BASIC POLYNOMIAL INVARIANTS, FUNDAMENTAL REPRESENTATIONS AND THE CHERN CLASS MAP

S. BAEK, E. NEHER, AND K. ZAINOULLINE

Introduction

Consider a crystallographic root system Φ together with its Weyl group W acting on the weight lattice Λ of Φ. Let ℤ[Λ]^{W} and S^{∗}(Λ)^{W} be the W -invariant subrings of the integral group ring ℤ[Λ] and the symmetric algebra S^{∗}(Λ). A celebrated theorem of Chevalley says that ℤ[Λ]^{W} is a polynomial ring over ℤ in classes of fundamental representations ρ_{1}, …, ρ_{n} and S^{∗}(Λ)^{W} ⊗ ℚ is a polynomial ring over ℚ in basic polynomial invariants q_{1}, …, q_{n}, where n = rank(Φ).

In the present paper we establish and investigate the relationship between ρ_{i}’s and q_{i}’s. To do this we introduce an equivariant analogue of the Chern class map ϕ_{i} that provides an isomorphism between the truncated rings ℤ[Λ]/I_{m}^{j} and S^{∗}(Λ)/I_{a}^{j} modulo powers of the respective augmentation ideals. This allows us to express basic polynomial invariants in terms of fundamental representations and vice versa, hence relating the geometry of the variety of Borel subgroups X with representation theory of the respective Lie algebra 𝔤.

A multiple of ϕ_{i} restricted to the respective cohomology (K_{0} and CH^{∗}) of X gives the classical Chern class map c_{i}: K_{0}(X) → CH^{i}(X). This geomeric interpretation provides a powerful tool to compute the annihilators of the torsion of the Grothendieck γ-filtration on K_{0} of twisted forms of X as well as a tool to estimate the torsion part of its Chow groups in small codimensions.

The paper is organized as follows. In the first section we introduce the I-adic filtrations on ℤ[Λ] and S^{∗}(Λ) together with an isomorphism ϕ_{i} on their truncations. Then we study the subrings of invariants and introduce the key notion of an exponent τ_{i} of a W -action on a free abelian group Λ. Roughly speaking, the integers τ_{i} measure how far is the ring S^{∗}(Λ)^{W} (with integer coefficients) from being a polynomial ring in q_{i}’s. In section 5 we compute all the exponents up to degree 4 and show that they all coincide with the Dynkin index of the Lie algebra 𝔤. In section 6 we apply the obtained results to estimate the torsion in Grothendieck γ-filtration of some twisted flag varieties. In section 7 we compute the second exponent τ_{2} for a non-crystallographic group H_{2}. Acknowledgments. The first author has been partially supported from the NSERC grants of the other two authors and from the Fields Institute. The second author gratefully acknowledges support through NSERC Discovery grant 8836-20121. The last author has been supported by the NSERC Discovery grant 385795-2010, Accelerator Supplement 396100-2010 and an Early Researcher Award (Ontario).

2000 Mathematics Subject Classification. Primary 13A50; Secondary 14L24.

Key words and phrases. Dynkin index, polynomial invariant, fundamental representation, invariant theory, Chern class map, finite reflection group.

1. Two filtrations

Consider the two covariant functors S^{∗}(−) and ℤ[−] from the category of abelian groups to the category of commutative rings

S^{∗}(−): Λ ↦ S^{∗}(Λ) and ℤ[−]: Λ ↦ ℤ[Λ]

given by taking the symmetric algebra of an abelian group Λ and the integral group ring of Λ respectively. The ith graded component S^{i}(Λ) is additively generated by monomials λ_{1}λ_{2} … λ_{i} with λ_{j} ∈ Λ and the ring ℤ[Λ] is additively generated by exponents e^{λ}, λ ∈ Λ.

The trivial group homomorphism induces the ring homomorphisms

ϵ_{a}: S^{∗}(Λ) → ℤ and ϵ_{m}: ℤ[Λ] → ℤ

called the augmentation maps. By definition ϵ_{a} sends every element of positive degree to 0 and ϵ_{m} sends every e^{λ} to 1. Let I_{a} and I_{m} denote the kernels of ϵ_{a} and ϵ_{m} respectively. Observe that I_{a} = S^{>0}(Λ) consists of elements of positive degree and I_{m} is generated by differences (1 − e^{−λ}), λ ∈ Λ. Consider the respective I-adic filtrations:

S^{∗}(Λ) = I_{a}^{0} ⊇ I_{a} ⊇ I_{a}^{2} ⊇ … and ℤ[Λ] = I_{m}^{0} ⊇ I_{m} ⊇ I_{m}^{2} ⊇ …

and let

gr_{a}^{∗}(Λ) = ⨁_{i≥0} I_{a}^{i}/I_{a}^{i+1} and gr_{m}^{∗}(Λ) = ⨁_{i≥0} I_{m}^{i}/I_{m}^{i+1}

denote the associated graded rings. Observe that gr_{a}^{∗}(Λ) = S^{∗}(Λ).

1.1. Example. If Λ ≃ ℤ, then the ring S^{∗}(Λ) can be identified with the polynomial ring in one variable ℤ[ω], where ω is a generator of Λ and the ring ℤ[Λ] can be identified with the Laurent polynomial ring ℤ[x, x^{−1}] where x = e^{ω}. The augmentations ϵ_{a} and ϵ_{m} are given by

ϵ_{a}: ω ↦ 0 and ϵ_{m}: x ↦ 1.

We have I_{a} = (ω) and I_{m} is additively generated by differences (1 − x^{n}), n ∈ ℤ.

