Math. Ann. DOI 10.1007/s00208-014-1125-4

Mathematische Annalen

Schur asymptotics of Veronese syzygies Mihai Fulger · Xin Zhou

Received: 31 May 2013 / Revised: 19 July 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract We study the asymptotic behavior of Veronese syzygies as representations of the general linear group. For a fixed homological degree p of the syzygies, we describe the exact asymptotic growth for the number of distinct irreducible representations and for the number of irreducible representations also counting multiplicities. This shows that asymptotically Veronese syzygies have a very rich algebraic and representation-theoretic structure as the degree of the embedding grows. 1 Introduction Turning to details, let P = P(V ) be a complex projective space. Recall that the d-th Veronese embedding of P is the embedding: P∼ = X d → P(Symd V ) defined by all monomials of degree d. It is a classical problem to understand the defining equations of X d and the syzygies among them. Specifically, write:

M. Fulger · X. Zhou Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA X. Zhou e-mail: [email protected] M. Fulger Institute of Mathematics of the Romanian Academy, P. O. Box 1-764, Bucharest 014700, Romania Present address: M. Fulger (B) Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA e-mail: [email protected]

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S=

∞ i=0

Symi Symd V,

R=

∞ i=0

Symid V

for the coordinate ring of the Veronese ambient space and the homogeneous coordinate ring of the Veronese image respectively. Then we may form the graded free S-resolution: 0 ← R ← F0 ← F1 ← · · · ← Fr ← 0, where F p = ⊕ j S(−a p, j ) are free graded modules over S. In order to keep track of the gradings, it is convenient to introduce finite dimensional vector spaces K p,q (d) and write: F p = ⊕q K p,q (d) ⊗ S(− p − q). The geometric study of syzygies was initiated by Green [7] who showed that for the dth Veronese embedding, K p,q (d) = 0, if q ≥ 2, and d ≥ p. As q indicates the weight of syzygies, the result says that the first d modules of syzygies lie in the lowest weights (for the dth Veronese embedding). This is usually referred to as property Nd and is of wide interest [16,17,21], etc. Contrary to the naive extrapolation of this result, Ein and Lazarsfeld [4] proved that syzygies of high degree embeddings are asympotically as complicated as possible with respect to the grading. Roughly speaking, essentially all the K p,q that are not forced to vanish for reasons of Castelnuovo–Mumford regularity are in fact non-zero. In other words, the correct expectation in the study of syzygy modules of an arbitrarily high degree embedding is that it is quite complicated. In the case of Veronese embeddings, the syzygies K p,q (d) are representations of the group G L(V ). It is natural to ask how the syzygies decompose into irreducible representations. This has been studied by many authors [2,3,19–22] and it seems that it is closely related to the well known intractable problem of plethysm [5, Exer. 6.17]. It does not seem realistic to expect an exact decomposition of K p,q (d) in general. Instead we give asymptotic measures of the complexity of its decomposition. Because of Green’s result above and other elementary considerations, the whole of the pth syzygy space K p (d) =

∞ q=−∞

K p,q (d)

is captured by the weight q = 1 for d ≥ p. Our main theorem describes the asymptotics of K p,1 (d): Theorem Fix p ≥ 1 and assume dim V ≥ p + 1. Then as d grows, K p,1 (d) contains

