGL], which gives a full answer to Question1.1(i) in the special case λ = ( m r , s ). The argument of [EGL] uses representation theory ofCherednik algebras, while an argument based on classical methods is unknown. This is soeven in the case r = 1, addressed in [BGS]. Also, Question 1.1(ii) for partitions of the form( m r , p s ) is asked in [SV1].We also point out the paper [DS], in which the Cohen–Macaulay property of X λ for thespecial case λ = ( m, s ) is used to calculate character formulas for modular irreduciblerepresentations of Cherednik algebras. A curious property of these subspace arrangements isthat they are defined over Z (and flat over Z ) but the Cohen–Macaulay property can dependon characteristic (see [BGS, Example 5.2]). It would be of interest to understand what isfailing in small characteristics, but in this paper we only work over characteristic 0.While the full answer to either of the Questions 1.1(i),(ii) is unknown to us, we answerthem in many cases, combining the methods of representation theory of Cherednik alge-bras, standard methods of commutative algebra (such as Reisner’s theorem, [Re]), and com-puter calculations in Macaulay2 [M2]. In particular, we show that the answer to Questions1.1(i),(ii) is “no” if λ has at least four distinct parts, or three distinct parts such that thelargest one is not equal to the sum of two smaller ones. We also formulate some conjecturesfor the remaining cases based on computational evidence. Finally, we settle a conjecturefrom [SV1] (end of Section 4) stating that the algebra generated by deformed Newton sums a ( y i + · · · + y ir ) + ( z i + · · · + z is ) , i = 1 , , . . . is CM for generic complex a . Finally, we discuss the set of values of a for which this propertyfails.Our main conclusion is that the CM property is rather rarely satisfied for X λ and X λ /S n ,and whenever it is, there is often some structure behind it, coming from representation theoryand integrable systems, which is instrumental in the proof of the CM property. Namely, ourmain results are summarized by the following theorem. Theorem 1.2. If all parts of λ are equal, i.e., λ = ( m r ) , then X λ and hence X λ /S n isCM (Theorem 2.1).
2. (a) If λ = ( m r , s ) , where m > , r > , s > , then X λ and X λ /S n are CM if and only ifeither m = 2 or s < m (Theorem 2.1, Proposition 4.7). (b) Let λ = ( m r , p s ) , m > p > , r > , s > , and let m = kb , p = kc , with k ∈ Z > and gcd( b, c ) = 1 . If c ≤ r , ≤ b ≤ s , then X λ /S n and hence X λ is not CM (Proposition4.7). (c) If c = 1 and b = 2 or b > s then X λ /S n is CM (Theorem 2.1). N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 3 (d) For fixed r > , s > , X λ /S n is CM for all but finitely many pairs ( b, c ) (Theorem4.4). (e) X (4 , , is CM, while X (3 r , s ) is not CM for r ≥ , s ≥ , r + s ≥ and X (5 r , s ) is notCM for r ≥ , s ≥ (Proposition 5.1).
3. (a) If λ has three distinct parts a > b > c and a = b + c then X λ /S n and hence X λ is notCM (Theorem 7.1). (b) If λ = ( b + c, b r , c s ) then X λ is not CM (Theorem 7.2). If λ has at least four distinct parts, then X λ /S n and hence X λ is not CM (Theorem 7.1). Unfortunately, the theorem still leaves open infinite families of cases when λ has 2 or 3distinct parts. We give some conjectures for next steps in § Supporting files.
A few of the proofs in this paper reduce to a finite computation, which weperform with the computer algebra system Macaulay2 [M2]. We have included the scriptsto perform these calculations as ancillary files in the arXiv submission of this paper. Inthe proofs where they are used, we have included a description of the calculations beingperformed.
Acknowledgments.
P. E. is grateful to A. Polishchuk for discussions that led to this re-search, and to A. Veselov and M. Feigin for explanations concerning the paper [SV1]. We alsothank Eric Rains who verified Propositions 5.13 and 7.5 in this paper using MAGMA. Thework of P. E. was partially supported by the NSF grant DMS-1000113. The work of A. B.and D. C. was done in the Summer Program of Undergraduate Research at the Mathematicsdepartment of MIT. The work of S. S. is supported by a Miller research fellowship.2.
Preliminaries and known results
Recall that a finitely generated commutative C -algebra R (or, equivalently, the scheme X = Spec( R )) is Cohen–Macaulay (CM) if R is a finitely generated (locally) free moduleover some polynomial subalgebra C [ u , . . . , u r ] ⊂ R . In this case, R is a (locally) free moduleover any such subalgebra over which it is finitely generated (as a module). For more detailsabout the CM property see [Eis, Chapter 18].The following theorem was proved in [EGL, Proposition 3.11]: Theorem 2.1.
Suppose that λ = ( m r , s ) (i.e., r copies of m and s copies of ). Then X λ is CM if and only if either m > s , or m ≤ . The proof is based on the representation theory of rational Cherednik algebras and Reis-ner’s theorem, [Re].Also, note that if λ = ( p, q ) then X λ is automatically CM, since X λ is 1-dimensional.Let us now consider the quotient X λ /S n . The algebra of functions on this variety isdescribed by the following proposition. Proposition 2.2.
The algebra O ( X λ /S n ) is the subalgebra of C [ y , . . . , y r ] generated by theNewton λ -sums P i,λ ( y , . . . , y r ) := λ y i + · · · + λ r y ir . Proof.