Note that the rings ℤ[ω] and ℤ[x, x^{−1}] are not isomorphic, however they become isomorphic after the truncation. Namely for every i ≥ 0 there is ring isomorphism

ϕ_{i}: ℤ[x, x^{−1}]/I_{m}^{i+1} →^{≃} ℤ[ω]/I_{a}^{i+1}

defined by ϕ_{i}: x ↦ (1 − ω)^{−1} = 1 + ω + … + ω^{i} with the inverse defined by ϕ_{i}^{−1} : ω ↦ 1 − x^{−1}. It is useful to keep the following picture in mind

\xymatrix{ℤ[x, x^{−1}] \ar@{->>}[d] \ar[dr] & ℤ[ω] \ar@{->>}[d] \ar[l] \\ ℤ[x, x^{−1}]/I_{m}^{i+1} \ar[r]^{ϕ_{i}}_{≃} & ℤ[ω]/I_{a}^{i+1}}

observing that the inverse ϕ_{i}^{−1} can be lifted to the map ℤ[ω] → ℤ[x, x^{−1}] but ϕ_{i} can’t.

The example can be generalized as follows:

1.2. Lemma. [GaZ, 2.1] Assume that Λ is a free abelian group of finite rank n. The rings ℤ[Λ] and S^{∗}(Λ) become isomorphic after truncation. Namely, if {ω_{1}, …, ω_{n}} is a ℤ-basis of Λ, then for every i ≥ 0 there is a ring isomorphism

ϕ_{i}: ℤ[Λ]/I_{m}^{i+1} →^{≃} S^{∗}(Λ)/I_{a}^{i+1}

defined by ϕ_{i}(1) = 1 and

ϕ_{i}(e^{∑_{j=1}^{n} a_{j}ω_{j}}) = ∏_{j=1}^{n}(1 − ω_{j})^{−a_{j}}

with the inverse defined by ϕ_{i}^{−1}(ω_{j}) = 1 − e^{−ω_{j}}.

Note that the map ϕ_{i} preserves the I-adic filtrations. Indeed, by definition ϕ_{i}(I_{m}^{j}) ⊆ I_{a}^{j} for every 0 ≤ j ≤ i. Moreover, we have the following

1.3. Lemma. The isomorphism ϕ_{i} restricted to the subsequent quotients I_{m}^{i}/I_{m}^{i+1} doesn’t depend on the choice of a basis of Λ. Hence, there is an induced canonical isomorphism of graded rings

ϕ_{∗} = ⊕_{i≥0}ϕ_{i} : gr_{m}^{∗}(Λ) ⟶^{≃} gr_{a}^{∗}(Λ) = S^{∗}(Λ).

Proof. Indeed, in this case we can define the inverse ϕ_{i}^{−1} : I_{a}^{i}/I_{a}^{i+1} → I_{m}^{i}/I_{m}^{i+1} by

ϕ_{i}^{−1}(λ_{1}λ_{2} … λ_{i}) = (1 − e^{−λ_{1}})(1 − e^{−λ_{2}}) … (1 − e^{−λ_{i}}).

It is well-defined since (1 − e^{−λ−λ′}) = (1 − e^{−λ}) + (1 − e^{−λ′}) modulo I_{m}^{2}. □

Consider the composite of the map ϕ_{i} with the projections

ϕ^{(i)} : ℤ[Λ] → ℤ[Λ]/I_{m}^{i+1} ⟶^{ϕ_{i}} S^{∗}(Λ)/I_{a}^{i+1} → S^{i}(Λ).

The map ϕ^{(i)}, and therefore ϕ_{i}, can be computed on generators e^{λ}, λ ∈ Λ as follows: Let f(z) = ∏_{j}(1 − ω_{j}z)^{−a_{j}}, where λ = ∑_{j} a_{j}ω_{j}. Then

ϕ^{(i)}(e^{∑_{j} a_{j}ω_{j}}) = \frac{1}{i!} \frac{d^{i}f(z)}{dz^{i}} |_{z=0}

To compute the derivatives of f(z) we observe that f′(z) = f(z)g(z), where g(z) =

∑_{j} a_{j}ω_{j}(1 − ω_{j}z)^{−1} and \frac{d^{i} g(z)}{d z^{i}} = ∑_{j} \frac{i! a_{j}ω_{j}^{i+1}}{(1−ω_{j}z)^{i+1}} . Hence, starting with g_{0} = 1 we obtain the following recursive formulas

\frac{d^{i} f(z)}{d z^{i}} = f(z) · g_{i}(z), where g_{i}(z) = g(z)g_{i−1}(z) + g′_{i−1}(z).

1.4. Example. For small values of i we obtain

\begin{matrix}i & i! · ϕ^{(i)}(e^{λ}) = \\ 1 & λ \\ 2 & λ^{2} + λ(2) \\ 3 & λ^{3} + 3λ(2)λ + 2λ(3) \\ 4 & λ^{4} + 6λ(4) + 6λ(2)λ^{2} + 8λ(3)λ + 3λ(2)^{2} \end{matrix}

where given a presentation λ = ∑_{j=1}^{n} a_{j,λ}ω_{j}, a_{j,λ} ∈ ℤ in terms of the basis {ω_{1}, ω_{2}, ….ω_{n}}, the character λ(m), m ≥ 1 is defined by

λ(m) = ∑_{j=1}^{n} a_{j,λ}ω_{j}^{m}.

2. Invariants and exponents

Let W be a finite group which acts on a free abelian group Λ of finite rank by ℤ-linear automorphisms. Consider the induced action of W on ℤ[Λ] and S^{∗}(Λ). Observe that it is compatible with the I-adic filtrations, i.e. W (I_{m}^{i}) ⊆ I_{m}^{i} and W (I_{a}^{i}) ⊆ I_{a}^{i} for every i ≥ 0.

Note that the isomorphisms ϕ_{i} and ϕ_{i}^{−1} are not necessarily W -equivariant. However, by Lemma 1.3 their restrictions to the subsequent quotients I_{m}^{i}/I_{m}^{i+1} and I_{a}^{i}/I_{a}^{i+1} = S^{i}(Λ) are W -equivariant and we have

(I_{m}^{i}/I_{m}^{i+1})^{W} ≃ (I_{a}^{i}/I_{a}^{i+1})^{W} .

