Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields
In this paper we explore a relationship between the topology of the complex hyperplane complements in type B/C and the combinatorics of certain spaces of degree– polynomials over a finite field . This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras , and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over with nonzero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as –algebras finitely generated in –degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.
- 1 Introduction
- 2 Foundations: -modules and character polynomials
- 3 Convergence and nonconvergence results
- 4 Statistics on squarefree polynomials and hyperplane arrangements of type
Hyperplane complements and statistics on
Define complex hyperplane complements
The Grothendieck–Lefschetz trace formula in -adic cohomology and results of Lehrer [Leh92] and Kim [Kim94] imply an amazing relationship between the complex cohomology of these hyperplane complements and point counts for certain spaces of polynomials over a finite field . Church, Ellenberg, and Farb [CEF14] describe this relationship for the hyperplane complement , and use it to relate stability results for the cohomology groups as symmetric group representations to stability results for polynomial statistics on the set of monic squarefree degree– polynomials in , in the limit as tends to infinity [CEF14, Theorem 3.7 and Theorem 1].
One goal of this paper is to prove the analogues of Church, Ellenberg, and Farb’s results in type B/C. We investigate the relationship between the complex cohomology of the hyperplane complement and statistics on the set of monic squarefree degree– polynomials in with nonzero constant term. The space has an action of the hyperoctahedral group (Definition 2.1), and the following theorem – a twisted version of the Grothendieck–Lefschetz trace formula, specialized to the scheme – relates the structure of the cohomology groups as –representations to point-counts on .
Theorem 4.1 (A point-counting formula for ). Let be an integral power of a prime number and let be a class function on . Then for each we have
Section 4.2 describes how to interpret a class function as a function on polynomials in , by considering the action of the Frobenius morphism on an associated set .
In previous work the second author showed that there is a sense in which the –representations stabilize as grows [Wil15, Theorem 5.8], using a description of these cohomology groups due to Orlik and Solomon [OS80]. In Theorem 4.9 we use this representation stability result for the complex cohomology groups and a combinatorial result, Theorem 3.8, to prove asymptotic stability for certain polynomial statistics on the polynomials .
Theorem 4.9 (Stability for polynomial statistics on ). Let be an integral power of an odd prime. For any hyperoctahedral character polynomial the normalized statistic converges to a limit as . In fact
and the series in the right hand side converges.
The functions and are signed-cycle-counting class functions on (Definition 2.9). Given a polynomial in these functions, the values encode data on the irreducible factors of the polynomial and the square roots of their zeroes in , as we summarize on the upcoming page. Precise descriptions of character polynomials are given as class functions in Section 2.2 and as statistics on in Section 4.2.
Convergence and nonconvergence results: algebras in degree
To prove Theorem 4.9 we study the algebraic structure of the cohomology rings of the family . The proof is combinatorial, and the approach was motivated by our general interest in understanding what are the combinatorial features of the generators and relations of a sequence of algebras that allow for convergence results of the form of [CEF14, Theorem 1] and our Theorem 4.9. The cohomology rings of both the families and have the structure of –algebras finitely generated in -degree , and a second goal of this paper is to investigate the asymptotic properties implied by this structure.
Let generically denote either the family of symmetric groups or the family of hyperoctahedral groups . In Section 2, we review the definitions of –modules and –algebras, and the algebraic framework they provide for studying sequences of –representations. In earlier work [JRW17], the authors prove that any –algebra finitely generated in –degree 0 or 1 satisfies the desired convergence condition; this result is stated in Theorem 3.1. In particular, consider the sequence of commutative or graded-commutative algebras with an action of by permuting the subscripts and possibly negating the variables. Then for any integer and any type character polynomial , the following limit converges:
This same convergence result holds if we take the sequence of (graded)-commutative algebras in finitely many collections of variables
with an action of by permuting the subscripts and possibly negating the variables.
A natural question, then, is whether convergence is again automatic for for –algebras finitely generated in -degree . Specifically, suppose is the sequence of polynomial algebras
with an action of by permuting the indices and possibly negating the variables. We may assume the variables satisfy any one of , or . Then the question is, will this sequence of algebras be convergent in the sense of Equation (1)?
In Theorem 3.4, we show that this is not the case; it fails even for the trivial characters .