It is easy to see that the map E λ → X λ /S n is surjective, so the corresponding pullbackdefines an inclusion of O ( X λ /S n ) into C [ E λ ]. The ring C [ E λ ] can be realized as C [ y , . . . , y r ],where y is the value of x i for 1 ≤ i ≤ λ , y is the value of x i for λ < i ≤ λ + λ , etc. AARON BROOKNER, DAVID CORWIN, PAVEL ETINGOF, AND STEVEN V SAM
Now, O ( X λ /S n ) is generated by ordinary power (=Newton) sums p i = x i + · · · + x in . Uponthe embedding into C [ y , . . . , y r ], we get p i ( x ) = P i,λ ( y ). This implies the statement. (cid:3) Corollary 2.3. X λ /S n ∼ = X kλ /S kn for any k ≥ , where kλ := ( kλ , . . . , kλ r ) . Thus, X λ /S n is CM if λ = ((2 k ) r , k s ) or λ = (( mk ) r , k s ) if s < m .Proof. The first statement follows from Proposition 2.2, as P i,kλ = kP i,λ . The second state-ment follows from the first one and Theorem 2.1. (cid:3) Remark 2.4.
The variety Y λ in the case λ = (2 , n − ) is considered in [KMSV]. Also, inthis paper the authors propose the problem to study Y λ for general λ .Proposition 2.2 motivates the following definition. Definition 2.5.
Let λ = ( λ , . . . , λ r ) be a collection of variables. The algebra A r of Newton λ -sums is the subalgebra of C [ λ ± , . . . , λ ± r , y , . . . , y r ] generated over C [ λ ± , . . . , λ ± r ] by P i,λ for i = 1 , , . . . For any subset S ⊂ { , . . . , r } , let λ S = P i ∈ S λ i . Let A r, loc = A r [ λ − S , S = ∅ ] be thelocalization of A r obtained by inverting λ S for all S = ∅ .The following proposition is a straightforward generalization of [SV1, Theorem 5]. Proposition 2.6.
The algebra C [ λ i , λ − S , y , . . . , y r ] and hence its subalgebra A r, loc are finitelygenerated modules over the subalgebra C [ λ i , λ − S , P i,λ , i = 1 , . . . , r ] . In particular, A r, loc is afinitely generated algebra. The same statements hold for C [ y , . . . , y r ] and the specialization A λ of A r at any λ ∈ C r such that λ S = 0 for any nonempty S .Proof. It suffices to show that if λ S = 0 for nonempty S then the system of equations λ y i + · · · + λ r y ir = 0 , i = 1 , . . . , r has only the zero solution. Suppose ( y , . . . , y r ) is a solution. Let S j be the level sets of thefunction y k of k , and y k = z j if k ∈ S j (so that z j are distinct). Then our equations become λ S z i + · · · + λ S m z im = 0 , i = 1 , . . . , r. Since m ≤ r , by the Vandermonde determinant formula this implies that λ S j z j = 0 for all j .Since λ S j = 0 for any j , we get that z j = 0 for all j , as desired. (cid:3) Remark 2.7. (1) If λ i are positive integers (enumerated in decreasing order), then thefact that A λ is a finitely generated algebra also follows from the fact that A λ = O ( X λ /S n ) and Hilbert’s theorem on invariants.(2) We do not know the smallest N for which the polynomials P i,λ for i ≤ N generate A λ for generic λ (as a module over C [ P ,λ , . . . , P r,λ ] or as a ring).(3) If λ S = 0 for some S = ∅ (but λ i = 0 for all i ) then A λ is not finitely generated. Tosee this, let us first see that A , − is not finitely generated. This algebra is generatedby the polynomials F i := y i − z i , i = 1 , , . . . . Setting y = z + w , we find that F i = z i − w + · · · . So in each monomial in F i , the z -degree is at most i − w -degree, which implies that F i +1 cannot be expressed via F , . . . , F i , i.e., A , − isinfinitely generated. Now, if λ i + · · · + λ i k = 0, we can set x i = x i = · · · = x i k − = y , x i k = z , and all other x i = 0, and obtain A , − as a quotient of A λ , which implies thestatement.For a similar reason, in this case A λ is not Noetherian (the infinite chain of ideals I m := ( F , F , . . . , F m ) in A , − is strictly ascending) and thus not CM. N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 5 Assume that λ S = 0 for S = ∅ , let Y λ = Spec( A λ ) be the variety corresponding to A λ ,and let S λ be the subgroup of S r preserving λ . Then by Proposition 2.6, we have a naturalfinite morphism π : C r /S λ → Y λ , induced by the embedding A λ ⊂ C [ y , . . . , y r ]. It is easy tosee that π is surjective (as it is finite and dominant), and birational (i.e., is a normalizationmorphism). On the other hand, π is not always injective, e.g., for r = 3 and λ = (3 , , S λ = 1), the points ( t, ,
0) and (0 , t, t ) in C have the same image under π (as thepolynomials P i,λ have the same values at these points).The following question generalizes Question 1.1(ii): Question 2.8.
For which λ ∈ C n with λ S = 0, S = ∅ is the variety Y λ (or, equivalently, thealgebra A λ ) CM?Let A λ be the quotient of A λ by the ideal generated by P ,λ , and Y λ = Spec( A λ ). Byusing the variables ¯ y i = y i − P λ i y i P λ i we see that Y λ = Y λ × C , so Y λ is CM if and only if so is Y λ . Thus, for n = 2, Y λ is alwaysCM, since Y λ is a curve.3. The formal neighborhoods method
In spite of the results of the previous section, it turns out that X λ and even X λ /S n israther rarely CM. Let us discuss a method of proving that these varieties are not CM forparticular λ – the method of formal neighborhoods.This method is adopted from the proof of [EGL, Proposition 3.11]. Namely, arguing as inthis proof, one can show: Proposition 3.1.
