BUNDLES, COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS

ALEJANDRO ADEM^{∗} AND ZINOVY REICHSTEIN^{∗∗}

Abstract. The cohomology of the classifying space BU(n) of the unitary group can be identified with the the ring of symmetric polynomials on n variables by restricting to the cohomology of BT, where T ⊂ U(n) is a maximal torus. In this paper we explore the situation where BT = (ℂP^{∞})^{n} is replaced by a product of finite dimensional projective spaces (ℂP^{d})^{n}, fitting into an associated bundle

U(n) ×_{T} (𝕊^{2d+1})^{n} → (ℂP^{d})^{n} → BU(n).

We establish a purely algebraic version of this problem by exhibiting an explicit system of generators for the ideal of truncated symmetric polynomials. We use this algebraic result to give a precise descriptions of the kernel of the homomorphism in cohomology induced by the natural map (ℂP^{d})^{n} → BU(n). We also calculate the cohomology of the homotopy fiber of the natural map E S_{n} ×_{S_{n}}(ℂP^{d})^{n} → BU(n).

Contents

1. Introduction 1 2. Bundles and symmetric invariants 3 3. Cohomology calculations when n! is invertible 6 4. The orthogonal groups and more calculations at p = 2 11 5. Truncated symmetric polynomials 12 6. Regular sequences 14 Acknowledgments 17 References 18

1. Introduction

One of the nicest calculations in algebraic topology is that of the cohomology of the classifying space BU(n) of the unitary groups as the ring of symmetric polynomials on n variables (see [3]). In fact the restriction map identifies H^{∗}(BU(n), ℤ) with the invariants in the cohomology of the classifying space BT of a maximal torus under the action of the Weyl

Date: June 25, 2009. 2000 Mathematics Subject Classification. 55R35, 05E95. Key words and phrases. Classifying space, bundle, cohomology, symmetric polynomial, regular sequence. ∗Partially supported by NSERC and NSF. ∗∗Partially supported by NSERC.

group S_{n}. This leads to a beautiful description of the cohomology of the flag manifold U(n)/T and more specifically a detailed understanding of the fibration U(n)/T → BT → BU(n).

In this paper we explore the situation where BT = (ℂP^{∞})^{n} is replaced by a product of finite dimensional projective spaces (ℂP^{d})^{n}, fitting into an associated bundle

U(n) ×_{T} (𝕊^{2d+1})^{n} → (ℂP^{d})^{n} → BU(n).

This requires an analysis of truncated symmetric invariants and in particular a precise description of the kernel I(n, d) of the algebra surjection H^{∗}(BU(n), 𝔽) → H^{∗}((ℂP^{d})^{n}, 𝔽)^{S^{n}}. The purely algebraic version of this problem is studied in §5 and §6. In particular, Theorem 5.1 allows us to exhibit an explicit set of generators for I(n, d) as follows.

Theorem 1.1. Let 𝔽 be a field and I(n, d) be the kernel of the map H^{∗}(BU(n), 𝔽) → H^{∗}((ℂP^{d})^{n}, 𝔽).

(a) If n! is invertible in 𝔽 then I(n, d) is generated by the elements P_{d+1}, P_{d+2}, …, P_{d+n} (b) If n < 2 char(𝔽)−1 then I(n, d) is generated by P_{d+1}, P_{d+2}, …, P_{d+n} and P^{d + 1, …, d + 1}_{p times}.

For the definition of P_{d+i} and P\underbrace{d + 1, …, d + 1}_{p^{i} times}, see §5. Note that the degree of P_{d+i} is 2(d + i)

and the degree of P\underbrace{d + 1, …, d + 1}_{p times} is 2p(d + 1).

If n! is invertible in a field 𝔽, then we show that the elements P_{d+i}, 1 ≤ i ≤ n form a generating regular sequence for I(n, d). In contrast, using Theorem 6.1 we show that in most other cases I(n, d) cannot be generated by a regular sequence:

Theorem 1.2. If n ≥ char(𝔽) > 0 and d > 1, then I(n, d) cannot be generated by a regular sequence.

There is a free action of S_{n} on the fiber space W (n, d) = U(n) ×_{T} (𝕊^{2d+1})^{n} arising from the normalizer of the maximal torus in U(n). The orbit space X(n, d) can be realized as the fiber of the natural map E S_{n} ×_{S_{n}}(ℂP^{d})^{n} → BU(n). Our algebraic calculations allow us to calculate the cohomology of this space in good characteristic.

Theorem 1.3. If 𝔽 is a field where n! is invertible, then the cohomology of X(n, d) is an exterior algebra on n generators

H^{∗}(X(n, d), 𝔽) ≅ Λ_{𝔽}(E_{d+1}, …, E_{d+n})

where E_{j} is a cohomology class in dimension 2j − 1.

This has an interesting computational consequence.

Theorem 1.4. For any field 𝔽 of coefficients, the Serre spectral sequence for the fibration (𝕊^{2d+1})^{n} → W (n, d) → U(n)/T collapses at E_{2} if and only if d ≥ n − 1. Consequently, we obtain an additive calculation

H^{∗}(W (n, d), 𝔽) ≅ H^{∗}(U(n)/T) ⊗ H^{∗}((𝕊^{2d+1})^{n}, 𝔽)

whenever d ≥ n − 1. In particular if n! is invertible in 𝔽, then

H^{∗}(X(n, d), 𝔽) ≅ [H^{∗}(U(n)/T) ⊗ H^{∗}((𝕊^{2d+1})^{n}, 𝔽)]^{S^{n}} ≅ Λ_{𝔽}(E_{d+1}, …, E_{d+n}) .

These results follow from a general theorem about the cohomology of fibrations which, although “classical” in nature, seems to be new.

Theorem 1.5. Let 𝔽 be a field and let π : E → B denote a fibration with fiber F of finite type such that B is simply connected. Assume

• H^{∗}(B, 𝔽) is a polynomial algebra on n even dimensional generators,

• π^{∗} : H^{∗}(B, 𝔽) → H^{∗}(E, 𝔽) is surjective,

• the kernel of π^{∗} is generated by a regular sequence u_{1}, …, u_{n}, where |u_{i}| = 2r_{i}. Then H^{∗}(F, 𝔽) is an exterior algebra on n odd dimensional generators e_{1}, …, e_{n}, where |e_{i}| = 2r_{i} − 1.

It is natural to ask whether the results of this paper can be extended to compact Lie groups, other than U(n). We thus conclude this introduction with the following open problem.

Problem: Let G be a compact Lie group with maximal torus T of rank n and Weyl group W. Describe generators for the kernel I_{G}(n, d) of the natural map H^{∗}(BG, 𝔽) → H^{∗}((ℂP^{d})^{n}, 𝔽) and use this to describe the cohomology of the homotopy fiber of (ℂP^{d})^{n} → BG when |W | is invertible in 𝔽.

Theorems 5.1(a) and 6.1(a) have been independently proved in a recent preprint [4] by A. Conca, Ch. Krattenthaler, J. Watanabe. We are grateful to J. Weyman for bringing this preprint to our attention.