Let I_{m}^{W} denote the ideal of ℤ[Λ] generated by W -invariant elements from the augmentation ideal I_{m}, i.e., by elements from ℤ[Λ]^{W} ∩I_{m}. Similarly, let I_{a}^{W} denote the ideal of S^{∗}(Λ) generated by W -invariant elements from I_{a}, i.e., by elements from S^{∗}(Λ)^{W} ∩ I_{a}.

For each χ ∈ Λ let ρ(χ) = ∑_{λ∈W (χ)} e^{λ} denote the sum over all elements of the W -orbit of χ. Every element in I_{m}^{W} can be written as a finite linear combination with integer coefficients of the elements ρ̂(χ) = ρ(χ) − ϵ_{m}(ρ(χ)), χ ∈ Λ. Therefore, the ideal I_{m}^{W} is generated by the elements ρ̂(χ), i.e.,

I_{m}^{W} = ⟨ρ̂(χ) | χ ∈ Λ⟩.

The image of I_{m}^{W} by means of the composite

ℤ[Λ] → ℤ[Λ]/I_{m}^{i+1} ⟶^{ϕ_{i}} S^{∗}(Λ)/I_{a}^{i+1} .

is an ideal in S^{∗}(Λ)/I_{a}^{i+1} generated by the elements ϕ_{i}(ρ̂(χ)), χ ∈ Λ. Therefore, the image of I_{m}^{W} in S^{i}(Λ) is the ith homogeneous component of the ideal generated by ϕ^{(j)}(ρ̂(χ)), where 1 ≤ j ≤ i, χ ∈ Λ, i.e.

ϕ^{(i)}(I_{m}^{W}) = ⟨f · ϕ^{(j)}(ρ̂(χ)) | 1 ≤ j ≤ i, f ∈ S^{i−j}(Λ), χ ∈ Λ⟩_{ℤ} .

We are ready now to introduce the central notion of the present paper:

2.1. Definition. We say that an action of W on Λ has finite exponent in degree i if there exists a non-zero integer N_{i} such that

N_{i} · (I_{a}^{W})^{(i)} ⊆ ϕ^{(i)}(I_{m}^{W}),

where (I_{a}^{W})^{(i)} = I_{a}^{W} ∩ S^{i}(Λ). In this case the g.c.d. of all such N_{i}s will be called the i-th exponent of the W -action and will be denoted by τ_{i}.

Observe that if ϕ^{(i)}(I_{m}^{W}) is a subgroup of finite index in (I_{a}^{W})^{(i)}, then τ_{i} is simply the exponent of ϕ^{(i)}(I_{m}^{W}) in (I_{a}^{W})^{(i)}. Note also that by the very definition τ_{0} = 1 and τ_{i} | τ_{i+1} for every i ≥ 0.

3. Essential actions

In the present section we study W -actions that have no W -invariant linear forms, i.e. we assume that Λ^{W} = 0. In the theory of reflection groups such actions are called essential (see [B, V, §3.7] or [H]). Note that this immediately implies that τ_{1} = 1.

3.1. Lemma. For every χ ∈ Λ and m ∈ ℕ_{+} we have ∑_{λ∈W (χ)} λ(m) = 0.

Proof. Let ω_{1}, ω_{2}, ….ω_{n} be a ℤ-basis of Λ. For m ∈ ℕ_{+} we have

∑_{λ∈W (χ)} λ(m) = ∑_{λ∈W (χ)} ( ∑_{j=1}^{n} a_{j,λ}ω_{j}^{m}) = ∑_{j=1}^{n} ( ∑_{λ∈W (χ)} a_{j,λ})ω_{j}^{m}.

In particular, for m = 1 we obtain

∑_{λ∈W (χ)} λ = ∑_{j=1}^{n} ( ∑_{λ∈W (χ)} a_{j,λ})ω_{i}.

Since Λ^{W} = 0, we have ∑_{λ∈W (χ)} λ = 0. Since ω_{j}, 1 ≤ j ≤ n are ℤ-free, we have ∑_{λ∈W (χ)} a_{j,λ} = 0 for all 1 ≤ j ≤ n. □

3.2. Corollary. For every χ ∈ Λ we have

ϕ^{(2)}(ρ(χ)) = \frac{1}{2} ∑_{λ∈W (χ)} λ^{2}.

In particular, the quadratic form ϕ^{(2)}(ρ(χ)) is W -invariant, i.e.

ϕ^{(2)}(ρ(χ)) ∈ S^{2}(Λ)^{W} .

Proof. By the formula for ϕ^{(2)} in Example 1.4 and by Lemma 3.1 we obtain that

ϕ^{(2)}( ∑_{λ∈W (χ)} e^{λ}) = \frac{1}{2} ∑_{λ∈W (χ)}(λ^{2} + λ(2)) = \frac{1}{2} ∑_{λ∈W (χ)} λ^{2}. □

3.3. Corollary. If S^{2}(Λ)^{W} = ⟨q⟩ for some q, then ϕ^{(2)}(I_{m}^{W}) is a subgroup of finite index in (I_{a}^{W})^{(2)}.

Proof. The image of the ideal I_{m}^{W} is generated by ϕ^{(1)}(ρ(χ)) and ϕ^{(2)}(ρ(χ)). Since Λ^{W} = 0, ϕ^{(1)}(ρ(χ)) = ∑_{λ∈W (χ)} λ = 0 and by Corollary 3.2, ϕ^{(2)}(I_{m}^{W}) is generated only by the W -invariant quadratic forms ϕ^{(2)}(ρ(χ)). For every χ ∈ Λ let

ϕ^{(2)}(ρ(χ)) = N_{χ} · q, N_{χ} ∈ ℕ.  (1)

Then the subgroup ϕ^{(2)}(I_{m}^{W}) is a subgroup of (I_{a}^{W})^{(2)} of exponent

τ_{2} = gcd_{χ∈Λ} N_{χ}. □

We now investigate the invariants of degree 3 and 4.