Theorem 3.4. (Nonconvergence for symmetric –algebras generated in –degree 2). Let be a subfield of , and an integer. Let be a graded –module over supported in positive grades containing a free –module on a representation of . Let be an –algebra containing the free symmetric algebra on . Then there exist character polynomials for which the following series does not converge:
Theorem 3.4 shows that the convergent results of [CEF14, Theorem 1] and Theorem 4.9 do not follow formally from the combinatorics of the generators of these algebras. In Proposition 3.6 we give a new proof of [CEF14, Theorem 1], and we extend this strategy to prove Theorem 4.9. These proofs examine how combinatorial aspects of the relations of these algebras can drive their convergence behaviour.
Statistics on squarefree polynomials and hyperplane complements of type B/C
Recall that denotes the set of monic squarefree polynomials in with nonzero constant term. It is natural to ask about the distribution of irreducible degree– factors of these polynomials. Roots of any degree– irreducible factor lie in , and it is a more subtle question to ask about the nature of the square roots of these roots. These data are encoded by the action of the Frobenius morphism on the set , and allows us to interpret the hyperoctahedral signed-cycle-counting class functions and (Definition 2.9) as the following functions on .
Details are given in Sections 4.1 and 4.2. With these definitions, Theorem 4.9 then states that for an odd prime power, given any polynomial in the class functions and , the limits in Formula (8) exist and are equal. It implies in particular that the expected value of on converges.
Corollary 4.10. (Stability for the expected value of polynomial statistics on ). Let be an integral power of an odd prime. For any polynomial the expected value of on converges as tends to infinity, and its limit is
To illustrate these results, in Section 4.4 we evaluate Formula (8) in some specific examples. We first review a result of Douglass [Dou92, Formula (1.1)] on a decomposition of the cohomology groups as –representations. Then, using results of Brieskorn [Bri73, Théorème 7] and Douglass, we compute the stable inner products on the right-hand side of Formula 8 for the character polynomials , (Lemma 4.15) and (Lemma 4.16). We then show that, from these computations, we can deduce the following (well-known) statistics on the polynomials over :
(Proposition 4.14). The number of degree– monic squarefree polynomials in with is
(Proposition 4.17). The expected number of linear factors in a random degree– monic squarefree polynomial in with converges to
(Proposition 4.18). The expected number of roots that are quadratic residues over for a random degree– monic squarefree polynomial in with converges to
These three statistics can be computed directly by combinatorial methods on , for example, they are a special case of a computation done by Alegre, Juarez, and Pajela [AJP15, Theorem 20] using the generating function techniques of Fulman [Ful16]. We have taken this unconventional approach to computing these statistics in Section 4.4 in order to showcase the extraordinary result of Grothendieck, Lehrer, Kim, and others that we can extract combinatorial data on polynomials over finite fields from topological properties of complex hyperplane complements, and vice versa.
New work of Matei
After posting a preprint of this paper, we learned of recent work of Vlad Matei [Mat]. Matei independently proves Theorem 4.1 [Theorem 4, Matei], also using methods inspired by Church, Ellenberg, and Farb’s proof of their Theorem 3.7 [CEF14]. He combines this result with a description of the cohomology groups due to Henderson [Hen06] to prove a theorem [Mat, Theorem 1] on the distribution of monic degree– polynomials of the form
New work of Casto
While finishing this paper, we learned of new work of Casto [Cas17] establishing a general asymptotic stability result for statistics on the th roots of the zeroes of polynomials over , and the associated orbit configuration spaces. Casto [Cas17] generalizes techniques of Farb and Wolfson [FW16] to FI–modules. He proves asymptotic stability by topological methods, showing that orbit configuration spaces have convergent cohomology [Cas17, Theorem 1.3 & Section 3]. His results [Cas17, Theorem 1.4 and 1.5] recover Theorems 4.1 and 4.9 as the special case .
Other related work
Convergence of the left hand side of Formula (8) in Theorem 4.9 above could also be proven using the generating function techniques that have been employed in recent work such as Fulman [Ful16] and Chen [Che16]; see [Che16, Corollay 4 (b)]. These methods can also be used to obtain additional results on the stable values, for example, Chen [Che16] shows that the stable Betti numbers of unordered configuration spaces are quasipolynomial and satisfy linear recurrence relations. It should be possible to adapt these techniques for the type B/C analogues.
The use of point-counting over finite fields to obtain topological information about complex reflection group arrangements has appeared in the work of Kisin and Lehrer [KL02]. See also Chen [Che16] and Fulman–Jiménez Rolland–Wilson [FJRW17] for related work on the spaces of maximal tori in Lie groups of types A and B/ C, respectively.
Farb–Wolfson [FW16, Theorem B] prove étale homological and representation stability for the cohomology of configuration spaces of points on smooth varieties. They use the Grothendieck–Leftschetz formula to relate this result to point counts over finite fields, and they prove asymptotic stability for these point counts by establishing subexponential bounds on the growth of the unstable cohomology of those spaces [FW16, Theorems A and C].
Gadish [Gad17, Theorem A] derives a version of the Grothendieck trace formula for ramified covers that is suited to applications in representation stability. With this formula he performs explicit computations for the Vieta cover of the variety of polynomials, and describes factorization statistics of polynomials over finite fields [Gad17, Section 3].
We are grateful to Tom Church, Weiyan Chen, Kevin Casto, Sean Howe, and László Lovász for helpful conversations. We thank Benson Farb for suggesting this project to us.
2 Foundations: -modules and character polynomials
Church–Ellenberg–Farb [CEF15] introduced the theory of FI–modules as an algebraic framework for studying sequences of symmetric group representations. Their results were generalized by the second author to sequences of representations of the classical Weyl groups in type B/C and D [Wil14, Wil15]. In this section we summarize the relevant terminology and results.
Let denote the Weyl group in type , which we call the the hyperoctahedral group or signed permutation group. We define as the subgroup of permutations on the set given by
There is a natural surjection by forgetting signs of the elements of .
Throughout the paper we will use to generically denote either Weyl group family: the symmetric groups in type A or the signed permutation groups in type B/C.
2.1 The theory of –modules
(The category ; –modules.) Let be one of the families or . To each of these families we associate a category , defined as follows. Its objects are finite sets
where is shorthand for for any integer . The morphisms are generated by its endomorphisms and the canonical inclusions . An –module over a commutative, unital ring is a functor from to the category of –modules; its image is sequence of –representations with actions of the morphisms.
In type A, the category is equivalent to the category FI studied by Church–Ellenberg–Farb [CEF15]. In type we denote the category by .
(Graded –modules; graded –algebras.) A graded –module over a ring is a functor from to the category of graded –modules. A graded –algebra over a ring is a functor from to the category of graded –algebras. We will refer to as the graded–degree and as the –degree of the –module .
(Finite generation; degree of generation; finite type.) An –module is generated (as a module) by elements if is the smallest –submodule of containing . We call an (additive) generating set for . A graded –algebra is generated (as an algebra) by elements if is the smallest –subalgebra containing the elements . We call an (algebra) generating set for . An –module (respectively, a graded –algebra )is finitely generated as a module (respectively, as an algebra) if it has a finite generating set, and is generated in degree if it is generated by . A graded –module or algebra has finite type if each graded piece is finitely generated as an –module.
(The representable –modules .) Following the notation introduced by Church–Ellenberg–Farb [CEF15], we write
for the represented –module over generated in degree . An orbit-stabilizer argument shows that for each there is an isomorphism of –representations
We denote represented FI–modules by and represented –modules by .
(The free -modules .) Fix a nonnegative integer and a –representation . Define the –module
using the right action of on We call the free –module on . As –representations,
Over fields of characteristic zero the decomposition of into irreducible representations is described by the branching rules. If is an irreducible –representation associated to a partition of , then we may write in place of . Similarly, for the irreducible –representation associated to double partition we write for .
2.2 The representation theory of
Just as the irreducible complex representations of are in natural bijection with partitions of , the irreducible complex representations of are in bijection with double partitions of , that is, ordered pairs of partitions such that . We denote the corresponding representation by . See (for example) Geck–Pfeiffer [GP00] for a detailed development of the representation theory of these Weyl groups, or (for example) Wilson [Wil14, Section 2.1] for a summary suited to present purposes. Each complex irreducible representation of these groups is defined over the rational numbers, or any subfield of . For the remainder of this section we will take coefficients to be in a subfield of .
(Character polynomials for ). For each , MacDonald [Mac79, I.7 Example 14] defined a class function that take a permutation to the number of –cycles in its cycle type. A character polynomial for with coefficients in a ring is a polynomial in these class functions . The degree of a character polynomial is defined by assigning for . A character polynomial defines a class function on for all ; we denote its restriction to by .
(Signed cycle type of a signed permutation). The conjugacy classes of are classified by signed cycle type as follows. For , a signed permutation in is called an –cycle if it projects to an –cycle in . A positive -cycle is an –cycle that negates an even number of letters in ; these are elements of the form
when expressed in cycle notation as a permutation on A positive –cycle has order . A negative –cycle is a signed permutation that negates an odd number of letters; these have the form
A negative –cycle has order , and its power is a product of transpositions
Young [You30] proved that signed permutations factor uniquely as a product of positive and negative cycles, and two signed permutations are conjugate if and only if they have the same signed cycle type.
(Character polynomials for ). Given , let be the number of positive –cycles in its signed cycle type, and the number of negative –cycles. A hyperoctahedral character polynomial with coefficients in a ring is a polynomial . Each character polynomial defines a class function on ; we write to mean its restriction to . We define the degree of a character polynomial by setting .
(The inner product ). For a finite group , we write for the standard inner product on the -valued class functions on . By abuse of notation we may write either class functions or –representations in its argument; a representations should be taken to represent the corresponding character.
2.3 Finitely generated –modules are representation stable
A central result of the work of Church–Ellenberg–Farb [CEF15] and Wilson [Wil14, Wil15] are constraints on the structure of finitely generated –modules. Specifically, a finitely generated –module over characteristic zero is uniformly representation stable in the sense of Church–Farb [CF13, Definition 2.3], and its characters are eventually polynomial, in the sense of the following theorem.
(Constraints on finitely generated –modules). Let be an –module over a subfield of which is finitely generated in degree .
2.4 Existing asymptotic results
Below is a summary of results on asymptotics of character polynomials, which we will use to prove the results in Section 3.
(Asymptotic equivalence; asymptotic bounds; Big and little O notation). Let be functions. We say that is asymptotically equivalent to and write if
We say is asymptotically dominated by and write if
The function is order or asymptotically bounded above by if
equivalently, if for some constant and all sufficiently large. The function is order if
(Stability for characters of finitely generated –algebras). [JRW17, Lemma 3.3]
Let be a subfield of , and let represent one of the families or . Suppose that is an associative –algebra over that is generated as an –algebra by finitely many elements of positive graded-degree. Then for each and any character polynomial , the following limit exists:
For each there is a function that independent of and order such that
For each character polynomial there is a function that is independent of and order such that
3 Convergence and nonconvergence results
In recent work [JRW17] the authors prove that –algebras finitely generated in –degree zero or one satisfy a certain convergence result; the precise statement is given in Theorem 3.1 below. We use this theorem to investigate the structure of coinvariant algebras of type A and type B/C. Our results can be interpreted as asymptotic convergence results for ’polynomial statistics’ on maximal tori in the corresponding matrix groups over finite fields; see [CEF14, Theorem 5.6] and [JRW17, Theorem 4.3].
Many naturally arising –algebras, however, are finitely generated by elements in -degree two or higher, and so do not fall within the scope of Theorem 3.1. These include the main examples of Sections 3 and 4, the cohomology algebras of the hyperplane complements associated to braid arrangements in type A and B/C. Again we are faced with the question of whether these –algebras are convergent in the sense of Theorem 3.1: in Section 4 we describe how this result corresponds to convergence results for certain statistics on polynomials over finite fields.
An optimistic, if naive, conjecture is that all finitely generated –algebras satisfy the form of convergence of Theorem 3.1. Unfortunately, in Section 3.1, we show that this is not the case in general. In Sections 3.2 and 3.3, however, we develop combinatorial strategies for proving convergence in our specific examples, the cohomology of the hyperplane complements. Proposition 3.6 gives a new proof of convergence in type A, and Theorem 3.8 establishes convergence in type B/C.
3.1 Failure of convergence for free –algebras generated in degree 2
In earlier work [JRW17, Theorem 3.5] the authors prove a convergence result for free (graded)-commutative –algebras on generators in –degrees zero or one.
(Criteria for convergent –algebras). [JRW17, Theorem 3.5]. Let be one of the families or , and let be a subfield of . For nonnegative integers , define a graded –module over
with positive gradings. Let be the commutative, exterior, or graded-commutative algebra generated by . Let be any sequence of graded -representations such that is a subrepresentation of . Then for any character polynomial and integer , the following sequence converges asbolutely:
In this next proposition we show that if we replace by
which are in some sense the next ’smallest’ projective –modules, then the convergence result no longer holds.
(Nonconvergence for symmetric –algebras on or ). Let be a subfield of , and an integer. Let be the free –module on the trivial –representation , concretely, this is in type or in type B/C. Assume is graded by some positive grading. Let be an –algebra containing the symmetric algebra on . Then there exist character polynomials for which the following series does not converge:
Suppose that is concentrated in graded–degree . It is enough to consider the character polynomial and the case that is equal to the symmetric –algebra freely generated by . The inner product is the dimension of the –invariant subspace of ; this value is nonnegative and could only grow if were larger.
The graded –module is generated by a single generator in –degree and graded-degree . Then is finitely generated by as an –algebra, and Lemma 2.14 guarantees that the limit exists.