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(i) Exactly on the order of d p distinct irreducible representations. p+1 (ii) Exactly on the order of d ( 2 ) irreducible representations counting multiplicities. In contrast to [4, §6], which gives a range of parameters for which syzygies are nonzero, we focus on a fixed syzygy known to be nonzero and study the asymptotics of its decomposition as the degree of the embedding grows. The theorem is illustrated by the following example: Quadrics The ideal of the Veronese embedding, i.e. the kernel of R ← S, is generated as an ideal by degree 2 forms. These correspond to generators of K 1,1 (d), the kernel of the equivariant map Sym2d ← Sym2 Symd . Sym2 Symd is a known plethysm which decomposes multiplicity-free into representations with Young diagrams of 2d boxes, at most 2 rows and each row having an even number of boxes. The kernel consists of all these except the one rowed Young diagram corresponding to Sym2d . Since each Young diagram appears with multiplicity one, we have about d/2 many irreducible representations, all distinct. The general outline of the proof of the theorem is simple. The syzygy groups can be computed from a Koszul-type complex of equivariant spaces. The number of distinct types of irreducible subrepresentations and the sum of the multiplicities of irreducible subrepresentations are additive in short exact sequences of G L(V )-representations. The main result comes down to asymptotic estimates on plethysms that are carried out using techniques of convex geometry and invariant theory. In Sect. 5 we state mostly without proof several results about twisted syzygies. They suggest that new ideas are needed for some related but more subtle questions. 2 Notation and facts 0. Throughout the paper, we use standard limit notation. For g and f functions of d, one writes g ∈ ( f ) if there exist constants c and C such that for all d we have c· f (d) ≤ g(d) ≤ C· f (d). We say that f ∼ g, if limd→∞ f /g = 1. 1. We adopt the notation and definitions of [5] and [6] for the basic objects of the Representation Theory of the general  linear group. In particular, we write λ  n when λ = (λ1 ≥ λ2 ≥ . . .), with i≥1 λi = n, is a partition of n. The length of λ, denoted |λ|, is the number of its nonzero parts. We write (22 , 1) for the partition (2, 2, 1), we write λ + μ for the partition (λ1 + μ1 , λ2 + μ2 , . . .), and 2λ for the partition (2λ1 , 2λ2 , . . .), etc. 2. We write S d for Symd . The symmetric group on p elements is denoted by S p .

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3. In this paper we work with functors F : Vect C → Vect C of finite dimensional complex vector spaces that have unique finite direct sum decompositions1 F=



MF ,λ ⊗C Sλ ,

λ

where MF ,λ are complex vector spaces, and Sλ is the Schur functor corresponding to the partition λ. The dimensions (F, λ) =def dim MF ,λ are the multiplicities of Sλ in F. The total multiplicity of F is N (F) =def



(F, λ).

λ

The complexity of F is the number of distinct types of Schur functors appearing in its decomposition, i.e., c(F) =def #{λ : (F, λ) = 0}. If V is a complex vector space of finite dimension, then F(V ) is naturally a G L(V )-representation. We will often use this to reduce questions about functors to questions about representations of the general linear group. 4. It is an elementary consequence of Pieri’s rule that ( p S d , λ) is equal to the number of semistandard Young tableaux ofshape λ and weight μ = (d p ).2 In particular, the Schur functor Sλ appears in p S d if, and only if, λ is a partition of pd with at most p parts. 5. Let V be a G L n -representation. Denoting by Un the unipotent group, by [6, p144], the total multiplicity of V is equal to dim V Un (the space of Un -invariant vectors in V ). It is classical that Sλ (C p ) = 0 if and only if, |λ| ≤ p. If F : Vect C → Vect C is a decomposable functor (as in Fact 3), such that any Schur subfunctor corresponds to a partition with at most p rows, then the Schur decomposition of F can be read from the decomposition into irreducible G L p -subrepresentations of F(C p ). In particular, we can read the total multiplicity and the complexity of F by applying the functor to C p , i.e., N (F) = dim F(C p )U p . 3 Asymptotic plethysms In this section  we investigate the growth with  d of the total multiplicity, and of the complexity of p Symd , Sym p Symd , and p Symd when p is fixed. 1 Plethysm functors are polynomial. Rubei in [21, §2] checks that the general linear group decompositions of syzygies are functorial. 2 Such tableaux can be described by Gelfand–Tsetlin patterns (see [23, p133] for a definition).