Suppose that ℓ cannot be nontrivially written as a sum of some parts of λ (for example, ℓ ≤ λ r ). Then if X λ ⊂ C n is not CM, then X λ ∪ ℓ ⊂ C n + ℓ is not CM either.The same applies to X λ /S n and X λ ∪ ℓ /S n + ℓ .Proof. Consider the point x in X λ ∪ ℓ given by x = (1 , . . . , , , . . . , ℓ . It is easy to see that the formal neighborhood of x in X λ ∪ ℓ looks like the product ofthe formal neighborhood of zero in X λ with a formal disk (since, by the assumption on λ and ℓ , if the coordinates of x vary so that the point remains in X λ ∪ ℓ then the first ℓ coordinatesmust remain equal). The same applies to X λ /S n and X λ ∪ ℓ /S n + ℓ . This implies the statementsince a local ring is CM if and only if its completion is CM [Eis, Proposition 18.8]. (cid:3) Let us now adapt the formal neighborhoods method to the variety Y λ for complex λ (with λ S = 0 for S = ∅ ). Let λ ′ = ( λ , . . . , λ r − ). Proposition 3.2.
Assume that λ r is not a sum of a subset of λ , . . . λ r − . Then the formalneighborhood of the point π (0 , . . . , , in Y λ is isomorphic to the product of the formalneighborhood of π ( ) in Y λ ′ with a -dimensional formal disk. Thus, if Y λ ′ is not CM then Y λ is not CM either. To see that π is birational, note that when y i are distinct, the function P ∞ i =0 P i,λ t i = P j λ j − y j t determines y j uniquely up to the action of S λ . AARON BROOKNER, DAVID CORWIN, PAVEL ETINGOF, AND STEVEN V SAM
Proof.
Let y r = 1 + u , and let y , . . . , y r − , u be formal variables. We have equations P i,λ ′ ( y , . . . , y r − ) = Z i − λ r ((1 + u ) i − , where Z i := P i,λ ( y , . . . , y r ) − λ r , formal functions on Y λ near π (0 , . . . , ,
1) vanishing at thatpoint. We claim that from these equations we can express u as a formal series of the Z i .Indeed, by Proposition 2.6, for some N we have P N,λ ′ = F ( P ,λ ′ , . . . , P N − ,λ ′ ) , where F is a quasi-homogeneous polynomial of degree N (with deg P i,λ ′ = i ). Thus, Z N − λ r ((1 + u ) N −
1) = F ( Z − λ r u, . . . , Z N − − λ r ((1 + u ) N − − . This equation has the form Z N − λ r N u = · · · , where · · · stands for quadratic and higherterms in u and Z i . Such an equation can clearly be solved for u in the form of a power seriesin the Z i , as claimed (note that we have not yet used the condition on λ ).Now it remains to note that if λ r is not a sum of a subset of λ , . . . , λ r − , then the equation π ( x ) = π (0 , . . . , ,
1) has a unique solution x = (0 , . . . , , y r can be uniquely read off from the function ∞ X i =0 P i,λ ( y ) t i = r X j =1 λ j − y j t . This together with the above implies the statement. (cid:3)
Remark 3.3.
Note that if λ i are positive integers, Proposition 3.2 reduces to Proposition3.1 for X λ /S n . 4. The algebra of deformed Newton sums
Let a be a variable. Definition 4.1.
The algebra Λ r,s of deformed Newton sums is the subalgebra of C [ a ± , y , . . . , y r , z , . . . , z s ]generated over C [ a ± ] by the deformed Newton sums Q r,s,i := a ( y i + · · · + y ir ) + ( z i + · · · + z is ) . The localized algebra Λ r,s, loc is the localization Λ r,s, loc := Λ r,s ⊗ C [ a ± ] K , where K := C [ a ± , ( a + p/q ) − | ≤ p ≤ s, ≤ q ≤ r ] . Proposition 4.2 ([SV1, Theorem 5]) . The algebra K [ y , . . . , y r , z , . . . , z s ] and hence itssubalgebra Λ r,s, loc are finitely generated modules over K [ Q r,s,i , i = 1 , . . . , r + s ] . In particular, Λ r,s, loc is a finitely generated algebra. The same statements hold for C [ y , . . . , y r , z , . . . , z s ] and the specialization Λ r,s,a of Λ r,s at any a ∈ C ∗ such that a = − p/q , ≤ p ≤ s , ≤ q ≤ r .Proof. This is a special case of Proposition 2.6, for λ = ( a r , s ). (cid:3) Remark 4.3.
Note that Λ r,s,a = A ( a r , s ) .The following theorem was conjectured by Sergeev and Veselov in [SV1] (end of § Theorem 4.4.
The algebra Λ r,s,a is Cohen–Macaulay for all but finitely many values of a ∈ C . N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 7 Proof.
By Proposition 4.2, Λ r,s,a has the same Hilbert series for almost all values of a . Also,by Theorem 2.1, it is CM for integers a > s (as in this case by Proposition 2.2, Λ r,s,a = O ( X λ /S n ), where λ = ( a r , s )). Since in flat families, CM-ness is an open condition, weconclude that Λ r,s,a is CM for all but finitely many values of a , as desired. (cid:3) Remark 4.5.