2. Bundles and symmetric invariants

A classical computation in algebraic topology is that of the cohomology of the classifying space BU(n) where U(n) is the unitary group of n × n matrices. We briefly recall how that goes; details can be found, e.g., in the survey paper [3] by A. Borel. Let T = (𝕊^{1})^{n} ⊂ U(n) denote the maximal torus of diagonal matrices in U(n); its classifying space is BT = (ℂP^{∞})^{n}. The inclusion T ⊂ U(n) induces a map between the cohomology of BU(n) and the cohomology of BT. Note that the normalizer NT of the torus is a wreath product 𝕊^{1} ≀ S_{n}, where the symmetric group S_{n} acts by permuting the n diagonal entries. Thus the Weyl group NT/T is the symmetric group S_{n}. Recall that H^{∗}(BT, ℤ) ≅ ℤ[x_{1}, …, x_{n}] where the x_{1}, …, x_{n} are 2-dimensional generators.

Theorem 2.1. The inclusion T ⊂ U(n) induces an inclusion in cohomology with image the ring of symmetric invariants in the graded polynomial algebra

H^{∗}(BU(n), ℤ) ≅ H^{∗}(BT, ℤ)^{S^{n}} = ℤ[x_{1}, …, x_{n}]^{S^{n}}

where the action of S_{n} arises from that of the Weyl group. □

Now recall that the complex projective space ℂP^{d} is a natural subspace of ℂP^{∞}; this induces a map

F̃(n, d) : (ℂP^{d})^{n} → BT → BU(n).

The permutation matrices S_{n} ⊂ U(n) act via conjugation on U(n); this restricts to an action on the diagonal maximal torus T which permutes the factors. Applying the classifying space functor yields actions of S_{n} on BT and BU(n) which make the map F̃(n, d) equivariant. Note however that the conjugation action on U(n) is homotopic to the identity on BU(n). We conclude that F̃(n, d) induces the natural map

F̃(n, d)^{∗} : H^{∗}(BU(n), ℤ) ≅ ℤ[x_{1}, …, x_{n}]^{S^{n}} → ℤ[x_{1}, …, x_{n}]/(x_{1}^{d+1}, … x_{n}^{d+1})

in integral cohomology whose image is precisely the ring of truncated symmetric invariants. We should also note that the map F̃(n, d) is (up to homotopy) the classifying map for the n–fold product of the canonical complex line bundle over ℂP^{d}.

To make this effective geometrically, we need to describe the map F̃(n, d) explicitly as a fibration. The space (ℂP^{d})^{n} is a quotient of (𝕊^{2d+1})^{n} by the free action of the maximal torus T. Using a standard induction construction we can view our map as a fibration which lies over the classical fibration connecting U(n)/T, BT and BU(n). Indeed, the following commutative diagram has fibrations in its rows and columns:

[figure: (𝕊^{2d+1})^{n}; (𝕊^{2d+1})^{n}; W (n, d); U(n) ×_{T} (𝕊^{2d+1})^{n}; (ℂP^{d})^{n}; F̃(n,d); BU(n); U(n)/T; BT; BU(n)]

Note that we also have a bundle

U(n) → U(n) ×_{T} (𝕊^{2d+1})^{n} → (ℂP^{d})^{n}

and its classifying map is F̃(n, d).

In some of our applications it will also make sense to take a quotient by the action of the symmetric group S_{n}. However for technical reasons this requires taking a homotopy orbit space which we now define.

Definition 2.2. Let G denote a compact Lie group acting on a space X, its homotopy orbit space X_{hG} is defined as the quotient of the product space EG×X by the diagonal G–action, where EG is the universal G–space.

Remark 2.3. It should be noted that if G is a finite group, X is a G–space and |G| is invertible in the coefficients, then the natural projection X_{hG} → X/G induces an isomorphism in cohomology (this follows from the Vietoris-Begle theorem). Hence for example if |G| is invertible in a coefficient field 𝔽, then H^{∗}(X_{hG}, 𝔽) ≅ H^{∗}(X, 𝔽)^{G} (the algebra of invariants).

In our context, the symmetric group S_{n} acts by permuting the factors in (ℂP^{d})^{n} and we can consider the associated homotopy orbit space

(ℂP^{d})_{h S_{n}}^{n} = E S_{n} ×_{S_{n}}(ℂP^{d})^{n}.

More precisely, the map BT → BU(n) naturally factors through the classifying space of the normalizer NT, as we have T ⊂ NT ⊂ U(n). The space BNT can be identified with BT_{h S_{n}} = (ℂP^{∞})_{h S_{n}}^{n}, where S_{n} acts by permuting the factors, as before. This homotopy orbit space restricts to the truncated projective spaces, yielding a map

F (n, d) : (ℂP^{d})_{h S_{n}}^{n} → BU(n) ,

which is surjective in rational cohomology. We would also like to describe this map as a fibration.

The map (ℂP^{d})^{n} → BT is an S_{n}-equivariant fibration, with fiber (𝕊^{2d+1})^{n}. This arises from the free T –action on the product of spheres, which extends in the usual way to an action of the semidirect product NT. If we take homotopy orbit spaces we obtain a fibration sequence

(𝕊^{2d+1})^{n} → (𝕊^{2d+1})_{hNT}^{n} → BNT .

Dividing out by the free T –action we can identify (𝕊^{2d+1})_{hNT}^{n} ≃ (ℂP^{d})_{h S_{n}}^{n}. This makes the fiber of the map (ℂP^{d})_{h S_{n}}^{n} → BNT very explicit. As before, in order to describe the fibration with target BU(n), it suffices to induce up the action on the fiber to a U(n)–action by taking the balanced product Z = U(n) ×_{NT} (𝕊^{2d+1})^{n}. This yields a fibration sequence

Z → Z_{hU(n)} → BU(n) .

Note that

Z_{hU(n)} ≃ EU(n) ×_{NT} (𝕊^{2d+1})^{n} ≃ (𝕊^{2d+1})_{hNT}^{n} ≃ (ℂP^{d})_{h S_{n}}^{n}

where the last equivalence follows from taking quotients by the free T –action, as before. Our discussion is summarized in the following diagram of fibrations, analogous to the non– equivariant situation:

[figure: (𝕊^{2d+1})^{n}; (𝕊^{2d+1})^{n}; X(n, d); U(n) ×_{NT} (𝕊^{2d+1})^{n}; E S_{n} ×_{S_{n}}(ℂP^{d})^{n}; F (n,d); BU(n); U(n)/NT; BNT; BU(n)]

Hence we have

Proposition 2.4. Up to homotopy the map F̃(n, d) : (ℂP^{d})^{n} → BU(n) is a fibration with fiber the compact simply connected manifold

W (n, d) = U(n) ×_{T} (𝕊^{2d+1})^{n}

of dimension equal to n(n + 2d). There is a free S_{n}–action on this manifold, and its quotient

X(n, d) = U(n) ×_{NT} (𝕊^{2d+1})^{n}

is homotopy equivalent to the fiber of F (n, d) : (ℂP^{d})_{h S_{n}}^{n} → BU(n). □

Remark 2.5. Note that there are fibrations

(𝕊^{2d+1})^{n} → X(n, d) → U(n)/NT

and

U(n) → X(n, d) → (ℂP^{d})_{h S_{n}}^{n}

where the second one is obtained from pulling back the universal U(n) bundle over BU(n) using F (n, d).