3.4. Lemma. For every χ ∈ Λ we have

ϕ^{(3)}(ρ(χ)) = \frac{1}{6} ∑_{λ∈W (χ)}(λ^{3} + 3λ(2)λ).

Proof. By the formula for ϕ^{(3)} in Example 1.4 and by Lemma 3.1 we obtain that

ϕ^{(3)}(ρ(χ)) = \frac{1}{6} ∑_{λ∈W (χ)}(λ^{3} + 3λ(2)λ + 2λ(3)) = \frac{1}{6} ∑_{λ∈W (χ)}(λ^{3} + 3λ(2)λ). □

3.5. Lemma. For every χ ∈ Λ we have

ϕ^{(4)}(ρ(χ)) = \frac{1}{24} ∑_{λ∈W (χ)}[λ^{4} + 6λ(2)λ^{2} + 8λ(3)λ + 3λ(2)^{2}].

Proof. It follows from Example 1.4 and Lemma 3.1. □

4. The Dynkin index

In the present section we show that the action of the Weyl group W of a crystallographic root system Φ on the weight lattice Λ has finite exponent in degree 2 which coincides with the Dynkin index of the respective Lie algebra.

Let W be the Weyl group of a crystallographic root system Φ and let Λ be its weight lattice as defined in [H, §2.9]. Let {ω_{1}, …, ω_{n}} be a basis of Λ consisting of fundamental weights (here n is the rank of Φ).

The Weyl group W acts on λ ∈ Λ by means of simple reflections

s_{j}(λ) = λ − ⟨α_{j}^{∨}, λ⟩ · α_{j}, j = 1 … n

where α_{j}^{∨} is the j-th simple coroot and ⟨−, −⟩ is the usual pairing. Note that ⟨α_{j}^{∨}, ω_{i}⟩ = δ_{ij}, where δ_{ij} is the Kronecker symbol.

The subring of invariants ℤ[Λ]^{W} is the representation ring of the respective Lie algebra 𝔤. By a theorem of Chevalley it is the polynomial ring in fundamental representations ρ(ω_{j}) ∈ ℤ[Λ]^{W}, i.e.

ℤ[Λ]^{W} ≃ ℤ[ρ(ω_{1}), …, ρ(ω_{n})].

Observe that the dimension of the fundamental representation ρ(ω_{j}) equals to the number of elements in the orbit that is ϵ_{m}(ρ(ω_{j})).

Therefore, the ideal I_{m}^{W} is generated by the elements ρ̂(ω_{j}), j = 1 … n and its image ϕ^{(i)}(I_{m}^{W}) is the i-th homogeneous component of the ideal generated by ϕ^{(j)}(ρ(ω_{l})), 1 ≤ j ≤ i, l = 1 … n.

4.1. Lemma. We have Λ^{W} = 0 and hence also

ϕ^{(1)}(ℤ[Λ]^{W}) = ϕ^{(1)}(I_{m}^{W}) = 0.

Proof. Let η ∈ Λ^{W}. Since η = s_{α_{j}}(η) = η − ⟨η, αj^{∨}⟩α_{j} we have ⟨η, αj^{∨}⟩ = \frac{2(α_{j},η)}{(α_{j},α_{j})} = 0

for all simple roots α_{j} which implies that η = 0. □

4.2. Lemma. We have S^{2}(Λ)^{W} = ⟨q⟩.

Proof. By [GN, Prop. 4] there exists an integer valued W -invariant quadratic form on Λ which has value 1 on short coroots. As the group S^{2}(Λ)^{W} is identical to the group of all integral W -invariant quadratic forms on T_{∗} ⊗ ℝ, the result follows. □

4.3. Corollary. The image ϕ^{(2)}(I_{m}^{W}) is a subgroup of (I_{a}^{W})^{(2)} of finite index.

Proof. This follows from Corollary 3.3 and Lemma 4.1. □

We recall briefly the notion of indices of representations introduced by Dynkin [D, §2] (See also [BR]).

Let f : 𝔤 → 𝔤′ be a morphism between Lie algebras. Then there exists a unique number j_{f} ∈ ℂ, called the Dynkin index of f, satisfying

(f(x), f(y)) = j_{f}(x, y),

for all x, y ∈ 𝔤, where (–,–) is the Killing form on 𝔤 and 𝔤′ normalized such that (α, α) = 2 for any long root α. In particular, if f : 𝔤 → 𝔰𝔩(V) is a linear representation, j_{f} is a positive integer, called the Dynkin index of the linear representation f, defined by

tr(f(x), f(y)) = j_{f}(x, y).

The Dynkin index of 𝔤 is defined to be the greatest common divisor of all the Dynkin indices of all linear representations of 𝔤. By [D, (2.24) and (2.25)], the Dynkin index of 𝔤 is the greatest common divisor of the Dynkin index of its fundamental representations. Moreover, all the Dynkin indices of the fundamental representations were calculated in [D, Table 5].

Using the 𝔰𝔩_{2}-representation theory, the Dynkin index of a linear representation f : 𝔤 → 𝔰𝔩(V) can be described as follows. Let α be a long root. For the formal character ch(V) = ∑_{λ} n_{λ}e^{λ}, one has (see [LS, Lemma 2.4] or [KNR, 5.1 and Lemma 5.2])

j_{f} = \frac{1}{2} ∑_{λ}⟨λ, α^{∨}⟩^{2}.  (2)

4.4. Theorem. The integers N(ω_{j}) for the j-th fundamental weight as defined in (1) coincide with the Dynkin index of the fundamental representation with highest weight ω_{j}. In particular, the second exponent τ_{2} coincides with the Dynkin index of 𝔤.

