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Transactions of the American Mathematical Society Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities
Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities
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Transactions of the American Mathematical Society
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10.1090/s0002994708045005
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May, 2008
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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 10, October 2008, Pages 5155–5172 S 00029947(08)045005 Article electronically published on May 27, 2008 ASYMPTOTIC BEHAVIOUR OF CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES ALLAN BERELE AND AMITAI REGEV Abstract. Let A be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence cn (A) is asymptotic to a function of the form ang n , where ∈ N and g ∈ Z. The second author conjectured that if A is any p. i. algebra in characteristic zero, with cocharacter sequence cn (A), then the asymptotic behaviour of this sequence should be given by cn (A) a · nt · n , where√ is √ a nonnegative integer, t is an integer or a halfinteger, and a belongs to Q[ 2π, b], for some 0 < b ∈ Z. We call n the exponential part and a · nt the rational part of the codimension growth. Giambruno and Zaicev made progress on this conjecture by proving that for some nonnegative integer , f1 (n)n ≤ cn (A) ≤ f2 (n)n , where f1 (n) and f2 (n) are Laurent polynomials, but not necessarily of the same degree. In this paper we make some progress towards verifying the above conjecture by proving the following theorem. Theorem 1. Let A be a p.i. algebra with 1 satisfying a Capelli identity. Then cn (A) a · ng · n where, furthermore, g ∈ 12 Z. Also, the constant a here is a sum of Selberg–type integrals; see for example Theorem 36 below. This theorem can help to determine whether or not the generating function f (x) = n≥0 cn (A)xn of the codimensions is algebraic. For example, if g ∈ Z and is negative, then f (x) is not algebraic; see Lemma 3.2 in [1]. This application raises the following further question about Theorem 1: when is g < 0? The techniques of [6] imply the following: Let Aj , j = 1, 2, be p. i. algebras with T –ideals of identities id(Aj ) = Ij . This is known to be the case, for example, Received by the editors June 5, 2006. 2000 Mathematics Subject Classiﬁcation. Primary 16R; 10. Key words and phrases. Polynomial identities, cocharacter sequence. The work of the ﬁrst author was supported by both the Faculty Research Council of DePaul University and the National Security Agency, under Grant MDA904500270. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. The work of the second author was partially supported by ISF grant 94704. c 2008 American Mathematical Society 5155 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5156 ALLAN BERELE AND AMITAI REGEV if Aj is the algebra of n × n matrices. Assume cn (Ai ) aj · ngj · nj with g1 , g2 < 0. Then the algebras with T –ideals of identities equal to I1 I2 and I1 ∩ I2 also have that same property g < 0. However, the comment at the end of the introduction implies that one can construct many algebras with g ≥ 0. The Giambruno–Zaicev theorem was proven in two stages: In [10] they proved it in the special case that A is ﬁnitely generated, or more generally that A satisﬁes a Capelli identity, and in [11] they proved the general case. In the current paper we will prove a slightly weaker version of the conjecture in the case that A satisﬁes a Capelli identity and has a unit. We will show that cn (A) ant n , where is a nonnegative integer, t is an integer or half integer but we don’t have control over the constant a. Getting such control would involve evaluating a certain integral, which we describe in the second section. Under the weaker assumption that cn (A) is an increasing sequence, we prove that a1 nt n ≤ cn (A) ≤ a2 nt n . The proof combines and adapts ideas from the ﬁrst author’s paper [5] and the second author’s paper [14]. The main idea is that for any p. i. algebra, cn (A) = mλ dλ , where dλ is the degree of the Sn character corresponding to λ, and mλ is the multiplicity of that character in the cocharacter sequence. If A satisﬁes a Capelli identity, then mλ will be zero unless λ has height bounded by some constant k. For such a λ, the multiplicities were estimated in [5]. We think of the multiplicities as functions from Nk → N. In [5] we showed that Nk could be partitioned into a ﬁnite number of regions such that mλ was equal to a polynomial on each. In section 1 we show that as far as the asymptotic behaviour of cn is concerned, we need only consider regions of the form {v0 + α1 v1 + · · · + αr vr  α1 , . . . , αr ≥ 0} ∩ (i + (dZ)k ), where the vi are all partitions, all except v0 are linearly independent and have height at most the exponent ≤ k, and v1 is the partition (1 ), thought of as the vector (1, . . . , 1, 0, . . . , 0) ( 1’s and k − 0’s). In the next section, we compute mλ dλ over one such region, adapting the arguments from [14]. In that work β the second author computes the asymptotic behaviour of dλ , where λ runs over partitions of n with at most parts. The sum we need to compute diﬀers both in the region we are summing over and the function we are summing, but for all that, the computations from [14] turn out to be just what we need here. The main result of this section is that mλ dλ over such a domain is asymptotic to a function of the form ant n , where a, t and all have the required form. Finally, in the last section we deal with the technical