One of our main results in this paper will be to calculate the cohomology of the fibers W (n, d) and X(n, d) associated to the fibrations F̃(n, d) and F (n, d) respectively.

3. Cohomology calculations when n! is invertible

Our standing assumption in this section (unless stated otherwise) will be that 𝔽 is a field such that n! is invertible in 𝔽, and cohomology will be computed with 𝔽–coefficients. A good example is the field ℚ of rational numbers. In this situation we have H^{∗}(X(n, d), 𝔽) ≅ H^{∗}(W (n, d), 𝔽)^{S^{n}}; it is this cohomology algebra that we will be most interested in.

We begin by considering the limit case d = ∞. In this case X(n, ∞) = U(n)/NT and we are looking at the classical fibration

U(n)/NT → BNT → BU(n)

Proposition 3.1. The map BNT → BU(n) induces an isomorphism in cohomology and U(n)/NT is 𝔽–acyclic.

Proof. Indeed, both maps in the sequence

H^{∗}(BU(n), 𝔽) → H^{∗}(BNT, 𝔽) → H^{∗}(BT, 𝔽)^{S^{n}}

are isomorphisms. Since BU(n) is simply connected, this can only happen if U(n)/NT is acyclic. □

Note that this computation is very different from what the cohomology of the flag manifold U(n)/T looks like; when we divide out by the action of the symmetric group all the reduced cohomology vanishes.

We now consider the unstable case of this result, namely when d is finite. This is considerably more interesting, as we know that the cohomology must be non–trivial. This calculation will be a special case of a more general result about the cohomology of fibrations.

Theorem 3.2. Let π : E → B denote a fibration with fiber F of finite type such that B is simply connected and

• H^{∗}(B, 𝔽) is a polynomial algebra on n even dimensional generators,

• π^{∗} : H^{∗}(B, 𝔽) → H^{∗}(E, 𝔽) is surjective,

• the kernel of π^{∗} is generated by a regular sequence u_{1}, …, u_{n} where |u_{i}| = 2r_{i}.

Then H^{∗}(F, 𝔽) is an exterior algebra on n odd dimensional generators e_{1}, …, e_{n}, where |e_{i}| = 2r_{i} − 1.

Proof. The cohomology of the fiber F in a fibration

F → E → B

can be studied using the Eilenberg–Moore spectral sequence. We refer the reader to [7], Chapter VIII for details. It has the form:

E_{2}^{∗,∗} = T or_{H^{∗}(B,𝔽)}(𝔽, H^{∗}(E, 𝔽)).

On the other hand, the hypotheses imply that

H^{∗}(E, 𝔽) ≅ H^{∗}(B, 𝔽)/(u_{1}, … u_{n})

where u_{1}, …, u_{n} form a regular sequence of maximal length in H^{∗}(B, 𝔽), a polynomial algebra on n even dimensional generators. In other words the cohomology of B is free and finitely generated over 𝔽[u_{1}, …, u_{n}]. Thus the spectral sequence simplifies to

E_{2}^{∗,∗} = T or_{H^{∗}(B,𝔽)}(𝔽, H^{∗}(B, 𝔽) ⊗_{𝔽[u_{1},...,u_{n}]} 𝔽) ≅ T or_{𝔽[u_{1},...,u_{n}]}(𝔽, 𝔽)

This can be computed using the standard Koszul complex, yielding

E_{2} = Λ_{𝔽}(e_{1}, …, e_{n})

where the e_{i} are exterior classes in degree 2r_{i} − 1. There are no further differentials, as the algebra generators for E_{2}^{∗,∗} represent non–trivial elements in the cohomology of F which by construction must transgress to the regular sequence {u_{1}, …, u_{n}} in H^{∗}(B, 𝔽) in the Serre spectral sequence for the fibration

F → E → B.

Therefore the Eilenberg–Moore spectral sequence collapses at E_{2} = E_{∞}. Now this algebra is a free graded commutative algebra, hence there are no extension problems and it follows that

H^{∗}(F, 𝔽) ≅ Λ_{𝔽}(e_{1}, …, e_{n})

as stated in the theorem. □

We now apply this result to the spaces X(n, d).

Theorem 3.3. The cohomology of X(n, d) is an exterior algebra on n generators

H^{∗}(X(n, d), 𝔽) ≅ Λ_{𝔽}(E_{d+1}, …, E_{d+n}) ,

where E_{j} is a cohomology class in dimension 2j − 1.

Proof. As observed previously we have a fibration

X(n, d) → (ℂP^{d})_{h S_{n}}^{n} → BU(n) .

The Eilenberg–Moore spectral sequence can therefore be applied to compute the cohomology of X(n, d). The map F (n, d) : (ℂP^{d})_{h S_{n}}^{n} → BU(n) induces a surjection of algebras

H^{∗}(BU(n), 𝔽) → H^{∗}((ℂP^{d})_{h S_{n}}^{n}, 𝔽) → 0

which can be identified with the natural map

𝔽[x_{1}, …, x_{n}]^{S^{n}} → (𝔽[x_{1}, …, x_{n}]/(x_{1}^{d+1}, … x_{n}^{d+1}))^{S^{n}}.

The kernel of this map is precisely the ideal

I_{n,d} = (x_{1}^{d+1}, …, x_{n}^{d+1}) ∩ 𝔽[x_{1}, …, x_{n}]^{S^{n}}.

By Theorem 6.1(a), I_{n,d} is generated by a regular sequence of elements P_{d+1}, …, P_{d+n}. Here each P_{j} is a homogeneous polynomial in x_{1}, …, x_{n} of degree j; its degree as a cohomology class is 2j. These classes form a regular sequence of maximal length in the polynomial algebra H^{∗}(BU(n), 𝔽). Thus the hypotheses of Theorem 3.2 hold, and the proof is complete. □

Corollary 3.4. If d < ∞, then X(n, d) is a compact, connected, orientable manifold.

Proof. According to our calculation, for m = n(n + 2d) we have H^{m}(X(n, d), ℚ) ≅ ℚ. This is precisely the dimension of the compact manifold X(n, d) = U(n) ×_{NT} (𝕊^{2d+1})^{n}, whence the result follows. □

Remark 3.5. Note that as d gets large, the connectivity of the space X(n, d) increases; this is consistent with the stable calculation, namely the acyclicity of U(n)/NT. Also note that the manifold U(n)/NT is not orientable, as it is ℚ–acyclic.

For the case of W (n, d) we offer the following general result:

Theorem 3.6. For any field 𝔽 of coefficients, the Serre spectral sequence for the fibration (𝕊^{2d+1})^{n} → W (n, d) → U(n)/T collapses at E_{2} if and only if d ≥ n − 1, from which we obtain an additive calculation

H^{∗}(W (n, d), 𝔽) ≅ H^{∗}(U(n)/T) ⊗ H^{∗}((𝕊^{2d+1})^{n}, 𝔽).