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Schur asymptotics of Veronese syzygies

 Remark 3.1 From the literature on the asymptotics of the decomposition of p S d and other plethysms, we mention the following. (Note that when numerics are concerned, in our setting, the natural way to vary the parameters is subtly different from those in the references.)  • The work of Kaveh and Khovanskii in [12] applies to the behavior of p S d with fixed p and varying d. Their focus is on convexity properties such as Fujita-type approximation, the Brunn–Minkowski inequality, the Brion–Kazarnowskii formula, etc.  • In [11], Kaveh computes the dimension of the moment body, varying p, for p V , i.e., the growth with p of the number of its distinct subrepresentations, where V is a fixed G L n -representation.  • Tate and Zelditch [24] studied the asympototics of Kostka numbers of p Sλ where p varies and Sλ is a fixed representation. The idea behind our study is convex-geometric, using the methods of [13–15]. Some of the conclusions of this section may be known to some experts, but we were unable to find precise references. 3.1 Integral points and

p

Sd

 In this subsection, we show that the total multiplicity and the complexity of p S d are counted by the number of lattice points inside slices over d of two rational convex cones. The growth with d of the number of such integral points is a polynomial of degree equal to the dimension of the cross section of the corresponding cone. We determine these two cones and compute the corresponding dimensions.3 These are captured by the following theorem: Theorem 3.2 Fix p ≥ 1. Then  (i) limd→∞ c( p S d )/d p−1 is a finite positive number.  p (ii) limd→∞ N ( p S d )/d ( 2 ) is a finite positive number. of their d-th Proof We specify two graded sets Y• and Y• such that the cardinalities  graded pieces count the complexity and the total multiplicity of p S • , respectively. Complexity: The set  Yd =def (λ2 , . . . λ p , d) : λ1 ≥ λ2 ≥ · · · ≥ λ p ≥ 0, with  p λ1 := pd − λk k=2

⊆N

p−1

× {d}

is a parametrization of the partitions λ  pd with at most p parts, which  is the set of distinct types of Schur functors appearing in the decomposition of p S d by Fact 4 3 In the language of [12], we are determining the dimension of the moment body and of the multiplicity

body (which in our case is also the classical Gelfand–Tsetlin polytope) for the p-th product of a sufficiently large dimension projective space.

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 from §0. Therefore the complexity of p S d is the number of integral points with the p−1 last coordinate d inside the cone contained in R≥0 × R≥0 over the cross section: Y =



def

(λ2 , . . . λ p , 1) : λ1 ≥ λ2 ≥ · · · ≥ λ p ≥ 0, with λ1 := p −

p

k=2

 p−1 λk ⊂ R≥0 × {1}.

This is ( p − 1)-dimensional, which proves part (i). Total multiplicity: Guided by Fact 4, we seek to parameterize the set of semistandard Young tableaux T with pd boxes, at most p rows, and weight (d p ). A first parameterization is the p × p matrix ti j =def number of j  s in the i − th row of T. The semistandard tableau condition implies ti j = 0 when j < i. The restriction on the weight of T imposes tjj = d −

 j−1 k=1

tk j for all j ∈ {1, . . . , p}.

Hence T is determined by (ti j )i< j and by d. With this parameterization, the set of p all such T is in bijection with the set Yd of of points in N( 2 ) × {d} = {((ti j )i< j , d)} subject to the following conditions:  • Denoting tii := d − i−1 j=1 t ji , we ask that tii ≥ 0,

(3.1)

i.e., the number of j’s on the i-th row should be nonnegative even when j = i. • We also ask that  j−1 k=i

tik ≥

j k=i+1

ti+1,k , for all 1 ≤ i < p, 1 ≤ j ≤ p.

(3.2)

This is the tableau condition: we ask that labels greater than or equal to j may start to appear on the (i + 1)-st row of T only on columns to the left or below where labels smaller than or equal to j − 1 stopped on the i-th row. (Note that applying this with j = p implies that the shape of T is a Young diagram.) Similar to the complexity problem, Yd is the set of integral points inside the cone with ( 2p) the last coordinate d in R≥0 × R≥0 over the cross section:

Y :=

((ti j )i< j , 1) :  j−1 k=i

This convex body is

 0 ≤ tii := 1 − i−1 ( 2p) k=1 tki , for all 1 ≤ i ≤ p j × {1}. ⊂ R≥0 tik ≥ k=i+1 ti+1,k , for all 1 ≤ i < p, 1 ≤ j ≤ p

 p 4 2 -dimensional, and part (ii) follows.