Proof. To find the precise value of τ_{2} we use the explicit formula for ϕ^{(2)}, that is

ϕ^{(2)}(ρ(χ)) = \frac{1}{2} ∑_{λ∈W (χ)} λ^{2}.

We know that τ_{2} is the greatest common divisor of the integers N_{j} = N_{ω_{j}} using the notation of the proof of Corollary 3.3, where ω_{j} is the j-th fundamental weight of 𝔤. As the Dynkin index is the greatest common divisor of the Dynkin indices of the fundamental representations ω_{j}, it suffices to show that N_{j} coincides with the Dynkin index of the representation V_{j} corresponding to ω_{j}. We can view ϕ^{(2)}(ρ(χ)) for χ = ω_{j} as a function on the lattice 𝔥_{ℤ} = Span_{ℤ}{α^{∨} | α ∈ Φ long}. Since V_{j} has character ch(V_{j}) = ∑_{λ∈W (ω_{j})} e^{λ}, by (2) the Dynkin index of the representation V_{j} is \frac{1}{2} ∑_{λ∈W (ω_{j})}⟨λ, α^{∨}⟩^{2}, where α is any long root in Φ. Thus, ϕ^{(2)}(ρ(ω_{j})) is the constant function with value N_{j}. □

We note that a different proof of Theorem 4.4 was given in [GaZ, §2].

5. Exponents of degrees 3 and 4

In the present section we show that τ_{2} = τ_{3} = τ_{4} for all crystallographic root systems

Let S = {λ_{1}, …, λ_{r}} be a finite set of weights. We denote by −S the set of opposite weights {−λ_{1}, …, −λ_{r}}, by S_{+} the set of sums {λ_{i} + λ_{j}}_{i<j}, by S_{−} the set of differences {λ_{i} − λ_{j}}_{i<j} and by S_{±} the disjoint union S_{+} ⨿ S_{−}. By definition we have |S_{+}| = |S_{−}| = \binom{r}{2}.

Using the fact that (λ + λ′)(m) = λ(m) + λ′(m) for every λ, λ′ ∈ Λ and m ≥ 0 we obtain the following lemma which will be extensively used in the computations

5.1. Lemma. (i) For every integer m_{1}, m_{2}, x, y ≥ 0 and a finite subset S ⊂ Λ we have

∑_{λ∈S⨿−S} λ(m_{1})^{x}λ(m_{2})^{y} = (1 + (−1)^{x+y}) ∑_{λ∈S} λ(m_{1})^{x}λ(m_{2})^{y}.

In particular, ∑_{λ∈S⨿−S} λ(2)λ^{2} = 0.

(ii) For every subset S ⊂ Λ with |S| = r and for every m_{1}, m_{2} ≥ 0 we have

∑_{λ∈S_{+}} λ(m_{1})λ(m_{2}) = (r − 1) ∑_{λ∈S} λ(m_{1})λ(m_{2}) + ∑_{i≠j} λ_{i}(m_{1})λ_{j}(m_{2}) and

∑_{λ∈S_{−}} λ(m_{1})λ(m_{2}) = (r − 1) ∑_{λ∈S} λ(m_{1})λ(m_{2}) − ∑_{i≠j} λ_{i}(m_{1})λ_{j}(m_{2}).

In particular, this implies that ∑_{λ∈S_{±}} λ(m_{1})λ(m_{2}) = 2(r − 1) ∑_{λ∈S} λ(m_{1})λ(m_{2}).

A_{n}-case. Let Φ be of type A_{n} for n ≥ 3. We denote the canonical basis of ℝ^{n+1} by e_{i} with 1 ≤ i ≤ n + 1. According to [H, §3.5 and §3.12] the basic polynomial invariants of the W -action on Λ (algebraically independent homogeneous generators of S^{∗}(Λ)^{W} as a ℚ-algebra) are given by the symmetric power sums

q_{i} := e_{1}^{i} + ⋯ + e_{n+1}^{i}, 2 ≤ i ≤ n + 1.

Let s_{i} denote the ith elementary symmetric function in e_{1}, …, e_{n+1}. Using the classical identities

q_{1} = s_{1}, q_{i} = s_{1}q_{i−1} − s_{2}q_{i−2} + … + (−1)^{i}s_{i−1}q_{1} + (−1)^{i+1}i · s_{i}, 1 < i < n + 1

and the fact that s_{1} = 0, we obtain that

q_{2}/2 = −s_{2}, q_{3}/3 = s_{3}, and q_{4}/2 = s_{2}^{2} − 2s_{4}.

generate (with integral coefficients) the ideal I_{a}^{W} up to degree 4.

The fundamental weights of Φ can be expressed as follows

ω_{1} = e_{1}, ω_{2} = e_{1} + e_{2}, …, ω_{n−1} = e_{1} + … + e_{n−1}, ω_{n} = −e_{n+1},

where e_{1} + e_{2} + … + e_{n+1} = 0. The orbits of ω_{1}, ω_{1} + ω_{n}, ω_{n} and ω_{2}, ω_{n−1} under the action of the Weyl group W = S_{n+1} are given by

W (ω_{1}) = {e_{1}, …, e_{n+1}} = −W (ω_{n}), W (ω_{1} + ω_{n}) = {e_{i} − e_{j}}_{i≠j} and
W (ω_{2}) = {e_{i} + e_{j}}_{i<j} = −W (ω_{n−1}).

Therefore, W (ω_{1} + ω_{n}) = S_{−} ⨿ −S_{−} and W (ω_{2}) = S_{+}, where S = W (ω_{1}). Applying Lemma 3.5 and Lemma 5.1 we obtain that

ϕ^{(4)}(ρ(ω_{1}) + ρ(ω_{n})) = \frac{1}{12} ∑_{λ∈S}(λ^{4} + 8λ(3)λ + 3λ(2)^{2}) and

ϕ^{(4)}(ρ(ω_{1} + ω_{n}) + ρ(ω_{2}) + ρ(ω_{n−1})) = \frac{1}{24} ∑_{λ∈S_{±}⨿−S_{±}}(λ^{4} + 8λ(3)λ + 3λ(2)^{2}) =

= \frac{1}{24} ∑_{λ∈S_{±}⨿−S_{±}} λ^{4} + \frac{n}{6} ∑_{λ∈S}(8λ(3)λ + 3λ(2)^{2}).