In particular if n! is invertible in 𝔽, then

H^{∗}(X(n, d), 𝔽) ≅ [H^{∗}(U(n)/T) ⊗ H^{∗}((𝕊^{2d+1})^{n}, 𝔽)]^{S^{n}} ≅ Λ_{𝔽}(E_{d+1}, …, E_{d+n}) .

Proof. Consider the Serre spectral sequence with 𝔽 coefficients for the fibration (𝕊^{2d+1})^{n} → W (n, d) → U(n)/T. The base is simply connected and the cohomology of the fiber is generated by the natural generators for the 2d + 1–dimensional cohomology of each sphere. The first differential in the spectral sequence can be computed as follows: if e_{i} ∈ H^{2d+1}((𝕊^{2d+1})^{n}) is a natural generator then

d_{2d+2}(e_{i}) = [x_{i}^{d+1}] ∈ H^{∗}(U(n)/T) ≅ H^{∗}(BT)/(s_{1}, s_{2}, …, s_{n})

where the s_{1}, s_{2}, …, s_{n} are the symmetric polynomials. This follows from the diagram of fibrations in the previous section and the well-known calculation of the cohomology of (ℂP^{d})^{n} and U(n)/T as quotients of H^{∗}(BT). We now need the following algebraic lemma.

Lemma 3.7. Let 𝔽 be a commutative ring and I be the ideal of 𝔽[x_{1}, ..., x_{n}] generated by the elementary symmetric polynomials s_{1}, ..., s_{n} in x_{1}, ..., x_{n}. Then (a) x_{1}^{n} ∈ I but (b) x_{1}^{n−1} ∉ I.

Suppose Lemma 3.7 is established (we only need it in the case where 𝔽 is a field). Then we conclude that d_{2d+2}(e_{i}) = [x_{i}^{d+1}] = 0 in H^{∗}(U(n)/T) for all i = 1, …, n if and only if d ≥ n − 1. This implies that all the differentials in the spectral sequence are zero and so it collapses at E_{2}. The assertions of Theorem 3.6 follow from this and Theorem 3.3.

It thus remains to prove Lemma 3.7.

(a) Recall that x_{1}, …, x_{n} are, by definition, the roots of the polynomial

x^{n} − x^{n−1}s_{1} + x^{n−2}s_{2} − ... + (−1)^{n}s_{n} = 0 .

Thus x_{1}^{n} = x_{1}^{n−1}s_{1} − x^{n−2}s_{2} + ... − (−1)^{n}s_{n}, and since every term in the right hand side lies in I, part (a) follows.

(b) Assume, to the contrary, that

(3.1)  x_{1}^{n−1} = f_{1}s_{1} + ... + f_{n}s_{n}

for some polynomials f_{1}, ..., f_{n} ∈ 𝔽[x_{1}, …, x_{n}]. If such an identity is possible over 𝔽, and α: 𝔽 → L is a ring homomorphism then, applying α to each of the coefficients of f_{1}, …, f_{n}, we obtain an identity of the same form over L. Thus, for the purpose of showing that (3.1) is not possible, we may, without loss of generality, replace 𝔽 by L. In particular, we may take L to be the algebraic closure of the field 𝔽/M, where M is a maximal ideal of 𝔽. After replacing 𝔽 by this L, we may assume that 𝔽 is an algebraically closed field.

Equating the homogeneous terms of degree n−1 on both sides, we see that after replacing f_{1}, f_{2}, …, f_{n−1} by their homogeneous parts of degrees n − 2, n − 3, …, 0, respectively, we may assume that f_{n} = 0.

Since 𝔽 is an algebraically closed field, x^{n} − 1 factors into a product of linear terms

(3.2)  x^{n} − 1 = (x − ζ_{1})(x − ζ_{2}) · . . . · (x − ζ_{n}) .

for some ζ_{1}, …, ζ_{n} ∈ 𝔽. (As an aside, we remark that ζ_{1}, …, ζ_{n} ∈ 𝔽 are distinct if p = char(𝔽) does not divide n but not in general; at the other extreme, if n is a power of p then ζ_{1} = ⋯ = ζ_{n} = 1.) By (3.2)

s_{i}(ζ_{1}, …, ζ_{n}) = (−1)^{i} (coefficient of x^{n−i} in x^{n} − 1) = 0

for every i = 1, …, n − 1. Hence, substituting ζ_{i} for x_{i} in (3.1), and remembering that f_{n} = 0, we obtain ζ_{1}^{n−1} = 0, i.e., ζ_{1} = 0. Since ζ_{1} is a root of x^{n} − 1 = 0, we have arrived at a contradiction. This shows that (3.1) is impossible. The proof of Lemma 3.7 and thus of Theorem 3.6 is now complete. □

Calculations with field coefficients can be pieced together to provide information on the integral cohomology of X(n, d).

Proposition 3.8. The cohomology ring H^{∗}(X(n, d), ℤ) has no p–torsion if p > n.

Proof. By our previous results if p > n then

dim_{𝔽_{p}} H^{∗}(X(n, d), 𝔽_{p}) = dim_{ℚ} H^{∗}(X(n, d), ℚ) = 2^{n}.

Hence by the universal coefficient theorem, there can be no p–torsion in the integral cohomology of X(n, d). □

The situation is more complicated if n ≥ p = char(𝔽). In particular, we will show that in this case the kernel I(n, d) of the map H^{∗}(BU(p), 𝔽_{p}) → H^{∗}((ℂP^{d})^{p}, 𝔽_{p}) cannot be generated by a regular sequence for any d ≥ 2 (and, in most cases for d = 1 as well); see Theorem 6.1(b). We now provide an explicit calculation in the case where n = d = p = 2.

Example 3.9. Consider the map F̃(2, 2) : 𝕊^{2} × 𝕊^{2} → BU(2). Its fiber is

W (2, 2) = U(2) ×_{T} (𝕊^{3} × 𝕊^{3})

which itself fibers over U(2)/T = 𝕊^{2} with fiber 𝕊^{3} × 𝕊^{3}. Hence for dimensional reasons H^{∗}(W (2, 2), ℤ) ≅ H^{∗}(𝕊^{3} × 𝕊^{3} × 𝕊^{2}, ℤ). The S_{2}–action on this space exchanges the two 3spheres and applies the antipodal map on 𝕊^{2}. Thus the orbit space X(2, 2) will be rationally cohomologous to 𝕊^{3} × 𝕊^{5}, as predicted by Theorem 3.3. However, it can be shown that H^{∗}(X(2, 2), 𝔽_{2}) has Poincaré series

p(t) = 1 + t + t^{2} + t^{3} + t^{5} + t^{6} + t^{7} + t^{8} .

On the other hand, the corresponding Poincaré series for rational cohomology is

q(t) = 1 + t^{3} + t^{5} + t^{8}

which accounts for the torsion free classes in the integral cohomology. This example illustrates the presence of 2–torsion in the cohomology of X(2, 2). Of course in this case we have π_{1}(X(2, 2)) = ℤ/2, which accounts for the classes in degrees one and two in mod 2 cohomology, and by Poincaré duality for the classes in degrees six and seven.