 

Remark 3.3 The limits in Theorem 3.2 are at least algorithmically computable for each p, since they are the volumes of the convex bodies Y and Y respectively. 4 To see this, it is enough to produce a point of Y that satisfies strictly all the defining inequalities. Setting ti j = pi  for all i < j, and for sufficiently small , defines such a point.

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Schur asymptotics of Veronese syzygies

  Example 3.4 (The plethysm of 2 S d ) The decomposition of 2 S d consists of all Schur functors of type (2d − a, a) with 0 ≤ a ≤ d, each with multiplicity 1. The total multiplicity and the complexity are both d in this case.   Remark 3.5 Our constructions are similar to the classical Gelfand–Tsetlin patterns. We use this particular form in order to compute the wanted growth rates.   The following remark is a consequence of the proof of Theorem 3.2. It allows us to deem certain collections of Schur functors asymptotically insignificant: Remark 3.6 The coordinates corresponding to Young tableaux of weight (d p ) with at most p − 1 rows, and to Young diagrams with pd boxes and at most p − 1 rows lie in proper linear subspaces of the real vector spaces spanned by Y• and Y• respectively. It follows that asymptotically all the Young tableaux and diagrams parameterized by   Y• and Y• respectively have p rows. 3.2 Sμ S d via

p

Sd

We investigate the asymptotics of the Schur decomposition of Sμ S d when μ is a fixed partition of p. Denote by Vμ an irreducible complex S p -representation of weight μ. Theorem 3.7 Fix p ≥ 1, and let μ be a partition of p. Then  p d ( p) p dim V (i) limd→∞ N (Sμ S d )/d ( 2 ) = p! μ · limd→∞ N S /d 2 . d p−1 (ii) As d grows, c(Sμ S ) = (d ). Proof We have inclusions of graded (by d) vector spaces A, B, C (whose graded pieces are denoted Ad , Bd , Cd respectively): A = Aμ :=



(Vμ ⊗ Sμ S d C p )U p → B :=

d≥0

  p

Sd C p

d≥0

U p

→ C :=

 p

Sd C p .

d≥0

With μ being fixed, the first inclusion is induced by the functorial decomposition p

=



Vμ ⊗ Sμ .

μ p

Observe that C is the section ring on the p-fold product of P(C p ) of OP(C p )× p (1, . . . , 1), and B also has an algebra structure. The action of G L p on C p induces a G L p -representation  p d p structure on Cd for all d. The symmetric group S p also acts S C by permuting the factors of the tensor product. The two actions on Cd = commute, hence the S p -action restricts to B. We have the following: (1) By [8, Thm. 16.2], B is finitely generated. (2) No nontrivial σ in S p acts as a scalar (depending only on σ and d) on Bd for each d. Otherwise, by the commutativity of the actions of S p and G L p , it also acts as a scalar on the G L p -span of Bd in Cd , which is the entire Cd by Fact 5.

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In particular S p acts faithfully on B, and then by [10, Corollary on p.378] (or see [18, Lemma 4] and [1] for a geometric approach on an equivariant resolution of Proj B), we get that Bd as a S p -representation is asymptotically a multiple of the regular representation, and the error in approximation is o(dim Bd ). From the known isotropic decomposition of the regular representation, it follows that dim Ad ∼

(dim Vμ )2 · dim Bd . |S p |

By Theorem 3.2 we have part (i). Part (ii) is a consequence of (i) and Lemma 3.8.  Lemma 3.8 Fix p ≥ 1, and let Fd be a sequence of subfunctors of p S d so that, as p d grows, we have N (Fd ) ∈ (d ( 2 ) ). Then c(Fd ) ∈ (d p−1 ). Proof Consider the moment map p μ : R( 2 ) × R → R p−1 × R, μ((ti j )i< j , d) := (λ2 , . . . , λ p , d), where p i−1 ti j , for all 1 ≤ i ≤ p, and tii := d − tki , for all 1 ≤ i ≤ p. λi :=

j=i

k=1

By the constructions of the previous section, μ maps Y onto Y , and Yd onto Yd for all d. By assumption, there exists C1 > 0 such that for large d, N (Fd ) ≥ C1 · d ( 2 ) . p