problems involved in summing the mλ dλ from the various domains. Based on these computations we present a conjecture. Conjecture. Let f (x1 , . . . , xk )/g(x1 , . . . , xk ) be a rational function such that the numerator and denominator are each symmetric polynomials with integer coeﬃcients, and such that the denominator is a product of terms of the form (1 − xa1 1 · · · xakk ). We expand the fraction as a series in Schur functions, f (x1 , . . . , xk )/g(x1 , . . . , xk ) = ∞ mλ Sλ (x1 , . . . , xk ) n=0 λ∈Par(n) so that mλ is the coeﬃcient of the Schur function Sλ (x1 , . . . , xk ). Deﬁne cn to be λ∈Par(n) mλ dλ . Then there exists a modulus d and constants ai , ti and i where License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES 5157 i = 0, . . . , d − 1 such that, if we restrict n to be congruent to i mod d, then cn has asymptotic behaviour cn ai nti ni . Moreover, each ti will be either an integer or a halfinteger and each i will be the dth root of a rational number. We close this introduction with a lower bound on the degree of the rational part of cn (A). Theorem 2. Let A be a p. i. algebra satisfying a Capelli identity with cn (A) ant n . Let 1 ≤ s ≤ 4 be minimal such that can be written as a sum of s squares. Then t ≥ 32 s − 1 − 12 . Proof. Let I be the ideal of identities of A, and denote by Ia the ideal of identities of a × a matrices. It is implicit in [10] and explicit in Theorems 2.5 and 2.8 of [7] that there exist a1 , . . . , ak such that I ⊆ Ia1 · · · Iak and such that the generic algebra with identities Ia1 · · · Iak has the same exponential rate of growth as A. Since this algebra will have cocharacter sequence less than or equal to that of A, it suﬃces to prove our lower bound in the case of I = Ia1 · · · Iak . It follows from Theorem 1.4 of [6] that the exponential rate of growth of such an algebra is = a21 + · · · + a2k and that the degree of the rational part is 1 1 3 t=k−1+ − (a2i − 1) = k − 1 − . 2 2 2 We minimize this sum by making k as small as possible. It turns out that there is no upper bound on the degree of the rational part. Here is an example taken from [13]. It generalizes easily to any value of . Let A be the subalgebra of k × k matrices spanned by e11 and all eij with i < j. Then all identities of A are consequences of [x1 , x2 ]x3 · · · xk and cn (A) ank−1 . So, the exponent k − 1 can be made as large as desired while the exponential rate of growth is equal to 1. 1. The multiplicities mλ We ﬁrst describe the relevant theorem from [5]. The space Rk can be partitioned into a ﬁnite number of subsets such that the multiplicity function mλ restricted to each of these subsets is equal to a polynomial in the coordinates. In order to describe this decomposition, it is useful to have a deﬁnition. Deﬁnition 3. Given v1 , . . . , vr vectors in Rk the linear cone C(v1 , . . . , vr ) is deﬁned by C(v1 , . . . , vr ) = {α1 v1 + · · · + αr vr  α1 , . . . , αr ≥ 0}. Moreover, given an additional vector v0 the aﬃne linear cone C(v0 ; v1 , . . . , vr ) is deﬁned to be v0 + C(v1 , . . . , vr ). This is a translation of C(v1 , . . . , vr ) by the vector v0 . If in addition the vectors v1 , . . . , vr are linearly independent we call C(v1 , . . . , vr ) a simplicial cone and C(v0 ; v1 , . . . , vr ) a simplicial aﬃne cone, and we call v1 , . . . , vr the basis in either case. Finally, if the vi can all be taken to have coordinates in Q we call the resulting cone rational. Lemma 4. Any (rational or aﬃne) cone is a ﬁnite union of (rational or aﬃne) simplicial cones with disjoint interiors. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5158 ALLAN BERELE AND AMITAI REGEV Proof. Let d be the dimension of the span of v1 , . . . , vr . We use induction on d. If d = 1, then the cone is a ray and so is already simplicial. If d > 1, then the boundary of C is a union of (d − 1)dimensional cones, so it is a ﬁnite union of simplicial cones v0 + C1 , . . . , v0 + Cm . Let v be a vector with rational coordinates such that v0 + v is in the interior of C. Then each of the Ci together with v span a simplicial cone Ci . Since the Ci have disjoint interiors, so do the Ci ; and since C is convex, the union of the Ci gives all of C. Here then is the theorem we need from [5]. Theorem 5. Given a p. i. algebra A with cocharacter supported in the strip of height k, there exists a positive integer d and a partition of (R+ )k into regions R1 , . . . , Rm , each a ﬁnite intersection of rational aﬃne linear cones such that mλ is given by a polynomial on each Ra ∩ (i + (dZ)k ),where i ∈ {0, . . . , d − 1}k . Note that here and throughout we use R+ instead of R≥0 for the nonnegative real numbers. Here is an example to illustrate the theorem; see [4] and [9]. It involves trace cocharacters instead of ordinary cocharacters, but the theory is the same. Theorem. Let λ = (λ1 , λ2 ) be a partition of height at most 2. Then the multiplicity of χλ in the trace cocharacter sequence of 3 × 3 matrices depends on whether λ1 > 2λ2 . If this is the case, then mλ = 11λ1 λ62 71λ72 λ21 λ52 − + + O(n6 ) 17280 103680 1451520 and if not, then λ6 λ2 λ5 λ2 λ4 λ3 λ3 λ4 7λ2 λ5 7λ1 λ62 19λ72 λ71 − 1 + 1 2 − 1 2+ 1 2− 1 2+ − +O(n6 ). 