It is shown in [SV1, Theorem 3] that the Hilbert series of Λ r,s,a for generic a is h r,s ( t ) = 1(1 − t ) · · · (1 − t r ) s X i =0 t i ( r +1) (1 − t ) · · · (1 − t i ) . Therefore, by Theorem 4.4, for generic a the Hilbert series of the generators of Λ r,s,a as amodule over C [ Q r,s,i , i = 1 , . . . , r + s ] is e h r,s ( t ) = (1 − t r +1 ) · · · (1 − t r + s ) s X i =0 t i ( r +1) (1 − t ) · · · (1 − t i ) . For instance, if s = 1, we get e h r, ( t ) = (1 − t r +1 )(1 + t r +1 − t ) = 1 + t r +2 + · · · + t r +1 (see [SV1], end of § B ( r, s ) ⊂ C ∗ \ {− p/q | ≤ p ≤ s, ≤ q ≤ r } of “exceptional” values of a , for which Λ r,s,a fails to be CM (clearly, they may be nonemptyonly if r + s ≥ B ( r, s ) = B ( s, r ) − , as Λ r,s,a = Λ s,r,a − . Proposition 4.6. If a ∈ B ( r, s ) and a = − pr +1 with ≤ p ≤ s then a ∈ B ( r + 1 , s ) .Similarly, if a ∈ B ( r, s ) and a = − s +1 q with ≤ q ≤ r then a ∈ B ( r, s + 1) .Proof. Since B ( r, s ) = B ( s, r ) − , it suffices to prove the first statement. This follows fromProposition 3.2 unless a = k for integer 1 ≤ k ≤ s . But in this case, we know from Theorem2.1 that a ∈ B ( r + 1 , s ) unless k = 1 or k = 2, while a = 1 , B ( r, s ). Thisimplies the statement. (cid:3) Proposition 4.7.
Let a = p/q , where p, q are coprime positive integers, p > q , ≤ p ≤ s , q ≤ r , Then a ∈ B ( r, s ) and a − ∈ B ( s, r ) .Proof. By Proposition 4.6, it suffices to prove the statement when r = q and s = p , sothat a = s/r , r < s , s ≥
3. In this case, Λ r,s,a = A λ , where λ = ( s r , r s ), a partition of n = 2 rs . Consider the formal neighborhood of the point (0 , . . . , , , . . . ,
1) of X λ /S n wherethe number of zeros and the number of ones are both equal to rs . Let us generically deformthis point inside X λ /S n . This deformation can happen in two ways: either the zeros deforminto r -tuples of s equal coordinates and ones deform into s -tuples of r equal coordinates,or the other way. This gives two subspaces of dimension r + s in C rs whose intersectionis 2-dimensional (the space of vectors whose first rs and last rs coordinates are equal).Thus the formal neighborhood in question looks like a 2-dimensional formal disk times theformal neighborhood of the intersection point in the transversal intersection of two spacesof dimension r + s −
2. This is not CM by Reisner’s theorem, [Re] (or by direct verification)if r + s − ≥
2, i.e., for s ≥
3. This implies the statement. (cid:3)
AARON BROOKNER, DAVID CORWIN, PAVEL ETINGOF, AND STEVEN V SAM
This implies that the union of the sets B ( r, s ) over all r, s contains all positive rationalnumbers except 1 , , / B ( r, s ) ⊂ B ( r, s ) be the set of points provided by Proposition 4.7. For example: B (2 ,
1) = B (1 ,
2) = ∅ , B (3 ,
1) = { / } , B (2 ,
2) = ∅ , B (1 ,
3) = { } ,B (4 ,
1) = { / , / } , B (3 ,
2) = { / , / } , B (3 ,
2) = { , / } , B (1 ,
4) = { , } , etc. Conjecture 4.8.
One has B ( r, s ) = B ( r, s ).In particular, this would imply that B ( r, s ) ⊂ Q (from abstract nonsense, it is only clearthat B ( r, s ) ⊂ Q ).Computations in Mathematica have confirmed this conjecture for r + s ≤
5; moreover, for r + s = 3 we will prove theoretically below that B ( r, s ) = ∅ . Remark 4.9.
While 1 , , do not belong to the sets B ( r, s ) (as the corresponding varietiesare CM), computations show that at these points (for r ≥ / s ≥ r,s,a are smaller than those for generic a , so they alsoshould be viewed as exceptional.On the other hand, for the values a = − p/q , 1 ≤ p ≤ s , 1 ≤ q ≤ r (which we excluded),[SV1, Theorem 2] implies that Λ r,s,a has the same Hilbert series as generically, but it is notfinitely generated and not Noetherian, so not CM.Thus, the full exceptional set is of the form e B ( r, s ) = { b ∈ C ∗ | b = ± pq , ≤ p ≤ s, ≤ q ≤ r } . Remark 4.10.