On the other hand, denoting by m p,d the maximal multiplicity of a Schur functor in  p d S , which contains Fd , we have N (Fd ) ≤ c(Fd ) · m p,d . By Theorem 3.2 (i) we obtain c(Fd ) ≤ c( that

p

S d ) ≤ C2 · (d p−1 ), it is enough to show

m p,d ≤ C  · d (

p−1 2

)

p p for some C  > 0 independent of d. To see this, choose a basis for Z( 2 ) ⊂ R( 2 ) so that μ is the projection onto the last p coordinates. Then choose an integer l > 0 such that

Y ⊂ [−l, l]( Then Yd ⊂ [−dl, dl]( p−1 (3l)( 2 ) .

p−1 2

p−1 2

) × Y ⊂ R( 2p) × {1}.

) × Yd , so m p,d ≤ #([−dl, dl] ∩ Z)( p−1 2 ) . We can set C  =

Corollary 3.9 (of Theorem 3.7) Fix p ≥ 1. Then

 p  1  p d  N S ∼ N (S p S d ) ∼ N Sd . p!

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Schur asymptotics of Veronese syzygies

As with Theorem 3.7, these estimates essentially follow from Howe’s work in [10]. Proof This follows by applying Theorem 3.7 for the trivial and alternating representations of S p . In the next section we will also use the following result: Proposition 3.10 Fix p ≥ 1. Then, as d grows, N 

 p+1 d S . N

 p

Sd ⊗ Sd



∼

1 p!

·

Proof This is the same argument as for Corollary 3.9. Apply Theorem 3.7.(i) to the alternating representation of S p with the action on the first p tensor factors of  p+1 d S . 4 Asymptotic syzygy functors Classical considerations (see [4, Proposition/Definition 3.1]) realize the syzygy functor K p,q (d) of the d-th Veronese embedding as the middle cohomology of the functorial Koszul-type complex  p+1

S d ⊗ S (q−1)d →

p

S d ⊗ S qd →

 p−1

S d ⊗ S (q+1)d .

It is elementary to compute K p,0 (d) = 0, if p ≥ 1, K0,1 (d) = 0, and K0,0 (d) = C. A result of Green from [7] implies that K p,q (d) = 0, if q ≥ 2, and d ≥ p. The interesting case is K p,1 (d), which is the middle cohomology of  p+1

Sd →

p

Sd ⊗ Sd →

 p−1

S d ⊗ S 2d .

(4.1)

The asymptotics of the decomposition of K p,1 (d) are described by: Theorem 4.1 Fix p ≥ 1. As d tends to infinity:  p · N ( p+1 S d ). (i) N (K p,1 (d)) ∼ ( p+1)! (ii) c(K p,1 (d)) ∈ (d p ). Proof The first map in (4.1) is an inclusion. By Remark 3.6, the total multiplicity of the last term is asymptotically insignificant compared to that of p+1 S d . We obtain that   

 p  p+1 d N K p,1 (d) ∼ N Sd ⊗ Sd − N S

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once we have proved that the right hand side is in (N 3.9 and Proposition 3.10, N

 p

 p+1

 S d ). Using Corollary

       p+1 d p+1 d p+1 d 1 1 ·N Sd ⊗ Sd − N S = ·N S − S , p! ( p + 1)!