Then the difference

ϕ^{(4)}(ρ(ω_{1} + ω_{n}) + ρ(ω_{2}) + ρ(ω_{n−1})) − 2n · ϕ^{(4)}(ρ(ω_{1}) + ρ(ω_{n})) =
= \frac{1}{24} ∑_{λ∈S_{±}⨿−S_{±}} λ^{4} − \frac{n}{6} ∑_{λ∈S} λ^{4} =  (3)

is a symmetric function in e_{1}, …, e_{n+1} and, therefore, it can be always written as a polynomial in q_{i}s. Indeed, since

∑_{λ∈S_{±}⨿−S_{±}} λ^{4} = 2 ∑_{i<j}((e_{i} + e_{j})^{4} + (e_{i} − e_{j})^{4}) = 4n ∑_{λ∈S} λ^{4} + 24 ∑_{i<j} e_{i}^{2}e_{j}^{2},

the difference (3) equals

= ∑_{i<j} e_{i}^{2}e_{j}^{2} = (q_{2}^{2} − q_{4})/2.

5.2. Lemma. For a root system of type A_{n}, n ≥ 2, we have τ_{2} = τ_{3} = τ_{4} = 1.

Proof. It is enough to show that the generators q_{2}/2, q_{3}/3 and q_{4}/2 are in the ideal generated by the image of ϕ^{(i)}, i ≤ 4.

By Corollary 3.2 we have ϕ^{(2)}(ρ(ω_{1})) = \frac{1}{2} ∑_{λ∈S} λ^{2} = q_{2}/2. By Lemma 3.4 we have q_{3}/3 = ϕ^{(3)}(ρ(ω_{1}))−ϕ^{(3)}(ρ(ω_{n})) (see also [GaZ, §1C]). If Φ is of type A_{2}, then s_{4} = 0 and, hence, q_{4} = q_{2}^{2}/2. If Φ is of type A_{n}, n ≥ 3, then by (3) the generator q_{4}/2 belongs to the ideal generated by the images of ϕ^{(2)} and ϕ^{(4)}. □

5.3. Lemma. For any crystallographic root system Φ the third exponent τ_{3} of the W -action coincides with τ_{2} (the Dynkin index).

Proof. If Φ is of type A_{n}, this follows from Lemma 5.2; for the other types there are no basic polynomial invariants of degree 3 [H, §3.7 Table 1]. Therefore, τ_{3} = τ_{2}. □

B_{n}, C_{n} and D_{n} cases. Let Φ be of type B_{n} or C_{n} for n ≥ 2 or of type D_{n} for n ≥ 4. We denote the canonical basis of ℝ^{n} by e_{i} with 1 ≤ i ≤ n. By [H, §3.5 and §3.12] the basic polynomial invariants of the W -action on Λ are given by even power sums

q_{2i} := e_{1}^{2i} + ⋯ + e_{n}^{2i}, 1 ≤ i ≤ n.

The first two fundamental weights of Φ are given by ω_{1} = e_{1}, ω_{2} = e_{1} + e_{2} and their W -orbits are

W (ω_{1}) = {±e_{1}, …, ±e_{n}} and W (ω_{2}) = {±e_{i} ± e_{j}}_{i<j}.

Hence W (ω_{1}) = S ⨿ −S and W (ω_{2}) = S_{±} ⨿ −S_{±}, where S = {e_{1}, …, e_{n}}. Applying Lemma 3.5 and Lemma 5.1 we obtain that

ϕ^{(4)}(ρ(ω_{1})) = \frac{1}{12} ∑_{λ∈S} λ^{4} + \frac{1}{12} ∑_{λ∈S}(8λ(3)λ + 3λ(2)^{2}) and

ϕ^{(4)}(ρ(ω_{2})) = \frac{1}{24} ∑_{λ∈S_{±}⨿−S_{±}} λ^{4} + \frac{n−1}{6} ∑_{λ∈S}(8λ(3)λ + 3λ(2)^{2}).

Then similar to the A_{n}-case we obtain

ϕ^{(4)}(ρ(ω_{2})) − 2(n − 1)ϕ^{(4)}(ρ(ω_{1})) = (q_{2}^{2} − q_{4})/2,  (4)

where q_{i} = e_{1}^{i} + … + e_{n}^{i}.

5.4. Lemma. For a root system of type B_{n} or C_{n}, n ≥ 2 or D_{n}, n ≥ 4 we have τ_{4} = τ_{2}.

Proof. It is enough to show that q_{4}/2 is in the ideal generated by the image of ϕ^{(2)} and ϕ^{(4)}.

By Corollary 3.2 we have ϕ^{(2)}(ρ(ω_{1})) = ∑_{λ∈S} λ^{2} = q_{2}. Therefore, by (4)

q_{4}/2 = (q_{2}/2) · ϕ^{(2)}(ρ(ω_{1})) − ϕ^{(4)}(ρ(ω_{2})) + 2(n − 1)ϕ^{(4)}(ρ(ω_{1}))

and the proof is finished. □

5.5. Theorem. For any crystallographic root system Φ we have τ_{2} = τ_{3} = τ_{4}.

Proof. The equality τ_{2} = τ_{3} is proven in Lemma 5.3. If Φ is of type A_{n}, τ_{4} = 1 follows from Lemma 5.2. If Φ is of type B_{n}, C_{n} or D_{n}, τ_{4} = τ_{2} follows from Lemma 5.4. For all other types τ_{4} = τ_{2} since there are no basic polynomial invariants of degree 3 and 4 (see [H, §3.7 Table 1]). □

6. Torsion in the Grothendieck γ-filtration

The goal of the present section is to provide geometric interpretation (see (6)) of the map ϕ_{i} and the exponents τ_{i}.