We expect that a proof of Conjecture 4.8 or even the rationality statementfor the exceptional values of a should involve an interpolation of the representation theory ofrational Cherednik algebras to complex rank. More precisely, recall from [EGL] that if a ≥ r,s,a = O ( X ( a r , s ) /S ra + s ), carries an action of the sphericalrational Cherednik algebra e H /a ( ra + s ) e for the symmetric group S ra + s with parameter c = 1 /a . More precisely, it carries a faithful action of the simple algebra B r,s ( a ) := e H /a ( ra + s ) e /I max , where I max is the maximal ideal in e H /a ( ra + s ) e (which is known to be unique). Thealgebra B r,s ( a ) is generated by P i = e P x ij and P ∗ i = e P D ij , where D j are the Dunkloperators; when written in coordinates y k , P i become the polynomials P i,λ and P ∗ i becomethe quantum integrals of the deformed Calogero–Moser system of type A r ( s ) with couplingconstant a , see [SV1].In other words, the algebra B r,s ( a ) is generated by Λ r,s,a and the algebra Λ ∗ r,s,a of quan-tum integrals of the deformed Calogero–Moser system (which is a commutative algebra ofdifferential operators isomorphic to Λ r,s,a ). This definition actually makes sense for anycomplex a = 0, not only for positive integers, and gives rise to a flat family of algebras B r,s ( a ) parametrized by a complex parameter a . Exceptional values of a are likely those forwhich this algebra ceases to be simple, and its representations degenerate (at other valuesof a , we expect that the methods similar to ones of [EGL] should apply to establish the CMproperty).The algebras B r,s ( a ) for general a are quotients of interpolations of e H /a ( ra + s ) e = e H rn − s ( n ) e N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 9 to complex values of the rank n = ra + s (which is a kind of toroidal Yangian, similarto the deformed double current algebras from [Gu]). Such interpolations for the full (non-spherical) Cherednik algebras were considered in [EA, Et] (based on the Deligne categoryRep( S ν )). These interpolations have two parameters c, ν , where ν is an interpolation of n ,and the situation at hand corresponds to the hyperbola c = rν − s in the ( c, ν )-plane, which isa reducibility locus for the interpolated polynomial module, see [EA]. Further degenerationsoccur at intersection points of this hyperbola with other reducibility hyperbolas, which occursat rational values of ν . This should lead to a proof of rationality of exceptional values. Thedetermination of the exact set of exceptional values by this method would likely require amore detailed study of representations of the rational Cherednik algebra of complex rankalong the lines of [EA].We note that an approach to deformed Calogero–Moser systems of type A r ( s ) essentiallybased on interpolation to complex rank is also proposed in [SV2].5. The case λ = ( a, b, c ) . Let us now focus on the first nontrivial case λ = ( a, b, c ). By Theorem 2.1, X λ is CM if( a, b, c ) = ( a, , a, a, a, a, a ) for positive integer a .Also, we have the following result. Proposition 5.1. X λ is CM if λ = (4 , , but is not CM if λ ∈ { (3 , , , (3 , , , (5 , , } .Thus, X λ is not CM for λ = (3 r , s ) for r ≥ , s ≥ , r + s ≥ , and λ = (5 r , s ) , r ≥ , s ≥ .Proof. The second statement follows from the first one and Proposition 3.1. The first state-ment is proved by a Macaulay2 calculation (we do not have a theoretical proof of these facts)which is included in the file computations1.m2 and which we comment on now.To show that X (4 , , is CM, it suffices to find a regular sequence for its coordinate ring.In the file, we produce a collection of 3 random linear forms and test if they form a regularsequence by calculating the Hilbert series of the coordinate ring before and after quotientingby these forms. It suffices that this works for one example, and the random collection willhave the desired property with high probability.For the examples that are not CM, we calculate the ideal and observe that the numeratorof the Hilbert series has negative coefficients: h X , , ( t ) = 1 + 4 t + 10 t + 20 t + 35 t + 35 t + 14 t − t (1 − t ) ,h X , , ( t ) = 1 + 5 t + 15 t + 35 t + 70 t + 98 t + 70 t − t (1 − t ) ,h X , , ( t ) = 1 + 6 t + 21 t + 56 t + 126 t + 216 t − t (1 − t ) . One additional comment: in the file, the calculations are set up for a finite field of size32003 to significantly speed up calculations. One cannot use this to deduce that the varietyis not CM over a field of characteristic 0, but the calculations can also be performed in thissituation (with much more patience) and the Hilbert series stays the same. (cid:3)
To address the case λ = ( a, b, c ), we look at Y λ and try to identify cases when it is notCM. This problem is convenient to approach by computer calculation, since Y λ is a surface, and a , b , and c occur as parameters in its presentation, which can actually be taken to benonzero complex numbers.By rescaling ( a, b, c ) as in Corollary 2.3, we may assume that c = 1. Then the ring R a,b := A λ of functions on Y λ is generated inside C [ x, y ] by the homogeneous polynomials P i = P i,a | b := ax i + by i + ( − ax − by ) i , i ≥ . (note that P = 0). As shown in Proposition 4.2, R a,b is a finitely generated module over C [ P , P ] outside of the lines a + 1 = 0 , b + 1 = 0 , a + b = 0 , a + b + 1 = 0 . Let us now study the CM property of R a,b outside of these lines. It is clear that in thiscase, R a,b is CM if and only if R a,b is a free C [ P , P ]-module.5.1. The special case ( a, a, . Let us start with the case b = a ; in this case R a,b = R a,a =Λ , ,a , generated by deformed Newton sums P i = a ( x i + y i ) + ( − a ) i ( x + y ) i . Thus, we know from Theorem 4.4 that R a,a is CM for almost all a , but we want to prove astronger statement: Proposition 5.2.
The algebra R a,a is CM for all a = − , − / (i.e., in all cases when R a,a is finitely generated). Moreover, for such a it has the Hilbert series t + t (1 − t )(1 − t ) . Therefore, the same statements hold for R a, for a = − , − . Thus, B ( r, s ) = ∅ if r + s = 3 (i.e., Conjecture 4.8 holds in this case). The rest of the subsection is devoted to the proof of Proposition 5.2.First of all, note that R a,a is a subring of the ring of symmetric polynomials in x, y , i.e.,of C [ u, v ], where u = x + y , v = x + y . E.g., P = av + a u . Let us denote P by w . So v = a − w − au , and C [ u, v ] = C [ u, w ].Now let us express P as a linear combination of u and uw . We get(1) P = 32 uw − a ( a + 1)(2 a + 1) u . Now we will need the following sequence of lemmas.
Lemma 5.3.
Every element F ∈ R a,a is a quasi-invariant, in the sense that ∂F∂x = 0 when x = − a ( x + y ) .Proof. By the Leibniz rule, it is enough to show that P i are quasi-invariants, which is easy. (cid:3) Lemma 5.4.
Let S be the algebra of all quasi-invariants. The codimension of S d in C [ u, w ] d in any positive degree d is . In other words, the Hilbert series of S is h S ( t ) = 1(1 − t )(1 − t ) − t − t = 1 + t + t (1 − t )(1 − t )= 1 + t + t + 2 t + 2 t + 3 t + 3 t + · · · N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 11 Proof.
This follows from the fact that in every positive degree, quasi-invariance gives onenontrivial linear equation. In more detail, the Hilbert series of C [ u, v ] is − t )(1 − t ) (as wehave free generators u, v of degrees 1 , t − t = t + t + t + · · · . (cid:3) Lemma 5.5.