On the other hand, recall that if H^{∗}(BU(2), 𝔽_{2}) ≅ 𝔽_{2}[c_{2}, c_{4}] and H^{∗}(𝕊^{2}×𝕊^{2}, 𝔽_{2}) ≅ Λ(u_{2}, v_{2}) then F̃(2, 2)^{∗}(c_{2}) = u_{2}+v_{2} and F̃(2, 2)^{∗}(c_{4}) = u_{2}v_{2}. Thus we see that F̃(2, 2)^{∗} is not surjective and that its kernel is generated by the classes c_{2}^{2}, c_{2}^{3} + c_{2}c_{4}, c_{4}^{2}. These classes correspond to the symmetric polynomials P_{2} = x_{1}^{2} + x_{2}^{2}, P_{3} = x_{1}^{3} + x_{2}^{3} and P_{2,2} = x_{1}^{2}x_{2}^{2}. Note that if 2 is invertible in the coefficients then

P_{2,2} = \frac{P_{2}^{2} − (x_{1} + x_{2})P_{3} + (x_{1}x_{2})P_{2}}{2} ,

and the third generator is redundant.

More generally, using the algebraic calculations in Theorem 5.1, Theorem 6.1 and Corollary 6.3 we obtain the following.

Theorem 3.10. Assume that p ≤ n ≤ 2p−1 and d ≥ 2. Then the kernel of the map induced by F̃(n, d) in cohomology

F̃(n, d)^{∗} : H^{∗}(BU(n), 𝔽_{p}) → H^{∗}((ℂP^{d})^{n}, 𝔽_{p})

is generated by the following n + 1 elements:

• P_{d+i}, where 1 ≤ i ≤ n and |P_{j}| = 2j

• P\underbrace{d + 1, …, d + 1}_{p times} and |P\underbrace{d + 1, …, d + 1}_{p times}| = 2p(d + 1)

Moreover these elements cannot form a regular sequence, and this kernel cannot be generated by less than n + 1 elements. □

4. The orthogonal groups and more calculations at p = 2

The situation for p = 2 is somewhat different, as there are specific geometric models which are special to this characteristic. Here we consider the standard diagonal inclusion V = (ℤ/2)^{n} ↪ O(n) into the group of orthogonal n × n matrices. The group V is selfcentralizing in O(n); its normalizer NV is the wreath product NV = ℤ/2 ≀ S_{n}. The Weyl group W = NV/V of V in O(n) is thus isomorphic to S_{n}; it acts on V = (ℤ/2)^{n} by permuting the n factors of ℤ/2. The classifying space for V is BV = (ℝP^{∞})^{n}, its mod 2 cohomology is a polynomial algebra on n one dimensional generators 𝔽_{2}[x_{1}, …, x_{n}]. The inclusion induces a map from the cohomology of BO(n) to this algebra, which gives rise to an isomorphism onto the symmetric invariants. As before, the truncated projective space ℝP^{d} is a natural subspace of ℝP^{∞}, and Theorem 5.1 provides a description of the kernel of the homomorphism induced by the map H(n, d) : (ℝP^{d})^{n} → BO(n) for n = 1, 2, 3.

The classifying space for NV = ℤ/2 ≀ S_{n} is BNV = (ℝP^{∞})_{h S_{n}}^{n}. However, as our calculations are at p = 2 and | S_{n} | is even, the homotopy orbit space has a lot more cohomology than just the truncated symmetric invariants (for example, it contains a copy of H^{∗}(S_{n}, 𝔽_{2})). The wreath product NV acts on (𝕊^{d})^{n} extending the coordinatewise antipodal action of V. Thus we have a fiber bundle (𝕊^{d})^{n} → (ℝP^{d})_{h S_{n}}^{n} → BNV where we identify (𝕊^{d})_{hNV}^{n} ≃ (ℝP^{d})_{h S_{n}}^{n}.

Example 4.1. For n = 2 we can identify NV with the dihedral group D_{8} and its cohomology has generators e, u v in degrees 1, 1, 2 respectively with the single relation e · u = 0 (see [1]). The elements u, v can be identified with the standard symmetric generators x_{1} + x_{2} and x_{1}x_{2} in H^{∗}(V, 𝔽_{2})^{S^{2}} via the restriction map. In fact we have isomorphisms (see [1], page 118) H^{∗}(BD_{8}, 𝔽_{2}) ≅ H^{∗}(S_{2}, H^{∗}(V, 𝔽_{2})) and H^{∗}((𝕊^{d})_{hD_{8}}^{2}, 𝔽_{2}) ≅ H^{∗}(S_{2}, H^{∗}((ℝP^{d})^{2}, 𝔽_{2}). Using these descriptions and Theorem 5.1 it can be shown that the the kernel of the homomorphism H^{∗}(BD_{8}, 𝔽_{2}) → H^{∗}((𝕊^{d})_{hD_{8}}^{2}, 𝔽_{2}) is the ideal generated by the three elements P_{d+1} = x_{1}^{d+1} + x_{2}^{d+2}, P_{d+2} = x_{1}^{d+2} + x_{2}^{d+2} and P_{d+1,d+1} = x_{1}^{d+1}x_{2}^{d+1}. This ideal is called the Fadell–Husseini index (see [6]) of the D_{8}–space 𝕊^{d} × 𝕊^{d}; it has some interesting applications in topology and it has been fully calculated in [2].

Geometrically, the fibration which our mod 2 calculations can be applied to is described by the diagram:

[figure: (𝕊^{d})^{n}; (𝕊^{d})^{n}; Y (n, d); O(n) ×_{V} (𝕊^{d})^{n}; (ℝP^{d})^{n}; H(n,d); BO(n); O(n)/V; BV; BO(n)]

Here we recall some classical results. First, from the homotopy long exact sequence of the fibration we see that O(n)/V is path–connected because π_{1}(BV) → π_{1}(BO(n)) ≅ ℤ/2 is surjective (the dual map in mod 2 cohomology is injective). Its fundamental group acts homologically trivially on H^{∗}((𝕊^{d})^{n}, 𝔽_{2}), as it acts through its image in V. Therefore the Serre spectral sequence for the fibration (𝕊^{d})^{n} → Y (n, d) → O(n)/V has the form

E_{2}^{∗,∗} = H^{∗}(O(n)/V) ⊗ H^{∗}((𝕊^{d})^{n}, 𝔽_{2}) ⟹ H^{∗}(Y (n, d), 𝔽_{2}) .

Using Lemma 3.7, we see that this spectral sequence collapses at E_{2} if and only if d ≥ n − 1.

Theorem 4.2. If d ≥ n − 1 then we have an additive isomorphism

H^{∗}(Y (n, d), 𝔽_{2}) ≅ H^{∗}(O(n)/V) ⊗ H^{∗}((𝕊^{d})^{n}, 𝔽_{2}).