1451520 103680 17280 5184 2592 17280 34560 483840 Moreover, each of the two O(n6 ) terms is given by 36 (not necessarily distinct) polynomials, depending on λ1 and λ2 mod 6. mλ = Note that the set of partitions is divided into partitions with λ1 ≥ 2λ2 and λ1 ≤ 2λ2 . These two sets are each cones: The former is C ((2, 1), (1, 0)) and the latter is C ((2, 1), (1, 1)), and within each of these cones, the partitions are further divided up according to which lattice (a, b) + 6Z × 6Z they belong to. Deﬁnition 6. Given a real function f on D ⊆ Zk such that there exists a modulus d and such that f restricted to each D ∩ (i + (dZ))k is equal to a polynomial function for each i ∈ {0, . . . , d − 1}k , we will say that f is almost polynomial. In Theorem 5, each Ri is a ﬁnite intersection of aﬃne cones. Our next goal is to show that each such intersection can be written as a ﬁnite union of simplicial cones. Deﬁnition 7. A polyhedral set in Rk is an intersection of ﬁnitely many halfspaces. If the halfspaces can all be deﬁned by equations with rational coeﬃcients, we call the polyhedral set rational. Note that the intersection of the halfspaces a1 x1 +· · ·+ak xk ≥ b and a1 x1 +· · ·+ak xk ≤ b is an (n−1)dimensional hyperplane, and so a polyhedral set may be contained in a hyperplane of dimension less than k. The dimension of a polyhedral set is the dimension of the smallest hyperplane containing it. (We use the term hyperplane to refer to a translate of a subspace of Rk , not necessarily of dimension k − 1.) Note that an intersection of aﬃne cones License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p F p p −p ∪Pi p p p F p p p p F p p 1p p p p p p 1p p p p p p p p 1p P * p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p H p p p p p p p p p p p p p p p p p p H p p p p p p p p p qp p p p p p p p pF2p p p p p p p p p p p ⇒ pF2p P p 2 pp p p p p p p p ⇒ p p p p p F p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p P p p p p p p p p p p p p p p 3p p p p p P p 4p p p * * p @ p p p p p p p p p p p p @ p p p p p p p p p p p p F p 3p p p p p p F p 4p p p p F p 3p p p p p p F p 4p p p @ p p p@ p p p p p p p p p p p p@ p p p p p p p p p @ 5159 p p p p p p p p p p p p p p p p p p p p p F p p −p ∪Pi p p p F p p p p p p p p 1p * p p p p p p p p p p p p * p p p p p p p p p * p p p * * p p p p p p p p p p p p * * pF2p p p p p p p p p p p * * p p p p p p p p p p p p * * p p p p p p p p p p p p * * p p p p p p p p p p p p * * p @ p p p p p p p p pp p p F p 3p p p p p p F p 4p p p @ p p p@ p p p p p p p p p Figure 1. Proof of Lemma 10 is a polyhedral set. Given S ⊆ Rk we denote by conv(S) the convex closure of S, i.e., the smallest convex set containing S. Lemma 8. Let F ⊆ Rk be a ddimensional rational polyhedral set contained in the ddimensional hyperplane Π and p ∈ Rk − Π. Then there exist ﬁnitely many disjoint rational polyhedral sets each with the same dimension as F and together containing all of the integer points of conv(p, F ). Proof. Let Π be a (k − 1)dimensional rational hyperplane containing Π and not containing p. Let Π have equation a1 x1 + · · · + ak xk = b, where the ai and b are all integers and the ai are relatively prime. Let p = (p1 , . . . , pk ) and let a1 p1 + · · · + ak pk = b . Note that conv(p, F ) is contained between these two hyperplanes. But there are only ﬁnitely many integers bj between b and b (inclusive) and so only ﬁnitely many hyperplanes parallel to Π containing integer points. Each of these hyperplanes intersects conv(F, p) in a polyhedral set having the same dimension as F. We now need a general fact from convex set theory. Let S ⊆ Rk be any inﬁnite, closed, convex set, and let p and q be two points of S. Given a v ∈ Rk the ray p + R+ v will be contained in S if and only if the ray q + R+ v is contained in S. Based on this fact, the characteristic cone of S is deﬁned to be cc(S) = {v  p + R+ v ⊆ S}. An important theorem about cc(S) is that it is indeed a cone and if S is a polyhedral set, then it will be a linear cone; see for example section 2.5 of [12] or section 1.5 of [16]. We record this as a lemma. Lemma 9. Let S ⊆ Rk be any inﬁnite, closed, convex set, and let p be a point of S. Then the set of rays with endpoint p which are completely contained in S forms a cone. Lemma 10. Let F be a rational polyhedral set in Rk . Then there exist ﬁnitely many rational aﬃne linear cones whose interiors are disjoint and whose union contains all but ﬁnitely many of the integer points of F . Proof. We use induction on d = dim F to prove the lemma. The case of d = 1 is easy since a convex onedimensional set is a line segment or a ray. A line segment will contain only ﬁnitely many integer points, and a ray is an aﬃne linear cone. In the general case, let F be a rational polyhedral set of dimension d and let p be a rational point in the interior of F ; see Figure 1. The (d − 1)dimensional License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5160 ALLAN BERELE AND AMITAI REGEV faces of F are called facets and their union is the boundary of F . Let F1 , . . . , Fa be the facets of F and for each i let Pi = conv(Fi , p). For example, in Figure 1, the twodimensional ﬁgure F is bounded by the onedimensional facets F1 , . . . , F4 . Note that if the facet Fi is contained by the (d−1)dimensional hyperplane Πi , then p ∈ Rk − Πi ; hence Lemma 8 can be applied. By that lemma, the integer points of Pi are covered by ﬁnitely many (d − 1)dimensional rational polyhedral sets. So there exist polyhedral sets of dimension d − 1, Pij , such that ∪j Pij has the same integer points as Pi . By induction each Pij is the union of disjoint rational aﬃne cones. In the example, P2 and P3 are ﬁnite and will have only ﬁnitely many integer points, and these points are not covered by cones in this example. The regions P1 and P4 are inﬁnite, and Lemma 8 implies that in each there are a ﬁnite number of rays containing all of their rational points. However, the union of the Pi may not cover all of F . The points of F − ∪Pi consist of the union of all rays with endpoint p which do not intersect the boundary of F . Hence it is the union of all rays starting at p and completely contained in F . In the example, F − ∪Pi is a 2dimensional cone. In general, we may apply Lemma 9 to conclude that it is an aﬃne cone. Hence F will be the union of this cone with the Pi , and since there are ﬁnitely many aﬃne cones containing all but ﬁnitely many of the integer points of ∪Pi , this completes the proof. Corollary 11. Let F be a rational polyhedral set in Rk . Then there exist ﬁnitely many disjoint rational, aﬃne simplicial cones whose union contains all but ﬁnitely many of the integer points of R. Proof. In Lemma 10, if two of the cones intersect along their boundary we replace one by a slightly smaller cone and then use Lemma 4 to replace the cones by simplicial ones. In light of this corollary, we may now reformulate Theorem 5. Corollary 12. Given a p. i. algebra A with cocharacter supported in the strip of height k, there exist a positive integer d and a partition of (R+ )k into disjoint rational, aﬃne simplicial cones C1 , . . . , Cm and a ﬁnite number of points such that mλ is given by a polynomial on each Ca ∩ (i + (dZ)k ), where i ∈ {0, . . . , d − 1}k . One of the main results of [14] is that if λ has all parts close to equal, then dλ has exponential behaviour kn . Lemma 16 below shows that if none of the basis vectors is (1, . . . , 1), then the exponential behaviour of dλ is strictly less. Remark 13. Given v ∈ Rk , denote v = (v (1) , . . . , v (k) ). It is a partition if v ∈ Zk and v (1) ≥ · · · ≥ v (k) ≥ 0. Given u = (u(1) , . . . , u(k) ) ∈ Rk with u(1) , . . . , u(k) ≥ 0, denote u = u(1) + · · · + u(k) . Let C = C(v0 ; v1 , . . . , vr )be a cone, (w.l.o.g.) v1 , . . . , vr = 0, and let u = (u(1) , . . . , u(k) ) ∈ C, so u = v0 + ri=1 ti vi ; if v0 , . . . , vr are partitions, then u(1) ≥ · · · ≥ u(k) ≥ 0 and u = v0  + ri=1 ti vi . Moreover, since v1 , . . . , vr = 0, u → ∞ whenever i ti → ∞. If the partition vi is not (k) proportional to (1, . . . , 1), then there exist 0 ≤ δi < 1 such that vi ≤ (δi /k)vi . Lemma 14. Let C = C(v0 ; v1 , . . . , vr ) ⊆ Rk be a cone, v1 , . . . , vr = 0 partitions where none is proportional to (1, . . . , 1). There exist 0 ≤ δ < 1 such thatfor all r (1) (k) u = (u , . . . , u ) = v0 + i=1 ti vi ∈ C, u(k) ≤ (δ/k)u, provided that i ti is large enough. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES (1) (k) (1) 5161 (k) Proof. Write vi = (vi , . . . , vi ), so vi  = vi + · · · + vi , i = 1, . . . , r. For each (k) i there exists 0 ≤ δi < 1 such that vi ≤ (δi /k)vi . some δ < δ < 1, for Let δ = max{δi  i = 1, . . . , r}, so 0 ≤ δ < 1. Choose r (1) (k) example δ = (δ + 1)/2. Now let u = (u , . . . , u ) = v0 + i=1 ti vi ∈ C. Then (k) u(k) = v0 + r (k) ti vi (k) ≤ v0 + i=1 r r δi δ δ (k) (k) ti vi  ≤ v0 + ti vi  = v0 + u. k k i=1 k i=1 r (k) If i=1 ti is large, then also ri=1 ti vi  becomes large and v0 is negligible, so (k) v0 + δk u ≤ kδ u. Therefore u(k) ≤ kδ u. Lemma 15. Let 0 ≤ δ < 1 and consider partitions λ = (λ1 , . . . , λk ) satisfying λk ≤ (δ/k)λ. Then, as λ → ∞, the exponential growth of dλ (= f λ ) is bounded by (k − )λ for some > 0. Proof. Let n = λ. By the hook formula, (1) n! n! . ≤ dλ = hij λ1 ! · · · λk ! Write λk = ( k1 − x) · n, so 0 < (1 − δ)/k ≤ x. Clearly, λ1 + · · · + λk−1 = n − λk = ( k−1 k + x) · n. Fixing λk , it is well known that the product λ1 ! · · · λk−1 ! minimizes x when λ1 ≈ · · · ≈ λk−1 , so when λ1 ≈ · · · ≈ λk−1 ≈ ( k1 + k−1 ) · n, in which case n! n! ≤ hij λ1 !···λk ! maximizes. Hence, by (1), dλ ≤ n! , λ1 ! · · · λk ! 1 1 x + − x · n. · n, λk = k k−1 k √ √ m m to the r.h.s. of (2). After discarding Apply Stirling’s formula m! ≈ 2π·e−m m√ a few terms of the order of magnitude of n we have dλ (1/g(x))n where x ( k1 −x) (k−1)( k1 + k−1 ) 1 1 x g(x) = + −x . k k−1 k (2) λ1 ≈ · · · ≈ λk−1 ≈ It is easy to show that g(x) monotonically increases: let h(x) = ln g(x); then x )/( k1 − x)] > 0. Thus 1/g(x) decreases with x. Since g(0) = 1/k, h (x) = ln[( k1 + k−1 therefore for 0 < x ≤ 1/k, g(x) > 1/k and we can write 1/g(x) = k − for some > 0. As a corollary of Lemmas 14 and 15 we now have Lemma 16. Let C = C(v0 ; v1 , . . . , vr ) ⊆ Rk be an aﬃne linear cone in which the basis vectors are all partitions and none is a multiple of (1, . . . , 1). Then there exists an > 0 and a polynomial p(x1 , . . . , xk ) such that for every partition λ ∈ C, dλ ≤ p(λ)(k − )λ . Given 0 ≤ ≤ k we let Rk, ⊆ Rk be the vectors with the last k − coordinates all equal to zero and we let 1k, be the vector in Rk, with ﬁrst coordinates equal to 1. We now have this corollary. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5162 ALLAN BERELE AND AMITAI REGEV Corollary 17. Let C = C(v0 ; v1 , . . . , vr ) be an aﬃne linear cone in Rk in which v1 , . . . , vr ∈ Rk, and none of v1 , . . . , vr is a multiple of 1k, . Then there exists an > 0 and a polynomial p(x1 , . . . , xk ) such that for every partition λ ∈ C, dλ ≤ p(λ)( − )λ . In [10] Giambruno and Zaicev proved that for any p. i. algebra A satisfying a Capelli identity of degree k, the sequence { n cn (A) } has a limit = e(A) which is a nonnegative integer less than or equal to k. (Gaimbruno and Zaicev denote the limit by e for exponential rate of growth, but we need to reserve e for 2.71 · · · .) In the course of proving this (Theorem 3 in their paper), they also proved that there was a constant K (in their paper K = dq − d) such that for every n, the cocharacter sequence contains χλ , λ ∈ Par(n), with nonzero multiplicity and λ close to a n rectangle of height , in the sense that i> λi ≤ K, i=1 λi −  ≤ K, and λ and that it doesn’t contain any χ with λ = (λ1 , λ2 , . . .) and with λ+1 