Let G be a simple simply-connected Chevalley group over a field k. We fix a maximal split torus T of G and a Borel subgroup B ⊃ T. Let Λ be the group of characters of T. Since G is simply-connected, Λ coincides with the weight lattice of G.

Let X denote the variety of Borel subgroups of G (conjugate to B). Consider the Chow ring CH^{∗}(X) of algebraic cycles modulo rational equivalence and the Grothendieck ring K_{0}(X). Following [De74, §1] to every character λ ∈ Λ we may associate the line bundle L(λ) over X. It induces the ring homomorphisms (called the characteristic maps)

𝔠_{a}: S^{∗}(Λ) → CH^{∗}(X) and 𝔠_{m}: ℤ[Λ] ↠ K_{0}(X)

by sending λ ↦ c_{1}(L(λ)) and e^{λ} ↦ [L(λ)] respectively. Note that the map 𝔠_{a} is an isomorphism in codimension one, hence, giving

𝔠_{a}: S^{1}(Λ) = Λ →^{≃} P ic(X) = CH^{1}(X)

and the map 𝔠_{m} is surjective. Let W be the Weyl group and let I_{a}^{W} and I_{m}^{W} denote the respective W -invariant ideals. Then according to [De73, §4 Cor.2,§9] and [CPZ, §6]

ker 𝔠_{m} = I_{m}^{W}  (5)

and ker 𝔠_{a} is generated by elements of S^{∗}(Λ) such that their multiples are in I_{a}^{W}.

Consider the Grothendieck γ-filtration on K_{0}(X) (see [GaZ, §1]). Its ith term is an ideal generated by products

γ^{i}(X) := ⟨(1 − [L_{1}^{∨}])(1 − [L_{2}^{∨}]) · . . . · (1 − [L_{i}^{∨}])⟩,

where L_{1}, L_{2}, …, L_{i} are line bundles over X. Consider the ith subsequent quotient γ^{i}(X)/γ^{i+1}(X). The usual Chern class c_{i} induces a group homomorphism c_{i}: γ^{i}(X)/γ^{i+1}(X) → CH^{i}(X).

6.1. Proposition. For every i ≥ 0 there is a commutative diagram of group homomorphisms

\xymatrix{I_{m}^{i}/I_{m}^{i+1} \ar[r]^{(−1)^{i−1}(i−1)!·ϕ_{i}} \ar@{->>}[d]^{𝔠_{m}} & S^{i}(Λ) \ar[d]^{𝔠_{a}} \\ γ^{i}(X)/γ^{i+1}(X) \ar[r]^{c_{i}} & CH^{i}(X)}  (6)

Proof. Indeed, the γ-filtration on K_{0}(X) is the image of the I_{m}-adic filtration on ℤ[Λ], i.e. γ^{i}(X) = 𝔠_{m}(I_{m}^{i}) for every i ≥ 0. The Proposition then follows from the identity

c_{i}((1 − [L_{1}^{∨}])(1 − [L_{2}^{∨}]) … (1 − [L_{i}^{∨}])) = (−1)^{i−1}(i − 1)! · c_{1}(L_{1})c_{1}(L_{2}) … c_{1}(L_{i}), where L_{1}, L_{2}, …, L_{i} are line bundles over X and L_{i}^{∨} denotes the dual of L_{i}. □

6.2. Remark. Note that ℤ[Λ] can be identitfied with the T -equivariant K_{0} of a point pt = Spec k and S^{∗}(Λ) with the T -equivariant CH of a point (see [GiZ]). The maps 𝔠_{a} and 𝔠_{m} then can be identified with the pull-backs K_{T} (pt) → K_{T} (G) and CH_{T} (pt) → CH_{T} (G) induced by the structure map G → pt.

In view of these identifications the map ϕ_{i} can be viewed as an equivariant analogue of the Chern class map c_{i}.

Consider the diagram (6) with ℚ-coefficients. In this case the Chern class map c_{i} will become an isomorphism (by the Riemann-Roch theorem), the characteristic map 𝔠_{a} will turn into a surjection and the map (−1)^{i−1}(i − 1)! · ϕ_{i} will be an isomorphism as well. In view of (5) we obtain an isomorphism

ϕ^{(i)} ⊗ ℚ: I_{m}^{W} ∩ I_{m}^{i}/I_{m}^{W} ∩ I_{m}^{i+1} ⊗ ℚ ⟶ (I_{a}^{W})^{(i)} ⊗ ℚ

on the kernels of 𝔠_{m} and 𝔠_{a}. By the very definition of the exponents τ_{i} this implies that

6.3. Corollary. The action of the Weyl group of a crystallograhic root system has finite exponent τ_{i} for every i.

We are now ready to prove the main result of this section

6.4. Theorem. The integer τ_{i} · (i − 1)! annihilates the torsion of the ith subsequent quotient γ^{i}(X)/γ^{i+1}(X) of the γ-filtration on K_{0}(X) for i = 3, 4.

6.5. Remark. Note that by [SGA, Exposé XIV, 4.5] for groups of types A_{n} and C_{n} the quotients γ^{i}(X)/γ^{i+1}(X) have no torsion.

Proof. Assume that α is a torsion element in γ^{i}(X)/γ^{i+1}(X). Then c_{i}(α) = 0 since CH^{i}(G/B) has no torsion. Let α̃ be a preimage of α via 𝔠_{m} in I_{m}^{i}/I_{m}^{i+1} ⊆ ℤ[Λ]/I_{m}^{i+1}. By the same analysis as in [GaZ, §1B, §1C] one can show that ker(𝔠_{a})^{(i)} = (I_{a}^{W})^{(i)} for i ≤ 4. By (6) we obtain that

(i − 1)! ϕ_{i}(α̃) ∈ (I_{a}^{W})^{(i)}

By definition of the index τ_{i} we have

τ_{i} · (i − 1)! ϕ_{i}(α̃) = ϕ_{i}(β), where β ∈ I_{m}^{W} /I_{m}^{i+1} ∩ I_{m}^{W} .