 S d ) and part (i) follows. Part (ii) is a consequence of  Lemma 3.8 after observing that K p,1 (d) is noncanonically a subfunctor of p+1 S d .  

which is in (N

 p+1

5 Further results and comments Now that we know how the p-th syzygies behave asymptotically it is tempting to approach more subtle questions with the same methods. A generalization of syzygies which appears naturally when one attempts proofs by restrictions to hyperplanes are the twisted syzygies. Given p, q, b, d ≥ 0, we define the twisted syzygy functor K p,q (b; d) of the d-th Veronese embedding as the cohomology of the functorial Koszul-type complex  p+1

S d ⊗ S (q−1)d+b →

p

S d ⊗ S qd+b →

 p−1

S d ⊗ S (q+1)d+b .

When b = 0, we have K p,q (0; d) = K p,q (d), which is the subject of the previous section. In particular we saw that it is only interesting when q = 1. When b ≥ 1, we will see that there is evidence that the picture may be quite different. For instance K p,0 (b; d) by a similar but more involved argument has the same growth as the middle term of its defining Koszul-type complex. Proposition 5.1 Fix p ≥ 1 and b ≥ 1. Then as d grows,

p  (i) N (K p,0 (b; d)) ∈  d ( 2 ) . (ii) c(K p,0 (b; d)) ∈ (d p−1 ). We do not expect the general methods of this paper to work for N (K p,1 (b; d)) when b ≥ 1. When b = 1, we can show that these groups do not have maximal growth. 

 p+1 d S ). Proposition 5.2 If p ≥ 1, then as d grows, N (K p,1 (1; d)) ∈ o(N Proof Consider the sequence  p+1

Sd ⊗ S1 →

p

S d ⊗ S d+1 →

 p−1

S d ⊗ S 2d+1 .

Its first two cohomology groups are by definition K p+1,0 (1; d) and K p,1 (1; d). The total multiplicity of the third term is asymptotically insignificant compared to the first two. Hence  

 p  p+1 d S ⊗ S1 − N S d ⊗ S d+1 , N (K p+1,0 (1; d)) − N (K p,1 (1; d)) ∼ N

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Schur asymptotics of Veronese syzygies

if we show that the latter difference is in (N tions of the previous section, one can show N

 p+1

 p+1

 

 p Sd ⊗ S1 − N S d ⊗ S d+1 ∼

 S d ). Similar to the computa-

  p+1 d 1 S . ·N ( p + 1)!

By [19, Thm. 6.4], the decomposition of K p+1,0 (1; d) is obtained from the decomposition of S p+1 S d−1 by replacing Schur subfunctors Sλ of the latter with Sλ+(1 p+2 ) . In particular, N (K p+1,0 (1; d)) = N (S p+1 S d−1 ) ∼

  p+1 d 1 S . ·N ( p + 1)!

But this is the same approximation as for N (K p+1,0 (1; d)) − N (K p,1 (1; d)).

 

When b ≥ 1, even though they do not seem to grow maximally, c(K p,1 (b; d)) and N (K p,1 (b; d)) still have large asymptotic growth. Proposition 5.3 Fix p ≥ 1 and b ≥ 1. Assume that p ≥ b + 1. Then as d goes to infinity, log c(K p,1 (b; d)) ∈ (log d),

log N (K p,1 (b; d)) ∈ (log d).

The proof of this statement uses a restriction argument suggested by Robert Lazarsfeld, and is based on other explicit combinatorial results on asymptotic plethysms that we will develop in an upcoming paper. Remark 5.4 A significant next step on asymptotics of syzygy functors is to consider K p,q (d) and let p grow with d appropriately. Ein [4] seems to suggest that “appropriately” means when p grows with d like d q−1 , the functors should be in an asymptotic family and a result generalizing Theorem 4.1 should be true. However, the “right way” to vary p with d is not exactly clear and the asymptotic plethysms are beyond our capabilities. Acknowledgments We have benefited from useful discussions with Samuel Altschul, Igor Dolgachev, Daniel Erman, William Fulton, Roger Howe, Thomas Lam, and Mircea Musta¸ta˘ . We thank Claudiu Raicu, David Speyer, and the anonymous referee for providing many comments, suggestions, and improvements. We are especially grateful to our advisor Robert Lazarsfeld for posing the problem, and for numerous suggestions and encouragements.

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