□

5. Truncated symmetric polynomials

We will now state and prove the algebraic results used in the previous sections. For the rest of the paper we will use the following notations:

n, d  positive integers
S_{n}  the symmetric group on n letters
𝔽  base field
x_{1}, …, x_{n}  independent variables over 𝔽
𝔽[x_{1}, …, x_{n}]  polynomial ring in n variables
R_{n} := 𝔽[x_{1}, …, x_{n}]^{S^{n}}  ring of symmetric polynomials in n variables
I_{n,d} := (x_{1}^{d+1}, …, x_{n}^{d+1}) ∩ R_{n} ideal of R_{n}.

If a_{1}, …, a_{n} are non-negative integers, we will write P_{a_{1},...,a_{n}} for the orbit sum of x_{1}^{a_{1}} … x_{n}^{a_{n}}. In other words, P_{a_{1},...,a_{n}} is the sum of monomials x^{a′_{1}} … x_{n}^{a′_{n}}, as a′_{1}, …, a′_{n} range over all

possible permutations of a_{1}, …, a_{n}. This sum has \frac{n!}{λ_{1}! ⋯ λ_{m}!} terms, where λ_{1}, …, λ_{m} is the

partition of n associated to a_{1}, …, a_{n}. (Recall that this means that that there are m distinct integers among a_{1}, …, a_{n}, occurring with multiplicities λ_{1}, …, λ_{m}, respectively.)

Permuting a_{1}, …, a_{n} does not change P_{a_{1},...,a_{n}}, so we will always assume that a_{1} ≥ ⋯ ≥ a_{n}. With this convention, the orbit sums P_{a_{1},...,a_{n}} clearly form a basis of R_{n} := 𝔽[x_{1}, …, x_{n}]^{S^{n}} as an 𝔽-module. The multiplication rule in this basis is given by

(5.1)  P_{a_{1},...,a_{n}} · P_{b_{1},...,b_{n}} = ∑ k_{c_{1},...,c_{n}}P_{c_{1},...,c_{n}} ,

where c_{1} ≥ … ≥ c_{n} and there are exactly k_{c_{1},...,c_{n}} different ways to write

(c_{1}, …, c_{n}) = (a′_{1}, …, a′_{n}) + (b′_{1}, …, b′_{n})

for some permutation a′_{1}, …, a′_{n} of a_{1}, …, a_{n} and some permutation b′_{1}, …, b′_{n} of b_{1}, …, b_{n}.

To make our formulas less cumbersome, we will often abbreviate P_{a_{1},...,a_{r},0,...,0} as P_{a_{1},...,a_{r}}. As long as the number of variables n is fixed, this will not lead to any confusion. For example, in this notation,

P_{i} = x_{1}^{i} + ⋯ + x_{n}^{i}

is the usual power sum of degree i and

(5.2)  \begin{matrix}P_{1} = x_{1} + ⋯ + x_{n}, \\ P_{1,1} = x_{1}x_{2} + ⋯ + x_{n−1}x_{n}, \\ … \\ P\underbrace{1, …, 1}_{n times} = x_{1}x_{2} … x_{n} \end{matrix}

are the elementary symmetric polynomials.

The main result of this section is the following theorem.

Theorem 5.1. Let 𝔽 be a field of characteristic p ≥ 0.

(a) If p = 0 or n < p then the ideal I_{n,d} := (x_{1}^{d+1}, …, x_{n}^{d+1}) ∩ 𝔽[x_{1}, …, x_{n}]^{S^{n}} of R_{n} := 𝔽[x_{1}, …, x_{n}]^{S^{n}} is generated by P_{d+1}, …, P_{d+n}.

(b) If n ≤ 2p − 1 then I_{n,d} is generated by P_{d+1}, …, P_{d+n} and P^{d + 1, …, d + 1}_{p times}.

The rest of this section will be devoted to proving Theorem 5.1. First we note that every element of I_{n,d} is an 𝔽-linear combination of orbit sums P_{a_{1},...,a_{n}}, where a_{1} ≥ d + 1. Thus in order to prove Theorem 5.1 it suffices to show that every P_{a_{1},...,a_{n}} lies in I. Our first step in this direction is the following lemma.

We define the weight of the orbit sum P_{a_{1},...,a_{n}} as the largest integer r ≤ n such that a_{r} ≥ 1. As mentioned above, we will abbreviate such an orbit sum as P_{a_{1},...,a_{r}}.

We define the leading multiplicity of P_{a_{1},...,a_{n}} as the largest integer s ≤ n such that a_{1} = ⋯ = a_{s}. Here, as always, we are assuming that a_{1} ≥ a_{2} ≥ … ≥ a_{n} ≥ 0.

Lemma 5.2. Let 𝔽 be a field and J_{n,d} be the ideal of R_{n} = 𝔽[x_{1}, …, x_{n}]^{S^{n}} generated by P_{d+1}, …, P_{d+n}. Then J_{n,d} contains every orbit sum P_{a_{1},...,a_{n}} with a_{1} ≥ d + 1, whose leading multiplicity is invertible in 𝔽.

The leading multiplicity of P_{a_{1},...,a_{n}} is, by definition, an integer between 1 and n. Theorem 5.1(a) is thus an immediate consequence of this lemma.

Proof. We will argue by induction on the weight r of P_{a_{1},...,a_{n}}. For the base case, let r = 1. That is, we claim that P_{i} ∈ J_{n,d} for every i ≥ d + 1. For i = d + 1, …, d + n this is given. Applying Newton’s identities,

P_{m+n+1} = P_{1} · P_{m+n} − P_{1,1} · P_{m+n−1} + ⋯ + (−1)^{n+1}P^{1, …, 1}_{n times} · P_{m+1}

recursively, with m = d, d + 1, d + 2, etc., we see that P_{m+n+1} ∈ J_{n,d} for every m ≥ d. This settles the base case.

For the induction step assume that r ≥ 2. By (5.1),

(5.3) P_{a_{1}} · P_{a_{2},...,a_{r}} = sP_{a_{1},a_{2},...,a_{r}} + P_{a_{1}+a_{2},a_{3},...,a_{r}} + P_{a_{1}+a_{3},a_{2},a_{4},...,a_{r}} + ⋯ + P_{a_{1}+a_{r},a_{2},a_{3},...,a_{r−1}} .

Each of the terms

P_{a_{1}+a_{2},a_{3},...,a_{r}}, P_{a_{1}+a_{3},a_{2},a_{4},...,a_{r}}, …, P_{a_{1}+a_{r},a_{2},a_{3},...,a_{r−1}}

is an orbit sum of leading multiplicity 1 and weight r − 1. By the induction assumption each of them lies in J_{n,d}. Since we also know that P_{a_{1}} ∈ J_{n,d}, equation (5.3) tells us that P_{a_{1},...,a_{r}} ∈ J_{n,d} whenever s is invertible in 𝔽. □

We are now turn to the proof of Theorem 5.1(b). In view of part (a), we may assume that p < n ≤ 2p − 1. Let I be the ideal of R_{n} = 𝔽[x_{1}, …, x_{n}]^{S^{n}} generated by the polynomials listed in the statement of Theorem 5.1(b). Recall that it suffices to show that

(5.4)  P_{a_{1},...,a_{n}} ∈ I whenever a_{1} ≥ d + 1.