Applying ϕ_{i}^{−1} to the both sides we obtain

τ_{i} · (i − 1)! · α̃ = β ∈ I_{m}^{W} /I_{m}^{i+1} ∩ I_{m}^{W}

Applying 𝔠_{m} to the both sides and observing that I_{m}^{W} = ker 𝔠_{m} we obtain that τ_{i} · (i − 1)! · α = 0. □

Let _{ξ}X be a twisted form of the variety X by means of a cocycle ξ ∈ Z^{1}(k, G). By [P, Thm. 2.2.(2)] the restriction map K_{0}(_{ξ}X) → K_{0}(X) (here we identify K_{0}(X) with the K_{0}(X ×_{k} k̄) over the algebraic closure k̄) is an isomorphism. Since the characteristic classes commute with restrictions, this induces an isomorphism between the γ-filtrations, i.e. γ^{i}(_{ξ}X) ≃ γ^{i}(X) for every i ≥ 0, and between the respective quotients

γ^{i}(_{ξ}X)/γ^{i+1}(_{ξ}X) ≃ γ^{i}(X)/γ^{i+1}(X) for every i ≥ 0.

In view of this fact Theorem 6.4 imply that 6.6. Corollary. Let G be a split simple simply connected group of type B_{n} (n ≥ 3) or D_{n} (n ≥ 4). Then for every ξ ∈ Z^{1}(k, G) the torsion in γ^{4}(_{ξ}X)/γ^{5}(_{ξ}X) is annihilated by 12.

Consider the topological filtration on K_{0}(Y) given by the ideals

τ^{i}(Y) := ⟨[O_{V}] | V ↪ Y, codim_{V} Y ≥ i⟩.

It is known (see [GaZ, §2]) that γ^{i}(Y) ⊆ τ^{i}(Y) for every i ≥ 0.

6.7. Corollary. In the notation of Corollary 6.6 assume in addition that the induced map

γ^{4}(_{ξ}X)/γ^{5}(_{ξ}X) → τ^{4}(_{ξ}X)/τ^{5}(_{ξ}X)

is surjective. Then the 2-torsion of CH^{4}(_{ξ}X) is annihilated by 8.

Proof. By the Riemann-Roch theorem [F, Ex.15.3.6], the composition

CH^{4}(_{ξ}X) ↠ τ^{4}(_{ξ}X)/τ^{5}(_{ξ}X) →^{c_{4}} CH^{4}(_{ξ}X)

is the multiplication by (−1)^{4−1}(4 − 1)! = −6, where the first map is surjective. Hence, the torsion subgroup of CH^{4}(_{ξ}X) is annihilated by 72 and so the result follows. □

7. ‘The Dynkin index’ in the H_{2} case

Note that the notion of an exponent τ_{i} can be defined over a unique factorisation domain in the same way. As an example we compute the second exponent τ_{2} for the action of the Weyl group of of the non-crystallographic root system H_{2} over the base ring ℤ[\frac{1+\sqrt{5}}{2}], hence, giving rise to an interesting question about its geometric/Lie algebra interpretation.

7.1. Theorem. For the non-crystallographic root system H_{2} := I_{2}(5), the second exponent τ_{2} is \sqrt{5}.

Proof. We follow the notations in [CMP]. In the root system H_{2}, the Weyl group W is the dihedral group of order 10 and M is the ℤ[τ]-lattice generated by two simple roots α_{1} and α_{2}, where τ = (1 + \sqrt{5})/2. Observe that ℤ[τ] is an Euclidean domain.

The dual basis {ω_{1}, ω_{2}} is defined by

\{ \begin{matrix}ω_{1} = \frac{1}{3−τ} (2α_{1} + τα_{2}) \\ ω_{2} = \frac{1}{3−τ} (τα_{1} + 2α_{2}) \end{matrix} or \{ \begin{matrix}α_{1} = 2ω_{1} − τω_{2} \\ α_{2} = −τω_{1} + 2ω_{2} \end{matrix}

One computes the orbits of ω_{1} and ω_{2} as follows:

W (ω_{1}) = {ω_{1}, −ω_{2}, −ω_{1} + τω_{2}, −τω_{1} + ω_{2}, τω_{1} − τω_{2}},
W (ω_{2}) = −W (ω_{1}).

As the action of W on M is essential, by Corollary 3.2, we have

ϕ^{(2)}(ρ(ω_{2})) = ϕ^{(2)}(ρ(ω_{1})) = \frac{1}{2}(ω_{1}^{2} + ω_{2}^{2} + (ω_{1} − τω_{2})^{2} + (τω_{1} − ω_{2})^{2} + (τω_{1} − τω_{2})^{2})
= (1 + τ^{2})ω_{1}^{2} + (1 + τ^{2})ω_{2}^{2} − (2τ + τ^{2})ω_{1}ω_{2}.  (7)

Since ϕ^{(2)}(ρ(ω_{2})) is W -invariant by Corollary 3.2, we have

τ_{2} = gcd(1 + τ^{2}, 2τ + τ^{2}) = gcd(2 + τ, 2τ − 1).

But 2τ − 1 = \sqrt{5} is a prime in ℤ[τ], and we have 2 + τ = (2τ − 1)τ proving that τ_{2} = \sqrt{5}. □

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Sanghoon Baek, Department of Mathematics and Statistics, University of Ottawa

   E-mail address: sbaek@uottawa.ca

Erhard Neher, Department of Mathematics and Statistics, University of Ottawa

   E-mail address: erhard.neher@uottawa.ca

Kirill Zainoulline, Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa ON K1N 6N5, Canada

   E-mail address: kirill@uottawa.ca