If for d ≥ , F ∈ S d is divisible by w , then F = wG , where G ∈ S .Proof. By Lemma 5.4, dim S d − = dim S d −
1, so wS d − has codimension 1 in S d . But thecondition of divisibility by w is one nontrivial linear equation. (cid:3) Lemma 5.6. If d ≥ then S d = wS d − + P S d − .Proof. By Lemma 5.5, S d − has an element H with a nonzero coefficient of u d − (as notall elements are divisible by w ). So P H ∈ S d has a nontrivial coefficient of u d , if a is not0 , − , − / (cid:3) Corollary 5.7.
We have the following: (i) S is generated by , P , P as a module over C [ P , P ] . (ii) R a,a = S .Proof. (i) Lemma 5.6 implies that S is generated over C [ P , P ] by S ≤ . This implies thestatement, as it is easy to check that all elements of S ≤ belong to C [ P , P ]( C · ⊕ C · P ⊕ C · P ).(ii) Since by Lemma 5.3, R a,a ⊂ S , this follows from (i). (cid:3) Since R a,a clearly has rank 3 over C [ P , P ], Corollary 5.7 implies that R a,a is freely gen-erated over C [ P , P ] by 1 , P , P , which implies Proposition 5.2.5.2. The case a = b , a = 1 , b = 1 . It remains to consider the case a = b , a = 1, b = 1. Weassume that a, b, a + b = 0 , − Proposition 5.8. If a = b , a = 1 , b = 1 then the rank of R a,b over C [ P , P ] is .Proof. This follows from the fact that the equations P ( x, y ) = c , P ( x, y ) = c have 6solutions for generic c , c (by Bezout’s theorem). (cid:3) Corollary 5.9. R a,b is CM if and only if the Hilbert polynomial Q ( t ) of the finite dimensionalalgebra R a,b / ( P , P ) satisfies the condition Q (1) = 6 .Proof. R a,b is CM if and only if it is free over C [ P , P ] (of rank 6). In this case, the polynomial Q is the Hilbert polynomial of the generators. Otherwise, Q (1) > (cid:3) Now we have the following proposition:
Proposition 5.10.
Write the Hilbert series of R a,b as h R a,b = q ( t )(1 − t )(1 − t ) . If a = b, a =1 , b = 1 and ( a, b ) = (3 , , (2 , , (1 / , / , (3 / , / , (1 / , / , (2 / , / , then q ( t ) = 1 + t + t + t + t + t + Ct + · · · where C = 1 if a = b + 1 , b = a + 1 , = a + b . Proof.
The proof is by a computer calculation in Macaulay2. We are trying to calculatethe Hilbert series of the image of a polynomial map, which is an implicitization problem.However, that problem is too difficult, and we only need partial information. A naiveapproach to compute the dimension of the degree d piece of R a,b is to compute the dimensionof the linear span of the space of all monomials of (weighted) degree d in the P i .This is what is done in computations2.m2 and we first work with generic values of a, b todetermine the generic dimensions of these spaces. For our result, we just need d = 1 , . . . , , , , , , , , , q ( t ) = 1 + t + t + t + t + t + t + · · · for generic a, b .This dimension count is actually determining the rank of a matrix: each monomial inthe P i is written as a vector whose entries are the coefficients (which live in C [ a, b ]) of itsexpression as a polynomial in x, y . To determine when this dimension can drop, we calculatethe ideal of minors of each rank and decompose this ideal to get the bad conditions on a, b as above. This works up to degree 8, but degree 9 is too hard to handle directly; insteadwe find a special 7 × Q [ a, b ] since this gives an upperbound for the bad locus. (cid:3) Corollary 5.11. R a,b is not CM outside of the lines a = b, a = 1 , b = 1 , a = b + 1 , b = a + 1 , a + b. In particular, X ( a,b,c ) is not CM if a > b > c and a = b + c .Proof. If { P , P } does not form a regular sequence in R a,b then R a,b is not CM. If it does,then q ( t ) = Q ( t ), and by Proposition 5.10, outside of the given lines we have Q (1) ≥
7, sothe module is not free. (cid:3)
The case a = b + 1 . Now we consider the case when a = b + 1 (the other remainingcases, b = a + 1 and 1 = a + b , are reduced to it by permutation of coordinates). Now wehave the only parameter b = 0, which should not equal to − , − , − / Proposition 5.12.
The Hilbert series of R , is q ( t )(1 − t )(1 − t ) where q ( t ) = 1 + t + t + t + t + t + t + · · · , so R , is not CM. This is done in computations3.m2 by explicit calculation.Also, we have the following proposition, conjectured by the authors in the original versionof this paper and proved by Eric Rains using MAGMA:
Proposition 5.13.
For b = 0 , ± , ± , ± , R b +1 ,b is CM with Q ( t ) = 1 + t + t + t + t + t . Proof.
Let us show that R b +1 ,b is generated as a module over C [ P , P ] by T = 1 , T = P , T = P , T = P , T = P , T = P . Since this module is of generic rank 6, this means that it is a free module, which implies thetheorem.
N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 13 It is easy to compute (e.g., using MAGMA) that each of the elements x, y, z is annihilatedby a degree 6 monic polynomial over C [ P , P ]; let us denote these polynomials by Q x , Q y , Q z . Let Q ( u ) := Q x ( u ) Q y ( u ) Q z ( u ) = X j =0 a j u j , a monic polynomial of degree 18 (i.e., a = 1). It is clear that the P i satisfy the linearrecursion X i =0 a i P n + i = 0 , n ≥ . So it suffices to check that P , . . . , P belong to the C [ P , P ]-module M generated by T , . . . , T , and that this module is in fact an algebra (i.e., T i T j ∈ M ). This is done us-ing MAGMA (the computation takes less than a second). (cid:3) Remark 5.14.