Denote the leading multiplicity of P_{a_{1},...,a_{n}} by s. We will now consider three cases.

Case 1. s ≠ p. Since we are assuming that p < n ≤ 2p − 1, this is equivalent to s being invertible in 𝔽. Clearly, J_{n,d} ⊂ I; Lemma 5.2 thus tells us that (5.4) holds.

Case 2. s = p and P_{a_{1},...,a_{n}} has weight p. In other words, we want to show that

(5.5)  P\underbrace{a, …, a}_{p times} ∈ I .

Let e = a − (d + 1). By (5.1) we see that

(5.6)  P\underbrace{d + 1, …, d + 1}_{p times} · P\underbrace{e, …, e}_{p times} = P\underbrace{a, …, a}_{p times} + Γ ,

where Γ is a positive integer linear combination of orbit sums of leading multiplicity ≤ p−1. By Lemma 5.2 Γ ∈ I. Since by definition, P\underbrace{d + 1, …, d + 1}_{p times} lies in I, the left hand side also lies

in I. Thus (5.5) holds as well.

Note that the above argument depends, in a crucial way, on our assumption that n ≤ 2p − 1. For n ≥ 2p the sum Γ in (5.6) would contain a term of the form P_{d+1,...,d+1,e,...,e} (or P_{e,...,e,d+1,...,d+1}, if e > d + 1), with each e and d + 1 repeating exactly p times. This orbit sum has leading multiplicity p, and in the case we cannot conclude that Γ ∈ I.

Case 3. s = p, general case. Denote a_{1} = ⋯ = a_{p} by a. Using formula (5.1) once again, we see that

P_{a_{1},...,a_{n}} = P\underbrace{a, …, a}_{p times} · P_{a_{p+1},...,a_{n}} + Δ ,

where Δ is an integer linear combination of orbit sums P_{c_{1},...,c_{n}} of leading multiplicity ≤ p−1. Note that P\underbrace{a, …, a}_{p times} ∈ I by Case 2 and Δ ∈ I by Lemma 5.2. We thus conclude that P_{a_{1},...,a_{p}} ∈ I

as well. This completes the proof of Theorem 5.1. □

6. Regular sequences

We now turn to the question of whether or not the ideal I_{n,d} = (x_{1}^{d+1}, …, x_{n}^{d+1}) ∩ R_{n} of R_{n} = 𝔽[x_{1}, …, x_{n}]^{S^{n}} can be generated by a regular sequence. Our goal is to prove the following theorem.

Theorem 6.1. Let 𝔽 be a field of characteristic p ≥ 0.

(a) If n! is not divisible by p then I_{n,d} is generated by the regular sequence P_{d+1}, …, P_{d+n} in R_{n}.

(b) Assume that n ≥ p > 0 and either (i) n ≢ −1 (mod p) and d ≥ 1 or (ii) n ≡ −1 (mod p) and d ≥ 2. Then I_{n,d} is not generated by any regular sequence in R_{n}.

The assumptions on d in part (b) cannot be dropped; see Remark 6.4. Our proof of Theorem 6.1 will rely on the following elementary lemma.

Lemma 6.2. (a) The elements P_{a_{1},...,a_{n}}, with d ≥ a_{1} ≥ ⋯ ≥ a_{n} ≥ 0 form a basis for R_{n}/I_{n,d} as an 𝔽-vector space.

(b) The Krull dimension of R_{n}/I_{n,d} is 0.

(c) Suppose I_{n,d} is generated by r_{1}, …, r_{m} ∈ R_{n}, as an ideal of R_{n}. Then m ≥ n. Moreover, r_{1}, …, r_{m} form a regular sequence in R_{n} if and only if m = n.

Proof. (a) The power sums P_{a_{1},...,a_{n}} with a_{1} ≥ ⋯ ≥ a_{n} ≥ 0 form an 𝔽-basis of R_{n}. The power sums P_{a_{1},...,a_{n}} with a_{1} ≥ ⋯ ≥ a_{n} ≥ 0 and a_{1} ≥ d + 1 form an 𝔽-basis of I_{n,d}, and part (a) follows.

(b) By part (a) R_{n}/I_{n,d} is a finite-dimensional 𝔽-vector space.

(c) Recall that R_{n} is a polynomial ring over 𝔽 generated by the elementary symmetric polynomials in x_{1}, …, x_{n}. In particular, R_{n} is a Cohen-Macauley ring. Part (c) now follows from part (b). □

Proof of Theorem 6.1. (a) If p = char(𝔽) does not divide n! then Theorem 5.1(a) tells us that I_{n,d} is generated, as an ideal of R_{n}, by the n elements P_{d+1}, …, P_{d+n}. By Lemma 6.2(c) these elements form a regular sequence in R_{n}.

(b) If I_{n,d} is generated by a regular sequence then Socle(R_{n}/I_{n,d}) is a 1-dimensional 𝔽vector space; see, e.g. [8, p. 144] or [5, Section 21.2]. It is an immediate consequence of the multiplication formula (5.1) that

P\underbrace{d, …, d}_{n times} ∈ Socle(R_{n}/I_{n,d})

for any 𝔽, d and n. Thus in order to show that I_{n,d} is not generated by a regular sequence it suffices to exhibit an orbit sum P_{a_{1},...,a_{n}} ∈ Socle(R_{n}/I_{n,d}), with (a_{1}, …, a_{n}) ≠ (d, …, d). (Indeed, P_{a_{1},...,a_{n}} and P\underbrace{d, …, d}_{n times} are 𝔽-linearly independent in R_{n}/I_{n,d} by Lemma 6.2(a).)

(i) Suppose d ≥ 1 and n = pq + r, where q ≥ 1 and r ∈ {0, 1, …, p − 2}. We claim that in this case P_{a_{1},...,a_{n}} lies in Socle(R_{n}/I_{n,d}), if

a_{1} = ⋯ = a_{pq−1} = d and a_{pq} = a_{pq+1} = ⋯ = a_{n} = d − 1.

In other words, we claim that

(6.1)  P_{a_{1},...,a_{n}} · P_{b_{1},...,b_{n}} ∈ I_{n,d}

for every b_{1} ≥ b_{2} ≥ ⋯ ≥ b_{n} ≥ 0, with b_{1} ≥ 1. To prove this claim, we will examine the product P_{a_{1},...,a_{n}} · P_{b_{1},...,b_{n}} using the multiplication formula (5.1).

First of all, note that we may assume without loss of generality that

(6.2)  (b_{1}, …, b_{n}) = (\underbrace{1, …, 1}_{s times}, 0, …, 0)

for some 1 ≤ s ≤ r + 1. For all other choices of (b_{1}, …, b_{n}) ≠ (0, …, 0) every term P_{c_{1},...,c_{n}} appearing in the right hand side of the formula (5.1) will have c_{1} ≥ d + 1 and thus will lie in I_{n,d} (for any base field 𝔽).