We first consider the FI–algebra in type . The FI–module has bases
Then is spanned by monomials in commuting variables , and we may index these monomials by graphs on vertices labelled by and an edge for each variable that occurs. The –orbits of these monomials are indexed by unlabelled graphs on edges.
For sufficiently large, say , the number of these unlabelled graphs is independent of ; we can simply consider those graphs with edges and without isolated vertices. We will, in fact, only need to consider monomials without repeated variables, so the corresponding graphs have no multi-edges.
Let denote the number of graphs on edges without loops, multi-edges, or isolated vertices. Lupanov [Lup59] showed that
It follows that the series does not converge for any or . For ,
and we conclude that the series
does not converge for the constant class function .
The result in type B/C follows the same argument; again
with the action of on the generators factoring through the quotient . Thus the –invariants are isomorphic to the –invariants above, and the same graph-theoretic bounds imply that the series
will not converge in the case that . ∎
(Nonconvergence for exterior –algebras on or ). We expect that the nonconvergence result of Proposition 3.2 would also hold for the free exterior –algebras on or . Heuristically, this is because there is an asymptotic sense in which graphs generically have trivial automorphism groups – and graphs with trivial automorphism groups will correspond to orbits of anticommutative monomials that are nonzero in the –quotient.
We can leverage the results of Proposition 3.2 to prove that convergence fails for any commutative –algebra generated by a free –module on a –representation.
(Nonconvergence for symmetric –algebras generated in –degree 2). Let be a subfield of , and an integer. Let be a graded –module over supported in positive grades containing a free –module on a representation of . Let be an –algebra containing the free symmetric algebra on . Then there exist character polynomials for which the following series does not converge:
In light of Proposition 3.2, it suffices to consider the cases that is equal to if it is type A, or equal to one of
if it is type B/C.
If is any of the –modules , , , or , then is spanned by elements of the form with , , and , where acts by permuting the indices and negating the variables. Specifically,
The squares span a copy of in type or in type B/C. Since these squares are algebraically independent, contains the polynomial algebra they generate, and the result follows from Proposition 3.2.
It remains to address the –module . In –degree this module is spanned by variables
and an action of by permuting the indices. But then the elements are algebraically independent and span a copy of , so again convergence must fail by Proposition 3.2. ∎
Theorem 3.4 suggests the following problem.
(Convergence criteria for –algebra with generators in –degree ). Let be a finitely generated –algebra with generators in –degree . Given a presentation for as an –algebra, find combinatorial criteria on the relations that guarantee convergence in the sense of Theorem 3.1.
3.2 The braid arrangement: an example of a convergent algebra
Church–Ellenberg–Farb [CEF14, Proposition 4.2] showed that convergence in the sense of Theorem 3.1 does hold for the anticommutative FI–algebra generated by and subject to the “Arnold relations”. Our Theorem 3.4 shows that finite generation as an FI–algebra in FI–degree 2 is not enough in general to ensure this form of convergence, and that this result should be viewed as a feature of the relations that define this FI–algebra.
Let be a subfield of . Consider the Arnold algebra over
which Arnold [Arn69] proved to be isomorphic to the cohomology ring of the pure braid group. This algebra is, equivalently, the cohomology of the complex hyperplane complement associated to the symmetric group’s reflecting hyperplanes. This hyperplane arrangement is sometimes called the braid arrangement. Church–Ellenberg–Farb show that has the structure of an FI-algebra finitely generated by [CEF15, Example 5.1.3].
Proposition 3.6 below gives a simplified proof of a result of Church–Ellenberg–Farb [CEF14, Proposition 4.2]. These authors established the result using an explicit decomposition of each graded piece as an –representation, a decomposition proven in significant work of Lehrer–Solomon [LS86, Theorem 4.5]. Our proof will serve as a warm-up to proving Theorem 3.8, the analogous result in Type B/C.
(Convergence for the braid arrangement, [CEF14, Proposition 4.2]). Let denote the th-graded piece of the Arnold’s algebra. For and any character polynomial , the sum
Our approach to the proofs of Proposition 3.6 and Theorem 3.8 is driven by our interest in Problem 3.5, and suggests the following partial solution: we can establish convergence in these cases because we can reduce the spanning sets for the graded pieces from a set of general labelled graphs to a set of decorated labelled trees. Our proof will use the following lemma on forest enumeration.
(Forest enumeration). Let denote the number of unlabelled unrooted forests on edges. Let denote the number of unlabelled forests of rooted trees with the property that roots occur only at leaves. Then and are order .
Otter [Ott48] proved that the number of unlabelled trees on vertices and edges is asymptotically
The number of rooted unlabelled trees on vertices and edges is asymptotically