Note that for b = 1, R b +1 ,b is still CM, but has a different Hilbert series.6. The CM property of X b + c,b,c Thus, to study the CM properties of X b + c,b,c , where b > c ≥
1, we need more information.These varieties turn out to be not CM, but the necessary information is obtained not fromthe ring of invariants (which is CM unless b = 2 c ), but rather from the isotypic componentof the reflection representation. Proposition 6.1.
The variety X b + c,b,c is not CM for any positive integers b > c .Proof. Since, as we saw above, A (3 , , is not CM, we may assume that b = 2 c . Let β = b/c .In this case, consider M := Hom S n ( h , O ( X λ )), where n = 2( b + c ) and h is the reflectionrepresentation of S n . It is easy to see from standard invariant theory for S n that M isgenerated over R β +1 ,β by the polynomials T i := n X j =1 x ij u j , where P nj =1 u j = 0. (For example, this follows from the fact that the invariants of S n in C [ x , . . . , x n , u , . . . , u n ] are generated by the polynomials P x ij y sj , a special case of the Weyl’sFundamental Theorem of Invariant Theory).Let x = ( x , . . . , x n ) ∈ E λ . Let x be the common value of the first b + c coordinates x j , y of the next b coordinates, and z for the last c coordinates. Let b + c X j =1 u j = u, b + c X j = b + c +1 u j = v, b + c ) X j =2 b + c +1 u j = w. Then u + v + w = 0, so w = − u − v , and T i = x i u + y i v + z i w = ( x i − z i ) u + ( y i − z i ) v, where z = − ( β + 1) x − βy . So we get T i = ( x i − ( − ( β + 1) x − βy ) i ) u + ( y i − ( − ( β + 1) x − βy ) i ) v. Thus, M is the module over R β +1 ,β generated by T i inside R β +1 ,β u ⊕ R β +1 ,β v . In particular, M is finitely generated. The rank of M over R β +1 ,β is clearly 2 (since we have two vectors u, v ), so rank of M over C [ P , P ] should be 6 · M is not a CM module over the CM algebra R β +1 ,β , i.e.,that it is not free as a module over C [ P , P ]. This will follow from the following lemma: Lemma 6.2.
Set deg( u ) = deg( v ) = 0 . Writing the Hilbert series of M as h M ( t ) = q ( t )(1 − t )(1 − t ) , for a = 0 , ± , ± , ± , we have q ( t ) = t + t + t + t + 2 t + 3 t + 3 t + t + · · · . Proof.
The argument is similar to that of Proposition 5.10. In this case, computations4.m2 computes dim M n , the dimension of the degree n part of M , for 1 ≤ n ≤
8, using thespanning set { P i P i · · · P i k T i k +1 | i + · · · + i k +1 = n } . We get the list { , , , , , , , } for the ranks in degrees n = 1 , . . . ,
8, which tells us that q ( t ) = t + t + t + t + 2 t + 3 t + 3 t + t + · · · for generic β .We then determine for which β these dimensions drop, i.e., the ranks of the correspondingmatrices drop. In degree up to 5 we get that the decomposition of the ideal of minors onlydegenerates for β = 0 , ±
1. Taking a specific minor from the matrices of degrees 6 , , { , ± , ± , ± , } , and a single check rules out β = 4 as a degenerate value.Hence outside of these values q ( t ) is generic. (cid:3) If { P , P } is not a regular sequence, then M is not CM. If it is a regular sequence, then theHilbert polynomial Q ( t ) of M equals q ( t ). In this case, we have just shown that Q (1) ≥ M is 12, so M cannot be CM. This proves the proposition. (cid:3) The non-CMness theorems
Theorem 7.1.
Let λ = ( λ , . . . , λ r ) be a partition of n . If λ has at least four distinct parts,or has three distinct parts a > b > c with a = b + c , then X λ /S n and hence X λ is not CM.Proof. The proof is by induction on r . For r = 3 (base of induction), the statement followsfrom Corollary 5.11. Assume that r ≥
4. If λ has four smallest distinct parts a > b > c > d ,then either a = b + c or a = b + d , so we can remove either d or c using Proposition 3.1,descending from r to r − a > b > c and a = b + c . In thiscase, since r ≥
4, one of these parts should occur more than once, and we can remove it usingProposition 3.1, descending from r to r − (cid:3) Theorem 7.2. If λ = ( b + c, b r , c s ) , b > c , then X λ is not CM.Proof. Using Proposition 3.1, we can reduce to the situation when λ = ( b + c, b, c ), in which X λ is not CM by Proposition 6.1. (cid:3) Conjecture 7.3. If λ = (( b + c ) q , b r , c s ), b > c , then X λ is not CM for q >
1. Thus, X λ isnot CM for any λ that has at least three distinct parts. N COHEN–MACAULAYNESS OF S n -INVARIANT SUBSPACE ARRANGEMENTS 15 It suffices to prove Conjecture 7.3 for the case r = s = 1, as one can reduce to this caseusing Proposition 3.1. However, we cannot reduce q by removing a copy of b + c , since itequals the sum of two other parts, b and c .Conjecture 7.3 is supported by computational evidence for q = 2, r = s = 1, i.e., λ =( b + c, b + c, b, c ), obtained using Macaulay2 (namely, we computed that Y λ is not CM for λ = ( b + 1 , b + 1 , b,
1) for random b ).Let us now discuss the remaining case of two distinct parts, i.e., λ = ( b r , c s ), b > c . Let b/c = b ′ /c ′ and gcd( b ′ , c ′ ) = 1. If c ′ ≤ r , 3 ≤ b ′ ≤ s , then we know from Proposition 4.7 that X λ /S n and hence X λ is not CM. Otherwise, if c = 1 (and b > s or b = 2), we know fromTheorem 2.1 that X λ is CM. Finally, by Proposition 5.1, X (4 , , is CM, while X (3 r , s ) is notCM for r ≥ s ≥ r + s ≥ X (5 r , s ) is not CM for r ≥ s ≥
2. Apart from thesecases, we do not know the answer. In particular, the answer is not known for λ = (4 , s ).The situation with Y λ for λ = (( b + 1) q , b r , s ) seems to be even more complicated. Com-putational evidence (see above) suggests a conjecture that Y λ is not CM if q ≥
2. However,we have the following conjecture about q = 1: Conjecture 7.4. If λ = ( b + 1 , b, s ) then Y λ is CM for generic complex b . Moreover, theexceptional values of b for such λ are b = 0 and b = ± p/q , where 1 ≤ p ≤ s + 1, 1 ≤ q ≤ b , s +1 ), see Remark 4.9).Note that Conjecture 7.4 holds for s = 1 by Proposition 5.13. It was also proved by EricRains for s = 2 using MAGMA: Proposition 7.5.
Conjecture 7.4 holds for s = 2 . More specifically, if b = 0 , ± , ± , ± , ± , ± ,then the algebra C [ Y b +1 ,b, , ] is CM with Hilbert series t + t + t + t + t + 2 t + 2 t + t + t (1 − t )(1 − t )(1 − t ) . Proof.
The proof is similar to the proof of Proposition 5.13. Let us show that C [ Y b +1 ,b, , ] isgenerated as a module over C [ P , P , P ] by T = 1 , T = P , T = P , T = P , T = P , T = P ,T = P , T = P , T = P , T = P P , T = P P , T = P P . Since this module is of generic rank 12, this means that it is a free module, which impliesthe theorem.It is easy to compute (e.g., using MAGMA) that each of the elements x, y, z, w is annihi-lated by a degree 12 monic polynomial over C [ P , P , P ]; let us denote these polynomials by Q x , Q y , Q z , Q w . Let Q ( u ) := Q x ( u ) Q y ( u ) Q z ( u ) Q w ( u ) = X j =0 a j u j , a monic polynomial of degree 48 (i.e., a = 1). It is clear that the P i satisfy the linearrecursion X i =0 a i P n + i = 0 , n ≥ . So it suffices to check that P , . . . , P belong to the C [ P , P , P ]-module M generated by T , . . . , T , and that this module is in fact an algebra (i.e., T i T j ∈ M ). This is done usingMAGMA (the computation takes under an hour). (cid:3) A similar conjecture applies, of course, to the case λ = ( b + 1 , b s , b /b and rescaling. However, we do not know whathappens for λ = ( b + 1 , b r , s ). References [BGS] Christine Berkesch Zamaere, Stephen Griffeth, Steven V Sam, Jack polynomials as fractional quan-tum Hall states and the Betti numbers of the ( k + 1)-equals ideal, Comm. Math. Phys. (2014),no. 1, 415–434, arXiv:1303.4126v2 .[DS] Sheela Devadas, Steven V Sam, Representations of rational Cherednik algebras of G ( m, r, n ) inpositive characteristic, J. Commut. Algebra (2014), no. 4, 525–559, arXiv:1304.0856v2 .[Eis] David Eisenbud, Commutative Algebra with a view toward Algebraic Geometry , Graduate Texts inMathematics , Springer, 1995.[EA] Inna Entova Aizenbud, On representations of rational Cherednik algebras in complex rank, arXiv:1301.0120v2 .[Et] Pavel Etingof, Representation theory in complex rank I,
Transform. Groups (2014), no. 2,259–281, arXiv:1401.6321v3 .[EGL] Pavel Etingof, Eugene Gorsky, Ivan Losev, Representations of rational Cherednik algebras withminimal support and torus knots, Adv. Math. , 124–180, arXiv:1304.3412v3 .[GW] A. V. Geramita, C. A. Weibel, On the Cohen–Macaulay and Buchsbaum property for unions ofplanes in affine space,
J. Algebra (1985), 413–445.[M2] Daniel R. Grayson, Michael E. Stillman, Macaulay 2, a software system for research in algebraicgeometry, Available at .[Gu] Nicolas Guay, Affine Yangians and deformed double current algebras in type A, Advances in Math. (2007), 436–484.[KMSV] M. Kasatani, T. Miwa, A. N. Sergeev, and A. P. Veselov, Coincident root loci and Jack andMacdonald polynomials for special values of the parameters,
Jack, Hall-Littlewood and Mac-donald Polynomials , Contemp. Math. , Amer. Math. Soc., Providence, RI, 2006, 207–225, arXiv:math/0404079v1 .[Re] Gerald A. Reisner, Cohen–Macaulay quotients of polynomial rings,
Advances in Math. (1976),30–49.[SS] Hal Schenck, Jessica Sidman, Commutative algebra of subspace and hyperplane arrangements, Commutative Algebra , 639–665, Springer, New York, 2013.[SV1] A. N. Sergeev, A. P. Veselov, Deformed quantum Calogero–Moser systems and Lie superalgebras,
Comm. Math. Phys. (2004), 249–278, arXiv:math-ph/0303025v1 .[SV2] A. N. Sergeev, A. P. Veselov, Dunkl operators at infinity and Calogero–Moser systems, arXiv:1311.0853v2 .[Y] Sergey Yuzvinsky, Cohen–Macaulay seminormalizations of unions of linear subspaces,
J. Algebra (1990), 431–445.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
E-mail address : [email protected] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
E-mail address : [email protected] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
E-mail address : [email protected] Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720,USA
E-mail address ::