If (b_{1}, …, b_{n}) is as in (6.2), the only orbit sum P_{c_{1},...,c_{n}} with c_{1} ≤ d appearing in the right hand side of (5.1), will have c_{1} = ⋯ = c_{pq+s−1} = d and c_{pq+s} = c_{pq+s+1} = ⋯ = c_{n} = d − 1. This sum will appear with coefficient k_{c_{1},...,c_{n}} = number of ways to write (c_{1}, …, c_{n}) as (a′_{1}, …, a′_{n}) + (b′_{1}, …, b′_{n}), where (a′_{1}, …, a′_{n}) is a permutation of (a_{1}, …, a_{n}) and (b′_{1}, …, b′_{n}) is a permutation of (b_{1}, …, b_{n}). We claim that k_{c_{1},...,c_{n}} is divisible by p and hence, is 0 in 𝔽; this will immediately imply (6.1). Indeed, in this case k_{c_{1},...,c_{n}} is simply the number of ways to specify which s of the elements b′_{1}, …, b′_{pq+s−1} should be equal to 1 (the remaining ones will be 0). Thus

k_{c_{1},...,c_{n}} = \binom{pq + s − 1}{s} .

Since q ≥ 1 and 1 ≤ s ≤ r + 1 ≤ p − 1, this number is divisible by p, as claimed.

(ii) Now suppose d ≥ 2 and n = pq+p−1, where q ≥ 1. We claim that in this case P_{a_{1},...,a_{n}} lies in Socle(R_{n}/I_{n,d}), if a_{1} = ⋯ = a_{pq−1} = d and a_{pq} = a_{pq+1} = ⋯ = a_{pq+p−2} = d − 1 and a_{pq+p−1} = d − 2.

Once again, we need to show that (6.1) holds for every b_{1} ≥ b_{2} ≥ ⋯ ≥ b_{n} ≥ 0, with b_{1} ≥ 1. The analysis of the product P_{a_{1},...,a_{n}} · P_{b_{1},...,b_{n}}, based on formula (5.1), is similar to part (i) but a bit more involved. First of all, we may assume without loss of generality that

(6.3)  (b_{1}, …, b_{n}) = (2, \underbrace{1, …, 1}_{t times}, 0, …, 0)

for some t ∈ [0, p − 1] or

(6.4)  (b_{1}, …, b_{n}) = (\underbrace{1, …, 1}_{s times}, 0, …, 0)

for some s ∈ [1, p]. For other (b_{1}, …, b_{n}) ≠ (0, …, 0), every orbit sum P_{c_{1},...,c_{n}} appearing in the right hand side of (5.1) will lie in I_{n,d}, so that (6.1) will hold over any base field 𝔽.

In case (6.3) the only term P_{c_{1},...,c_{n}} with c_{1} ≤ d appearing in the right hand side of (5.1) will have c_{1} = ⋯ = c_{pq+t} = d and c_{pq+t+1} = ⋯ = c_{pq+p−1} = d − 1. This term will appear with coefficient k_{c_{1},...,c_{n}} = number of ways to write (c_{1}, …, c_{n}) as (a′_{1}, …, a′_{n}) + (b′_{1}, …, b′_{n}), where (a′_{1}, …, a′_{n}) is a permutation of (a_{1}, …, a_{n}) and (b′_{1}, …, b′_{n}) is a permutation of (b_{1}, …, b_{n}). One of the elements b′_{1}, …, b′_{pq+t} should be equal to 2, t of these elements should be equal to 1, and the remaining pq − 1 should be equal to 0. Thus

k_{c_{1},...,c_{n}} = (pq + t)\binom{pq + t − 1}{t} .

If t = 0 then the first factor is divisible by p. If 1 ≤ t ≤ p − 1 then the second factor is divisible by p. Either way, k_{c_{1},...,c_{n}} = 0 in 𝔽, as desired.

In case (6.4) exactly two orbit sums will appear in the right hand side of (5.1), namely

P_{\underbrace{d, …, d,}_{pq + s − 2} \underbrace{d − 1, …, d − 1}_{p − s + 1}}

and

P_{\underbrace{d, …, d,}_{pq − 1 + s} \underbrace{d − 1, …, d − 1}_{p − s − 1}, d−2}

with coefficients

k_{\underbrace{d, …, d,}_{pq + s − 2} \underbrace{d − 1, …, d − 1}_{p − s + 1}} = \binom{pq + s − 2}{s − 1}(p − s + 1)

and

k_{\underbrace{d, …, d,}_{pq + s − 1} \underbrace{d − 1, …, d − 1}_{p − s − 1}, d−2} = \binom{pq + s − 1}{s} ,

respectively. (The second orbit sum does not occur if s = p.) Both of these coefficients are divisible by p and hence, are 0 in 𝔽. This completes the proof of Theorem 6.1. □

Corollary 6.3. Suppose (i) p ≤ n ≤ 2p − 2 and d ≥ 1 or (ii) n = 2p − 1 and d ≥ 2. Then the ideal I_{n,d} can be generated by n+1 elements of R_{n} but cannot be generated by n elements.

Proof. Theorem 5.1(b) tells us that I_{n,d} is generated by n + 1 elements. If I_{n,d} could be generated by n elements then by Lemma 6.2(c) these n elements would form a regular sequence in R_{n}, contradicting Theorem 6.1(b). □

Remark 6.4. The conditions that d ≥ 1 and d ≥ 2 in parts (i) and (ii) of Theorem 6.1(b) respectively, cannot be dropped. The same goes for conditions (i) and (ii) in Corollary 6.3.

Indeed, suppose d = 0. Recall that R_{n} = 𝔽[x_{1}, …, x_{n}]^{S^{n}} is a polynomial algebra 𝔽[s_{1}, …, s_{n}], where s_{1} = P_{1}, s_{2} = P_{1,1}, etc., are the elementary symmetric polynomials in x_{1}, …, x_{n}. I_{n,0} is clearly the maximal ideal of R_{n} generated by the regular sequence s_{1}, …, s_{n}. Thus Theorem 6.1(b) fails if d = 0.

Now suppose d = 1 and n = 2p − 1, where char(𝔽) = p. By Theorem 5.1(b), I_{n,1} is generated by the n + 1 elements P_{2}, …, P_{n−1}, P_{n+1} and P^{2, …, 2}_{p times}.

Since we are in characteristic p, P_{n+1} = P_{2p} = P_{2}^{p}, is a redundant generator. In other words, I_{n,1} is generated by the n elements P_{2}, …, P_{n−1}, P_{n} and P^{2, …, 2}_{p times}. By Lemma 6.2(c)

these elements form a regular sequence in R_{n}. This shows that Theorem 6.1(b) fails for d = 1 and n = 2p − 1. □

Acknowledgments

The authors are grateful to L. Avramov, W. Dwyer, G. Lyubeznik, B. Sturmfels and J. Weyman for their helpful comments.

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Department of Mathematics University of British Columbia, Vancouver BC V6T 1Z2, Canada

E-mail address: adem@math.ubc.ca

Department of Mathematics University of British Columbia, Vancouver BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca