Coincidence in Poincaré polynomials of projective homogeneous varieties

Justin Martel

January 2011

Abstract

These notes express some basic facts concerning the Poincaré polynomials of projective homogeneous varieties (PHVs) as arising from their Bruhat-Tits cellular decomposition.

1 Introduction

1.1. For G a semisimple linear algebraic group over an algebraically closed field and P a parabolic subgroup, one obtains a projective homogeneous G-variety X = G/P. The remarkable Bruhat-Tits theorem says that such a homogeneous algebraic space has a cellular decomposition into affine cells 𝔸^{n} parametrized by the corresponding subgroup W_{P} of the Weyl group W (where, as usual, W is the Weyl group associated to the underlying root system Φ(G)). Moreover, each affine cell C(w) has dimension equal to the length ℓ(w). So Bruhat-Tits says the cellular structure of a PHV is controlled by its Weyl group (and length function).

The Poincaré polynomial of X is that polynomial whose k-th coefficient is the number of k-dimensional affine pieces in its cellular decomposition (equivalently, the number of elements w ∈ W_{P} with ℓ(w) = k). We say projective homogeneous varieties (PHVs) X, Y are coincident when their Poincaré polynomials coincide and express their coincidence as X ∼_{c} Y. Coincidence is the central theme of these notes; it’s how one asks two PHVs to ‘be made of the same stuff’.

The main result of this paper is the determination of all coincidences among simple PHVs (ie. X ≃ G/P, for G simple): this classification is split into our first and second main theorems, (4.3) and (4.5).

This paper divides as follows: after this introduction, the second section on lists and cyclotomic polynomials presents some terminology and lemmas that will be basic to these notes; this section is elementary, obvious, and can be quickly read through. Our third section consists of Chevalley’s Factorization, which is the main stimulus to these notes; through it alone does the arithmetic of Poincaré polynomials arising from PHVs become appreciable. In our fourth section on coincidences we establish our first and second main theorems through two separate chains of elementary and tedious reasoning. Finally, in the last section we explain how these coincidences are related to the problem of the classification of Grothendieck-Chow motives of PHVs.

Acknowledgements The author was supported in part by an NSERC Undergraduate Summer Research Award 168534 and NSERC Discovery 385795-2010, during which time he is grateful to Kirill Zainoulline for various discussions on the subject of the current paper.

2 Lists and Cyclotomic Polynomials

We introduce the necessary definitions and lemmas to be used repeatedly throughout these notes.

2.1 Definition. A list is a set in which multiplicities are taken into account. ie. while the set {a, a} may be identified with {a}, the list [a] is to be distinguished from [a, a]. Some authors call our lists multisets.

A finite list A with objects a_{i} shall be denoted [a_{1}, …, a_{n}]. For our purposes a list shall always refer to a finite list of integers which are > 1. The notions of membership, containment, join (∨) and meet (∧) are evident, as is the difference A − B of lists whenever B ⊂ A. In practise we shall write the join of lists A, A′ as the concatentation (or product) AA′. Given lists A, B, the formal quotient A/B will be called a rational list. We identify a list A with the rational list A/∅, where ∅ is the empty list. A rational list A/B is said to be reduced if A ∧ B is the empty list. Identifying the lists A/B and A ∨ A′/B ∨ A′, we find that any rational list A/B can be identified with a reduced rational list. The following definitions are convenient:

2.2 Definition. A list consisting of consecutive (even) integers shall be called (even) contiguous.

2.3 Definition. For an integer a > 1, its divisors list Div^{+}[a] is that list of integers d > 1 for which d|a. An arbitrary list A = [a_{1}, …, a_{k}] has divisors list Div^{+}(A) = ⋁_{i} Div^{+}[a_{i}]. We set Div^{+}∅ = ∅.

Recall the nth cyclotomic polynomials Φ_{n}(t) are those polynomials irreducible in ℤ[t] and recursively satisfying Φ_{0} = Φ_{1} = 1 and Π_{d|n}Φ_{d}(t) = t^{n} − 1 for n ≥ 1.

2.4 Lemma. Let A, B be lists of integers > 1 such that

∏_{a∈A} \frac{1 − t^{a}}{1 − t} = ∏_{b∈B} \frac{1 − t^{b}}{1 − t} .

Then A = B.

Proof. By definition the above equality becomes Π_{Div^{+}(A)}Φ_{d} = Π_{Div^{+}(B)}Φ_{d}. The irreducibility of Φ_{d} then says Div^{+}(A) = Div^{+}(B). Considering maximal elements in Div^{+}(A), one can retrieve A. Similarly, one retrieves B. Being retrieved from the same divisors list, A = B.

We easily find the following useful

2.5 Lemma. Let A, B, A′, B′ be lists of integers > 1. Then

∏_{Div^{+}(A)} Φ_{d}/ ∏_{Div^{+}(B)} Φ_{d} = ∏_{Div^{+}(A′)} Φ_{d}/ ∏_{Div^{+}(B′)} Φ_{d}

iff A/B, A′/B′ reduce to identical rational lists.

Proof. One has Div^{+}(A)Div^{+}(B′) = Div^{+}(AB′) = Div^{+}(A′B) iff AB′ = A′B (refer to proof of (2.4). Having cancelled common factors between A, B and A′, B′, one then sees that the above equality occurs iff A = A′ and B = B′, qed.

3 Chevalley’s Factorization

3.1. To be concerned with G only up to isogeny, we shall without qualms consider G as determined by its underlying root system Φ(G).

To a given PHV X ≃ G/P, we associate the formal quotient

\frac{𝒟_{1} ⋯ 𝒟_{r}}{𝒟′_{1} ⋯ 𝒟′_{s}} ,

where 𝒟_{1} ⋯ 𝒟_{r} is the decomposition of Φ(G) into irreducible root systems, and 𝒟′_{1} ⋯ 𝒟′_{s} likewise the decomposition of Φ(P). Such decompositions will be called root system words. Through the Chevalley Factorization we associate a list A_{i} of so-called basic invariant degrees to each irreducible root system type 𝒟_{i} (cf.[5]).

3.2. Let (W, S) be a Coxeter system (cf.[1], [5]). Then a length function ℓ(·) is defined relative to S on W. We define the Poincaré polynomial P (W, t) of W to be

P (W, t) = ∑_{w∈W} t^{ℓ(w)}.

This definition is independent of S cf.[5].

The following proposition–due to Chevalley–is for us crucial.

3.3 Proposition (Chevalley’s factorization). Let (W, S) be an irreducible Coxeter system. Then

P (W, t) = ∏_{i=1}^{n} \frac{1 − t^{d_{i}}}{1 − t} ,

where [d_{1}, …, d_{n}] is a list of basic invariant degrees assigned according to the following table:

\begin{matrix}Type & d_{1}, …, d_{n} \\ A_{n} & 2, 3, …, n + 1 \\ B_{n}, C_{n} & 2, 4, 6, …, 2n \\ D_{n} & 2, 4, 6, …, 2n − 2, n \\ E_{6} & 2, 5, 6, 8, 9, 12 \\ E_{7} & 2, 6, 8, 10, 12, 14, 18 \\ E_{8} & 2, 8, 12, 14, 18, 20, 24, 30 \\ F_{4} & 2, 6, 8, 12 \\ G_{2} & 2, 6 \end{matrix}

3.4 Remark. (i) For our purposes the Coxeter system (W, S) will arise as W = W (Φ(G), for G a semisimple linear algebraic group, and S the simple reflections corresponding to a base Δ of Φ(G) (or equivalently, a choice B of Borel subgroup of G). (ii) Henceforth we shall denote by a_{n} the degree list [2, 3, …, n] of the root system A_{n−1}.

3.5. Now suppose 𝒟_{1}, …, 𝒟_{r}, 𝒟′_{1}, …, 𝒟′_{s} are irreducible root system types with associated basic invariant degree lists A_{1}, …, A_{r}, A′_{1}, …, A′_{s}. Set A = ∨_{i}A_{i} and A′ = ∨_{j}A′_{j}. For a (PHV) X ≃ 𝒟_{1} ⋯ 𝒟_{r}/𝒟′_{1} ⋯ 𝒟′_{s}, we set

P (X, t) := P (Π𝒟_{i}/Π𝒟′_{j}, t) = ∏_{Div^{+}(A)−Div^{+}(A′)} Φ_{d}.

As immediate corollary to (2.5) we obtain

3.6 Proposition. PHVs are coincident iff their reduced rational degree lists coincide.

4 Coincidences

The question of ‘coincidence’ is the question of “to what extent do Poincaré polynomials determine their PHVs?’ We use the following terminology to state our main results:

4.1 Definition. Let 𝒟 be a root systems and 𝒟′ ⊂ 𝒟 a sub-root system. Then the 𝒟′-reduction of 𝒟 is the formal quotient 𝒟/𝒟′. If 𝒟, 𝒟′ have rational degree lists A, A′, then 𝒟/𝒟′ has degree list A/A′. Finally, if 𝒟′′ is another sub-root system ⊂ 𝒟 and disjoint from 𝒟′, then the 𝒟′′-reduction of 𝒟/𝒟′ is the formal quotient 𝒟/𝒟′𝒟′′ of 𝒟 by the sum 𝒟′𝒟′′.

4.2 Definition. We say simple PHVs 𝒟/𝒟′ and 𝒟′′/𝒟′′′ are type-distinct if type(𝒟) ≠ type(𝒟′′ ), and we say they are type-similar otherwise.

Our first main result classifies all type-distinct coincidences among simple PHVs:

4.3 Theorem (First Main Theorem). The only type-distinct coincidences between simple PHVs are those between:

• A_{2ℓ−1}/A_{2ℓ−2} and B_{ℓ}/B_{ℓ−1} (ℓ > 1);

• D_{ℓ+1}/A_{ℓ} and B_{ℓ}/A_{ℓ−1} (ℓ > 1);

• G_{2}/A_{1} and A_{5}/A_{4}, B_{3}/B_{2}.

For the expression of our second main result we introduce the following device:

4.4 Definition. A swindle is the substitution of the coincidence B_{m}A_{2m−2} ∼_{c} B_{m−1}A_{2m−1} (for m ≥ 1) within a root system word 𝒟𝒟′ · · ·.

The swindle formalizes the fact that all type-similar coincidences are consequences of the same ‘trick’. On degree lists the swindle is just the relation na_{n−1} = a_{n}

We may now present our second main theorem classifying all type-similar coincidences among simple PHVs:

4.5 Theorem (Second Main Theorem). A type-similar coincidence between simple PHVs is either a coincidence B_{k}/𝒟 ∼_{c} B_{k}/𝒟′, where 𝒟 ∼_{c} 𝒟′ with 𝒟, 𝒟′ of the form B ∏ A; or a coincidence D_{k}/𝒟 ∼_{c} D_{k}/𝒟′, where 𝒟 ∼_{c} 𝒟′ with 𝒟, 𝒟′ of the form D ∏ A. Moreover, all semisimple coincidences B ∏ A ∼_{c} B ∏ A and D ∏ A ∼_{c} D ∏ A are generated by swindles.

In (4.6) is expressed an observation which organizes our casewise proofs of (4.3), (4.5):

4.6 Lemma. Let 𝒟 be a connected Dynkin diagram and 𝒟′ a connected subdiagram. Then either 𝒟′ has type A_{k}, with k ≤ rank(𝒟), or 𝒟′ is the unique subdiagram having rank ≤ rank(𝒟) and type(𝒟′) = type(𝒟).

Proof. By inspection.

Using (4.1) we may rephrase (4.6): a connected Dynkin diagram 𝒟 admits only A.-type reductions after at most one 𝒟-type reduction.

Proof of First Main Theorem. A coincidence between irreducible types comes in three ‘flavours’: classical-classical, classical-exceptional, and exceptionalexceptional. Correspondingly we split the proofs of our first main theorem (4.3) ‘flavourwise’ into some lemmas and sublemmas.

4.7 Sublemma. The only coincidences produced by A-type reductions between pairwise type-distinct PHVs among A, B/B_{k}, and D/D_{k′} (with k ≥ 2, k′ ≥ 4) is that between A_{2ℓ−1}/A_{2ℓ−2} and B_{ℓ}/B_{ℓ−1}, (ℓ > 1).

Proof. Any quotient A_{ℓ}/A_{k} has a degree list containing an odd integer unless ℓ is odd and k = ℓ−1. Since a nontrivial quotient B/B has a degree list consisting of even integers, coincidence only occurs when k = ℓ − 1. This produces the coincident pair A_{2ℓ−1}/A_{2ℓ−2}, B_{ℓ}/B_{ℓ−1}.

A coincidence between reductions of B/B and D_{ℓ}/D_{k} must see ℓ even. If 2k ≤ ℓ, then the even ℓ occurs twice in the numerator of D/D’s reduced rational degree list. By (3.6) D/D cannot coincide with any A-type reductions of B/B (since numerator of reduced rational degree list is without any repetitions). When 2k > ℓ, B/B is coincident with D_{ℓ}/D_{k} through A-type reductions only if B/B = B_{ℓ−1}/B_{k−1} (since we need maximal elements of the numerators in the reduced rational degree lists to coincide). But no further A-type reduction can produce the necessary k on the denominator since B_{ℓ−1}/B_{k−1} does not have an A_{k−1}-type connected component (because ℓ < 2k). Consequently no A-type reductions of B/B, D/D are coincident.

Finally, consider A-type reductions of A and D_{ℓ}/D_{k} (ℓ ≥ 4). If ℓ is even, then coincidence would see no odd elements in the numerator of any reductions of A, which means A/A = A_{2ℓ−3}/A_{2ℓ−4}. But then A/A has numerator [2ℓ-2] and admits no further A-type reduction, and hence not coincident with any A-type reduction of D/D. If ℓ is odd, a coincidence between A-type reductions of A, D/D would require the (reduced) numerator of D/D to be contiguous. However for ℓ ≥ 4 this does not occur.

4.8 Sublemma. The only type-distinct coincidences produced by A-type reductions between A, B, and D is that between D_{ℓ+1}/A_{ℓ} and B_{ℓ}/A_{ℓ−1}.

Proof. The reduced rational list of any A-type reduced B has an even-contiguous numerator. So by (3.6) it’s clear that no A-type reductions generates a coincidence between A and B. Likewise we see that no A-type reductions of A and D_{ℓ} are coincident for ℓ ≥ 4. Furthermore it’s clear by (3.6) that for k ≤ k′ no A-type reductions render either B_{k} and B_{k′} or D_{k} and D_{k′} coincident. Finally any A-type reduction of D_{ℓ+1} is coincident with an A-type reduction of B only if we first reduce D_{ℓ+1} to D_{ℓ+1}/A_{ℓ} by (3.6) and after which we may reduce no more; here we find the coincidence with B_{ℓ}/A_{ℓ−1}.

4.9 Lemma (Classical-Classical (4.3)).

Proof. All together (4.7)–(4.8) yield (4.3) in the classical-classical case.

4.10 Lemma (Classical-Exceptional (4.3)).

Proof. The exceptional types have reduced rational degree lists with numerators which are neither contiguous nor even-contiguous. Their reductions to quotients 𝒟/𝒟 have contiguous or even-contiguous numerator degree lists only in the case G_{2}/A_{1}.

4.11 Lemma (Exceptional-Exceptional (4.3)).

Proof. The basic invariant degree lists of the exceptional irreducible types have distinct max elements, except in the case of E_{6} and F_{4}. But E_{6} has an odd basic invariant degree, while F_{4} cannot.

Together (4.7)-(4.12) establish our First Main Theorem (4.3).

Proof of Second Main Theorem. We begin with some observations:

4.12 Sublemma. The simple PHVs 𝒟/𝒟′ and 𝒟′′/𝒟′′′ are type-similar coincident only if 𝒟 = 𝒟′′ and 𝒟′ ∼_{c} 𝒟′′′.

Proof. For type-similar irreducible root systems 𝒟, 𝒟′′, one observes the numerators of their reduced rational degree lists have distinct maximal elements unless 𝒟 = 𝒟′′. Now by (3.6) they must have been reduced by coincident PHVs.

4.13 Sublemma. There are no type-similar coincidence among reductions of the simple PHVs of type A, E, F, or G.

Proof. By (4.12) this follows by realizing there are no type-similar semisimple coincidences between root systems of the form ∏ A, E ∏ A, F ∏ A, or G ∏ A.

It is convenient now to introduce the following

4.14 Definition. We define the sup.rank of a product 𝒟_{1}𝒟_{2}𝒟_{3} ⋯ of irreducible root systems 𝒟_{i} to be sup_{i}rank(𝒟_{i}) .

Together with (4.12) and (4.13) the following lemma establishes (4.5).

4.15 Lemma. All semisimple coincidences B ∏ A ∼_{c} B ∏ A and D ∏ A ∼_{c} D ∏ A are generated by swindles.

Proof. We proceed recursively by reducing via swindles the sup.ranks of coincident semisimple pairs. This establishes that indeed all such coincidences are generated by swindles.

Proceeding first with the B-type case, we consider a coincidence

B_{k−1} ∏_{I} A_{i−1} ∼_{c} B_{m} ∏_{J} A_{j−1}.  (∗_{0})

With respect to degree lists (∗_{0}) becomes

∏_{I} a_{i} = [2k, …, 2m]_{even} ∏_{J} a_{j}.  (∗_{0})

After cancelling common factors, the maximal degree M occurring in (∗_{0}) is seen to be 2m (hence the sup.rank is finite). Consequently there is a unique factor of a_{M} = a_{2m} on the left-hand side of (∗_{0}), all of which may be rewritten as

a_{M} ∏_{≤M−1} a_{i} = [2k, …, M]_{even} ∏_{≤M−1} a_{j}  (∗_{1})

There are now two cases: If (a) 2k = M, then after the swindle a_{M} = Ma_{M−1} (∗_{1}) becomes

a_{M_{1}} ∏_{≤M−1} a_{i} = ∏_{≤M−1} a_{j}.

By (4.12), then I_{≤M−1}.M = J_{≤M−1}.

Otherwise, if (b) 2k < M, then 2k < M −1. As left-hand side of (∗_{1}) contains a M − 1 factor and [2k, …, M]_{even} does not, we find that after having cancelled common factors there is a unique a_{M−1} factor on the right-hand side. The swindle a_{M} = Ma_{M−1} then reduces (∗_{1}) to a coincidence with sup.rank < M.

Clearly, in either of the cases we reduce the sup.ranks of the coincidence via a swindle.

We proceed similarly in the D-type case, by reducing the sup.rank through the successive use of swindles. Except for these details (which the reader may provide for themselves), this completes our proof.

As corollary we’ve established our second main theorem.

Coincidence in Semisimple Case The question of coincidence among semisimple PHVs is not entirely resolved by Lemma 3.6. In addition to the coincidences generated by our main theorems there are many non-interesting coincidences which arise from products of A_{k}/A_{k−1} (a PHV with rational degree list [k + 1]). For instance, it is immediate that the exceptional root system types E, F, G are all coincident with products of the classical types A/A. One also has the obvious coincidence between E_{7}A_{29}/A_{28} and E_{8}A_{5}A_{9}/A_{4}A_{8}. Supposing one calls a coincidence interesting if not generated by PHVs of the form A_{k}/A_{k−1}, there is the obvious question: are all interesting coincidences generated by those coincidences of (4.3) and (4.5)?

While (3.6) does not resolve the question of semisimple coincidence, it does permit the following obvious conclusions in the exceptional-exceptional case:

4.16 Proposition. There are no coincidences between PHVs generated by the root system types E_{6}, E_{7}, E_{8} and those generated by the root system types F_{4}, G_{2}.

4.17 Proposition. There are no coincidences between PHVs generated by the root system types E_{i}, F_{4}, G_{2} and E_{j}, F_{4}, G_{2} for i ≠ j (unless of course there is no E-type factor, ie. the PHV was generated by F_{4}, G_{2}).

5 Application to motives

In the present section we relate our main theorems (4.3) and (4.5) with the problem of classification of Grothendieck-Chow motives of PHV. ^{1} Note that this question has been intensively studied during the last few years (see [3],[4], [7],[9],[10]).

Following (2.2) of [10], the Chow motive ℳ (X) of the PHV X = G/P satisfies an isomorphism

ℳ (X) ≃ ⨁_{i=0}^{dimX} ℤ(i)^{⊕a_{i}(X)},

1For an introduction to Chow motives, see [8]

where Z(i) denotes the Tate motive shifted by i and

∑_{i≥0} a_{i}(X)t^{i}

is the Poincare polynomial of X. By (2.3) of [10], motives are the same iff they have the same Poincaré polynomials. In view of our main theorems (4.3) and (4.5) we immediately obtain the

5.1 Corollary. Let X, Y be simple PHVs. Then X, Y have isomorphic motives iff X, Y belong to the coincidence classes given in our main theorems.

5.2 Remark. Observe that the coincidence of Poincare polynomials (and hence, their motives) doesn’t imply that the respective varieties are isomorphic. For instance, the variety from (4.3) of type A_{2l−1}/A_{2l−2} is a projective space of dimension 2l − 1 and the variety of type B_{l}/B_{l−1} is the projective quadric of the same dimension. Both varieties have the same Poincare polynomial \frac{1−t^{2l}}{1−t} but are not isomorphic.

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