large. Translating these two facts into the language of Theorem 5 yields the following lemma. We attribute it to Giambruno and Zaicev since it is immediate from their work. To state the lemma more easily, let C = C(v0 ; v1 , . . . , vr ) be a simplicial cone. We will say that C is in the support of mλ if mλ is a nonzero polynomial on the intersection of C with some lattice i = (dZ)k . Lemma 18 (Giambruno and Zaicev). Let A be a p. i. algebra which satisﬁes a Capelli identity and with exponential rate of growth of the codimensions. Let C be a simplicial cone in the support of mλ , as above. Then each of the basis elements of C is a partition of height at most . Moreover, there exists at least one simplicial cone C in the support of mλ with basis element (1 ). Proof. Let C = C(v0 ; v1 , . . . , vt ) be a simplicial cone in the support of mλ . So, on the intersection of C and a lattice, the multiplicity function equals a nonzero . , αt ) be the multiplicity of λ = α1 v1 + polynomial p(λ1 , . . . , λk ). Let q(α1 , . . th · · · + αt vt . Note that each λj equals coordinate i αi vj,i , where vj,i is the i of vj , and so q is obtained from p by a linear substitution. Hence q is also a nonzero polynomial. Assume by way of contradiction that v1 is a partition of height ≥ + 1. Write q as i α1i qi (α2 , . . . , αt ). For any choice of α2 , . . . , αt in the lattice, there would exist inﬁnitely many values of α1 in the lattice such that v0 + α1 v1 + · · · + αt vt is not a partition with λ+1 small. By Giambruno and Zaicev’s theorem, the multiplicity mλ would be zero in these cases. Hence, by a Vandermonde argument each qi would be zero. This would imply that mλ is identically zero. On the other hand, if there were no simplicial cones with (1 ) as a basis element, then the exponential rate of growth would be less than , by Corollary 17. Theorem 19. Let A be a p. i. algebra which satisﬁes a Capelli identity of degree k such that e(A) = . Then there exists a set of aﬃne simplicial cones C1 , . . . , Ca ⊆ Rk , each with 1k, as a basis vector and with all basis vector partitions in Rk, such that a {mλ dλ  λ ∈ Par(n) ∩ Ci } cn (A) i=1 and such that the multiplicity function mλ restricted to each Ci is almost polynomial. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES 5163 Proof. Combining Theorem 5 with Corollary 11 shows that Par(n) can be written as a union of a ﬁnite number of disjoint rational aﬃne simplicial cones {Ci } containing all partitions of height at most k such that mλ is almost polynomial in each one of them. Hence, {mλ dλ  λ ∈ Par(n) ∩ Ci }. cn (A) i If 1k, is not a basis element of any such Ci , then by Corollary 17 the sum over Ci is at most Cnt ( − )n for appropriate constants. But the Giambruno and Zaicev lemma shows that cn (A) is bounded below by a function of the form Cnt n and so the sum of over the Ci not containing 1k, may be ignored. 2. The sums mλ dλ in cones In this section we ﬁx an aﬃne simplicial cone C̄ = C(v0 ; v1 , . . . , vr ) ⊂ Rk whose basis vectors are partitions contained in Rk, and the ﬁrst one is v1 = 1k, , a lattice i + (dZ) , and a nonnegative polynomial function with rational coeﬃcients, mλ . It is our goal to estimate {dλ mλ  λ ∈ C̄ ∩ (i + (dZ) ) ∩ Par(n)}. For convenience we modify this slightly by letting C be the corresponding linear cone C = C(v1 , . . . , vr ) and replacing i by i − v0 . This changes the sum to (3) {dλ+v0 mλ+v0  λ ∈ C ∩ (i + (dZ) ) ∩ Par(n − v0 )}. Our computation of this sum will imitate the computations in sections 1 and 2 β of [14]. In that paper the asymptotic behavior of dλ is computed, where λ was restricted to have at most nonzero parts. In adapting the proof to the current situation we need to deal with the facts that our λ is additionally restricted to be in an aﬃne cone and a lattice, and that the sum is of a polynomial times dλ instead of a power of dλ . It is possible that some λi = λj for every λ ∈ C. As in [14], we deﬁne θ1 , . . . , θp to be maximal such that λ1 = · · · = λθ1 , λθ1 +1 = · · · = λθ1 +θ2 , · · · for all λ ∈ C. Remark 20. Note that the sum mλ dλ with the above restrictions on λ will be empty unless the sum of the parts of i is congruent to n mod d, which we take to be the case. (4) Generalizing the notation Λ (n) from [14] we deﬁne Λθ (n) to be the set of all partitions of n with height at most that satisfy equation (4). The degrees dλ of such partitions are estimated in [14], and we now recall some notation from that paper. √ Deﬁnition 21. Given a partition λ = (λ1 , . . . , λ ) of n, deﬁne ci by λi = n + ci n, i = 1, . . . , . Let D(c) = {(ci − cj )  i < j, ci = cj }, E(c) = {(i − j)  j < −1 1 2 i, ci = cj }, and γ = (2π)− 2 2 . Let λ ∈ Λθ (n) with each λθ1 +···+θi = λθ1 +···+θi +1 . Then D(c) = Dθ (c) = {(cθ1 +···+θi − cθ1 +···+θj )  i < j} and E(c) = (θ1 − 1)! · · · (θp − 1)! License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5164 ALLAN BERELE AND AMITAI REGEV Lemma 22 (= Generalization 1.2 of [14]). With notation as in the previous deﬁnition, assume that λ ∈ Λθ (n) is such that each ci  ≤ a and that each cθ1 +···+θi − cθ1 +···+θi+1 is greater than or equal to some ﬁxed δ > 0 . Then θ i 2 −1 √ ( )− 2 i(2) n dλ γλ Dθ (c) (θi − 1)! e− 2 c n− 2 n2 , where by c2 we mean c21 + · · · + c2 . In order to take the v0 into account we use this lemma. Lemma 23. Let 1 ≤ ≤ k, let ν be a ﬁxed partition of height k and denote by ν̃ the sum i> νi . Let λ ∈ Λθ (n) be such that each λθ1 +···+θi − λθ1 +···+θi +1 goes to inﬁnity. As n goes to inﬁnity, dλ+ν bnν̃ , dλ where b ∈ Q is a constant. Proof. By the Young–Frobenius formula (or by the hook formula), dλ = k λ! i=1 (λi + k − i)! 1≤i<j≤k λi − λj + j − i, and similarly dλ+ν = k λ + ν! i=1 (λi + νi + k − i)! 1≤i<j≤k λi + νi − λj − νj + j − i. Compare corresponding terms in the ratio dλ+ν /dλ . The fraction λ + ν!/λ! is asymptotic to nν ; for i = 1, . . . , the fraction (λi + k − i)!/(λi + νi + k − i)! is i ; and for i > it is the rational number (k − i)!/(νi + k − i)! asymptotic to λ−ν i (independent of λ). Since ci  < a, λi n/, i = 1, . . . , , so ν1 +···+ν −ν 1 · · · λ . λ−ν 1 n Multiplying by nν gives a rational number times nν̃ . Finally, estimate λi + νi − λj − νj + j − i . A := λi − λj + j − i 1≤i<j≤k If i ≤ , then λi − λj → ∞; hence λi + νi − λj − νj + j − i 1. λi − λj + j − i If i > , then λi = λj = 0, and the corresponding term is a rational number. It follows that A r for some r ∈ Q. The lemma now follows. We now recall and modify some more deﬁnitions from [14]. Deﬁnition 24. Let Λθ (n, a) be the partitions in Λθ (n) with each ci  ≤ a and Λθ (n, a, δ) be the partitions in Λθ (n, a) with each cθ1 +···+θi − cθ1 +···+θi +1 ≥ δ. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES 5165 Combining this deﬁnition with the previous two lemmas yields: Corollary 25. If λ ∈ Λθ (n, a, δ), then θ i 2 −1 √ ( )− 2 i(2) n dλ+v0 KDθ (c)e− 2 c nṽ0 − 2 n2 , √ where K is a constant in Q( π) depending only on , θ and v0 . Recall that C is the simplicial cone C = (v1 , . . . , vr ) where v1 = 1k, . Since v1 , . . . , vr are linearly independent and by Lemma 18 they belong to Rk, , r will be less than or equal to = dim Rk, , since elements of Rk, have k coordinates, but the last k − coordinates are = 0. Let λ ∈ C, so it is a positive combination of the vi ’s, and write that combination in the form √ √ √ n λ = ( + α1 n)v1 + α2 nv2 + · · · + αr nvr . In order and √ for the sum to be in C we require that αi ≥ 0 for i = 2,n. . . , r √ n + α n ≥ 0. Given the partition λ = (λ , . . . , λ ), write λ = + c 1 1 i i n. Comparing to the previous description we have √ √ √ √ √ n n n ( + α1 n)v1 + α2 nv2 + · · · + αr nvr = ( + c1 n, . . . , + cr n). Keeping in mind that v1 = 1k, , this implies that α1 v1 + · · · + αr vr = (c1 , . . . , c ) and so we deﬁne (5) T (α1 , . . . , αr ) = r αi vi = (c1 , . . . , c ). i=1 Let τn be given by τn (α) = r √ √ n n v1 + nT (α) = v1 + n αi vi . i=1 Since C(v1 , . . . , vr ) is assumed to be simplicial, the vi are linearly independent and so T and hence τn are onetoone. Note that v1  = and τn (α) will have parts sum to n precisely when αi vi  = 0, where by vi  we mean the sum of the parts of vi . This is equivalent to r vi  αi . α1 = − i=2 Deﬁnition 26. Deﬁne Q(a) ⊆ (R+ )r−1 as follows. Let (x2 , . . . , xr ) ∈ (R+ )r−1 r and denote x1 = x1 (x) = − i=2 vi  xi . By (5), T (x1 (x), x) = T (x1 , x2 , . . . , xr ) = (c1 , . . . , c ). Given a > 0, deﬁne Q(a) via: x = (x2 , . . . , xr ) ∈ Q(a) iﬀ T (x1 (x), x) = (c1 , . . . , c ) and all ci  ≤ a. Continuing in this vein, note that c1 = · · · = cθ1 , cθ1 +1 = · · · = cθ1 +θ2 , . . ., in the image of T and so deﬁne Q(a, δ) to be the subset of Q(a) with the additional conditions cθ1 +···+θi − cθ1 +···+θi +1 ≥ δ. Remark 27. Note that both Q(a) and Q(a, a−r ) tend to (R+ )r−1 as a → ∞. Deﬁnition 28. Let SC (n) = {mv0 +λ dv0 +λ  λ ∈ Par(n) ∩ C ∩ (i + (dZ) )}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5166 ALLAN BERELE AND AMITAI REGEV Two related sums are SC (n, a, θ) = and SC (n, a, δ, θ) = {mv0 +λ dv0 +λ  λ ∈ Λθ (n, a) ∩ C ∩ (i + (dZ) )} {mv0 +λ dv0 +λ  λ ∈ Λθ (n, a, δ, θ) ∩ C ∩ (i + (dZ) )}. Remark 29. Note by equation (3) that SC (n) represents the contribution of the characters in the cone C intersecting the lattice i + (dZ) to the codimension, and recall by Remark 20 that Par(n) ∩ (i + (dZ) ) will be empty unless n ≡ i mod d. Hence, if n ≡ i mod d, each of SC (n), SC (n, a), etc. equals zero. We note that mλ is assumed to be a nonnegative polynomial function of λ1 , . . . , it follows λ with rational coeﬃcients. Let the vector vi have coordinates vij . Then √ from the deﬁnition of τn that if λ = τn (α), then each λj equals v1j n + n i αi vij . Let the multiplicity function mλ be equal to the polynomial m(x1 , . . . , x ) on the intersection of the cone and the lattice. By substitution, √ n αi vi ), mv0 +λ = m(v0 + v1 + n and so each xj is replaced by v0j + v1j n + i αi vij and so mv0 +λ nµ g(α1 , . . . , αr ), where µ is an integer or a halfinteger and g is a polynomial with rational coeﬃcients. The following is a direct analogue of Lemma 2.2 of [14] and the proof is the same. Lemma 30. Referring to Remarks 20 and 29, let n ≡ i mod d. Then, as n → ∞ (namely, n = i + q · d and q → ∞), the sum SC (n, a, δ, θ) is asymptotic to θj 2 2 −1 √ ( )− j ( 2 ) n+v0  nµ f (α)e− 2 T (α) nν̃0 − 2 n2 , where the sum is over (α2 , . . . , αr ) ∈ Q(a, δ), α1 = − i + (dZ) , and where f (α) = g(α)KDθ (T (α)) . vi  αi such that τn (α) ∈ Note that Q(a) and Q(a, δ) are bounded (r − 1)dimensional surfaces and we can consider their volumes. The proofs of Lemmas 2.3 and 2.4 of [14] now go over essentially unchanged. Lemma 31 (= Lemma 2.4 of [14]). If δ = 1 ar , then lim volr−1 (Q(n, a) − Q(n, a, δ)) = 0. a→∞ Following [14] we let Dδ (c) equal {(ci − cj )  i < j, ci − cj ≥ δ}. We now record the analogues of the next three lemmas of [14]. The proofs are essentially unchanged, although since we are restricting to partitions in the intersection of the cone C with the lattice (i = (dZ) ) we replace the condition that √ n + cj n ≥ 0 must be an integer with τn (α) ∈ (i + (dZ) ) and, in order for the intersection to be nonempty, we require n ≡ i mod d as in the previous lemma. Lemma 32 (= Proposition 2.5 of [14]). Let n ≡ i mod d and let 2 u(n, a) = g(α)Da−r (T (α))e− 2 (T (α) ) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES 5167 vi  be summed over (α2 , . . . , αr ) ∈ Q(n, a) − Q(n, a, a−r ) and α1 = − αi such that τn (α) ∈ (i + (dZ) ). As n → ∞ (namely, n = j + q · d and q → ∞), u(n, a) lim lim √ r−1 = 0. a→∞ n→∞ n Now, we may choose a small cube U in Q(1) with volume V1 such that g(α)Da−r (T (α))e− 2 (T (α) 2 ) ≥ m > 0 on U. We now have this analogue of Lemmas 2.6 and 2.7 of [14]. Lemma 33 (= Lemma 2.6 of [14]). If a ≥ 1 and a is so large that Q(a, a−r ) ≥ mV1 , where contains U , then limn→∞ √v(n,a) n r−1 2 v(n, a) = g(α)Da−r (T (α))e− 2 (T (α) ) vi  summed over (α2 , . . . , αr ) ∈ Q(n, a, a−r ) and α1 = − αi such that τn (α) ∈ (i + (dZ) ). Lemma 34 (= Lemma 2.7 of [14]). Let 3 w(n, a) = g(α)Dδ (T (α))e− 3 (T (α) ) vi  be summed over (α2 , . . . , αr ) ∈ / Q(n, a) and α1 = − αi such that τn (α) ∈ (i + (dZ) ). Then w(n, a) lim lim √ r−1 = 0. a→∞ n→∞ n 2 Just as in [14] we can combine these lemmas together with Lemma 1.6 of that work to get the following analogue of Theorem 2.8 of [10]: SC (n) . Then, SC (n, a, a−r ) as n → ∞ (namely, n = j + q · d and q → ∞), lima→∞ limn→∞ σ(n, a) = 1. Theorem 35. Let n ≡ i mod d, as above, and let σ(n, a) = v(n,a) As in [14], the limit of √ , deﬁned in Lemma 33, converges to a certain nr−1 integral. To see this, we need some concepts from lattices. The volume of a lattice L = b0 + ba Z + · · · + bn Z ⊂ Rn is deﬁned to be the volume of the parallelepiped vol(L) = vol{x1 b1 + · · · + xn bn  0 ≤ xi ≤ 1, i = 1, . . . , n}. Now let D ⊆ Rn be a “nice” domain, f : D → R a continuous function, and L an ndimensional lattice with small volume. Then 1 f (x) ≈ f (x)dx, v R x∈L∩D and the two sides become equal as vol(S) → 0. To apply this theory to our case, we start with the lattice L0 = {(α2 , . . . , αr )  α1 v1 + · · · + αr vr ∈ (dZ) }, αi  where, as usual, α1 = − . Let vol(L0 ) = V. Next deﬁne L = {(α2 , . . . , αr )  τn (α) ∈ v + (dZ) }. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5168 ALLAN BERELE AND AMITAI REGEV Then vol(L) = 1 V Q(a,a−r ) where x1 = − 1 V and so limn→∞ v(n,a) √ r−1 n converges to the integral g(x1 , . . . , xr )Dθ (T (x1 , . . . , xr ))e− 2 T (x1 ,...,xr ) dx2 · · · dxr , 2 ··· I(C) = V √ r−1 n vi  ∞ xi . Taking the limit as a → ∞ gives the integral ∞ ··· 0 g(x1 , . . . , xr )Dθ (T (x1 , . . . , xr ))e− 2 T (x1 ,...,xr ) dx2 · · · dxr . 2 0 Then, analogously to Theorem 2.10 of [14], we have Theorem 36. Let n ≡ i mod d, as above. Then, as n → ∞, √ ()−i (θ2i )+r−1 ṽ0 − t2 −1 n 2 n . SC (n) I(C)K n 2 In particular, for these values of n, SC (n) is asymptotic to a constant times n to the power of an integer or halfinteger times a positive integer to the power of n. In order to get more information about the constant factor, we would need to know more about the integral I(C). We close this section with a conjecture. √ √ √ Conjecture. The integral I(C) is in Q[ a1 , . . . , an , 2π] for some integers a1 , . . . , an , under the hypothesis that g is a polynomial with rational coeﬃcients and that the vectors vi used to deﬁne T and θ have rational coordinates. 3. Last steps of the proof In the previous section, we estimated mλ dλ for λ in a simplicial cone C, assuming mλ to be polynomial. In general, cn (A) will be a sum of such and we now deal with the problem of taking the sum. Deﬁnition 37. Let w1 + C1 , . . . , wa + Ca be rational aﬃne simplicial cones as in Theorem 19. I.e., it is a maximal list such that each Cj contains 1k, as a basis element and mλ is almost polynomial on each Cj ; see Deﬁnition 6. By Theorem 19 those cones not containing 1k, in their basis don’t contribute to the asymptotics of cn (A) and so we may disregard them. Let d be a modulus such that mλ equals a polynomial on each wj +Cj ∩(i+(dN)k ), for i ∈ (dN)k . Denote this polynomial by m(i, j)(λ). As in Remark 29, this intersection of a cone and a lattice will contain partitions of n if and only if n ≡ i (mod d). Denote by S(n) the set of all i ∈ (Zd )k and 1 ≤ j ≤ a such that Par(n) ∩ (i + (dZ)k ) ∩ Cj = 0. Now, using this notation, {mλ dλ λ ∈ Par(n)∩(i+(dZ)k )∩Cj } = cn (A) (i,j)∈S(n) SCj (n−wj ). (i,j)∈S(n) Combining this with Theorem 36 now gives Lemma 38. Let d be as in Deﬁnition 37 so that mλ is a polynomial on each (i + (dZ)k ) ∩ Cj where 1 ≤ j ≤ a and i ∈ {0, . . . , d − 1}k . Let 0 ≤ m ≤ d − 1, and let n ≡ m (mod d) go to inﬁnity, i.e., n = qk + m and q → ∞. Then there are constants am and tm , the latter in 12 Z, such that, for these n, the codimension sequence of A satisﬁes cn (A) am ntm n . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES 5169 In order to prove that the tm are equal we need only assume that the cn are nondecreasing. This is true whenever the algebra A has 1, but it is also true under a weaker hypothesis, such as xA = 0 for every 0 = x ∈ A. Lemma 39. With notation as in the previous lemma, if xA = 0 for all 0 = x ∈ A, then all tm are equal. Proof. The map f (x1 , . . . , xn ) → f (x1 , . . . , xn )xn+1 takes nonidentities of A to nonidentities. It follows that the codimension sequence is nondecreasing. If not all tm were equal, there would be an m such that, say, tm > tm+1 . But this would imply that for all n ≡ m mod d, cn+1 (A) am+1 (n + 1)tm+1 n+1 −→ 0, cn (A) am ntm n contradicting the fact that the cn (A) are nondecreasing. In this case we have a1 nt n ≤ cn (A) ≤ a2 nt n for some 0 < a1 ≤ a2 . Drensky showed in [8] that the cocharacter sequence of A is Young derived, whenever A has a unit. We record that theorem by way of reminding the reader of the deﬁnition of Young derived sequences. Theorem 40. Let A be a p. i. algebra with unit. Then there exists a sequence ∞ ˆ i . Let each {ψ } such that for all n, χ (A) = χ(n−i) ⊗ψ of Sn characters n n=0 n λ λ χn (A) = mλ χ and each ψn = mλ χ . Then, using Young’s rule, the relation between χ(A) and {ψ} can be restated as mλ = {mν  ν ⊆ λ, λ/ν a horizontal strip}. Letting cn (A) be the degree of ψn we get (6) cn (A) = n n c i i i=0 We now need to verify that cn (A) has growth properties similar to those of cn (A) as described in Theorem 36. This will be Lemma 43. In order to prove this, we need to show that the sequence {ψn } has the same properties of χn (A) that we used in the proof of Theorem 36. First note that {ψn } is supported by partitions of height at most k, since the cocharacter sequence of A is. The next ingredient we need is Theorem 5. This theorem is a consequence of the fact that the Poincaré series for (the generic algebra of) A is a nice rational function, see [5], which is based on [3]. A nice rational function is one in which the numerator and denominator are polynomials with integer coeﬃcients and the denominator can be written as a product of terms of the form (1 − u), where u is a monic monomial. Section 2 of [5] shows that if a nice rational function is also a symmetric function, then the coeﬃcients λ of the Schur functions areλ as in Theorem 5. Let mλ denote the multiplicity of χ in {ψn }, so ψn = mλ χ . Deﬁne g(x1 , . . . , xk ) to be mλ Sλ (x1 , . . . , xk ), where Sλ (x1 , . . . , xk ) are Schur functions. Then Drensky proved that g is related to the Poincaré series of A via g(x1 , . . . , xk ) . Pk (A) = (1 − x1 ) · · · (1 − xk ) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5170 ALLAN BERELE AND AMITAI REGEV Since Pk (A) is a nice rational function, g(x) will be one also, and it follows from the techniques of [5] that the mλ satisfy the conclusion of Theorem 5. Lemma 41. Let mλ be as above. Then there exists an integer d and a partition of (R+ )k into regions R1 , . . . , Rm , each a ﬁnite intersection of rational aﬃne linear cones such that mλ is given by a polynomial on each Ra ∩ (i + (dZ)k ), where i ∈ {0, . . . , d − 1}k . We now prove these analogues of results that Giambruno and Zaicev proved about the cocharacter sequence of A; see Lemma 18 and the paragraph preceding it. Lemma 42. For e(A) = , the multiplicities mλ are zero for partitions with λ large, and, for every n, there are nonzero mλ with λ (a partition of n · ( − 1)) close to the rectangle of height (n−1 ). Proof. Theorem 40 says that mλ is the sum of all mν such that λ/ν is a horizontal strip. This condition is equivalent to λi+1 ≤ νi ≤ λi for all i. Now if ν = (ν1 , . . . , νk ) is a partition of height greater than or equal to with mν > 0, let λ be the partition obtained from ν by increasing ν+1 to ν . Then λ/ν is a horizontal strip and so mλ ≥ mν > 0. Since e(A) = , there is a constant K such that λ+1 ≤ K whenever mλ > 0 (see after Corollary 17). Then the above implies that, also, ν ≤ K if mν > 0. Likewise, since mλ is nonzero for some λ close to (n ), there must be a ν with mν nonzero and λ/ν a horizontal strip. Hence each νi − νi+1 is bounded, and by the ﬁrst part of the lemma, ν is bounded. Hence, ν must be close to n−1 . The results of sections 1 and 2 now apply to cn , yielding the following analogue of Lemma 38. Lemma 43. Let 0 ≤ m ≤ d − 1 and let n ≡ m (mod d) go to inﬁnity. Then there are constants bm and tm with tm ∈ 12 Z such that cn bm ntm ( − 1)n . Note that the cn (A) need not be increasing, and so the tm need not be equal, so we cannot invoke an analogue of Lemma 39 in this case. For example, if A is the inﬁnitedimensional Grassmann algebra, then cn (A) is 0 or 1 depending on whether n is odd or even, respectively. We now combine this lemma with equation (6) to prove that the am ntm are all equal. It follows from that equation that n d−1 n n cn (A) = cq = cq . q q 0≤q≤n q=0 m=0 q≡m(mod d) Lemma 44. The codimensions cn (A) are asymptotic to d−1 n bm · q tm ( − 1)q . q 0≤q≤n m=0 q≡m(mod d) Proof. Use Lemma 43. Given > 0 there exists q0 such that for q > q0 , (1 − )bm · q tm ( − 1)q ≤ cq (A) ≤ (1 + )bm · q tm ( − 1)q , License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse CODIMENSIONS OF P. I. ALGEBRAS SATISFYING CAPELLI IDENTITIES 5171 n t q where q ≡ m mod d, for any m. Now observe that q≤q0 q q ( − 1) and n q≤q0 q cq (A) are both bounded by a polynomial in n and so may be disregarded. n tm Lemma 45. For each 0 ≤ m ≤ d − 1, the sum ( − 1)q , summed q bm · q tm n over 0 ≤ q ≤ n, q ≡ m (mod d), is asymptotic to an , where a is a constant depending on bm , and tm . In particular the asymptotics do not depend on n mod d. d−1 Proof. Let ω be a primitive dth root of 1 and note that α=0 ω (q−m)α is zero if n tm q ≡ m (mod d) and is d if q ≡ m (mod d). So, the sum ( − 1)q , q bm · q summed over 0 ≤ q ≤ n, q ≡ m (mod d) can be rewritten as n n d−1 d−1 1 (q−m)α n 1 −mα n tm q ω ω bm q ( − 1) = bm q tm (ω α ( − 1))q . q q d q=0 α=0 d α=0 q=0 Now, using Lemma 1.1 of [2] we get that the inner sum is asymptotic to a constant times ntm (ω α ( − 1) + 1))n . The absolute value of (ω α ( − 1) + 1) is maximized when ω α ( − 1) is real and positive, which happens precisely when α = 0, and so these terms dominate the sum. The lemma now follows. Substituting Lemma 45 into Lemma 43 yields cn (A) d−1 am ntm n . m=0 If g is the maximum value of the tm and a = asymptotic to ang n and 2g is an integer. {bm tm = g}, then this sum is References [1] W. Beckner and A. Regev, Asymptotics and algebraicity of some generating functions, Adv. Math. 65 (1987), 1–15. MR893467 (88h:05008) [2] W. Beckner and A. 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