The K-Theory of Versal Flags

and Cohomological Invariants of Degree 3

Sanghoon Baek, Rostislav Devyatov, Kirill Zainoulline

Received: February 22, 2017 Revised: May 12, 2017

Communicated by Nikita Karpenko

Abstract. Let G be a split semisimple linear algebraic group over a field and let X be a generic twisted flag variety of G. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring K_{0}(X) in terms of generators and relations in the case G = G^{sc}/μ_{2} is of Dynkin type A or C (here G^{sc} is the simply-connected cover of G); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.

2010 Mathematics Subject Classification: 14M17, 20G15, 14C35 Keywords and Phrases: linear algebraic group, twisted flag variety, ideal of invariants, versal torsor, cohomological invariant

1 Introduction

Let G be a split semisimple linear algebraic group over a field F. Let U/G be a classifying space of G in the sense of Totaro [18, Rem.1.4], i.e. U is an open G-invariant subset in some representation of G with U(F) ≠ ∅ and U → U/G is a G-torsor. Consider the generic fiber U′ of U over U/G. It is a G-torsor over the quotient field F′ of U/G called the versal G-torsor [4, Ch.I, §5]. We denote by X the respective flag variety U′/B over F′, where B is a Borel subgroup of G, and call it the versal flag. The variety X appears in many different contexts, e.g. related to cohomology of homogeneous G-varieties (see [6] for an arbitrary oriented theory; Karpenko [7], [8], [9] for Chow groups; Panin [17] for K-theory) and cohomological invariants of G (see Merkurjev [14] and [5], [15]). It can be viewed as a generic example of the so called twisted flag variety.

In the first part of the paper (Sections 2-4) we give an explicit presentation of the ring K_{0}(X) in terms of generators modulo a finite number of relations in cases when G = G^{sc}/μ_{2}, where G^{sc} is the product of simply-connected simple groups of Dynkin types A or C and μ_{2} is a central subgroup of order 2. Observe that for simply-connected G the ring K_{0}(X) can be identified with K_{0}(G/B) (e.g., see Panin [17]), and by Chevalley theorems there is a surjective characteristic map c: R(T_{sc}) → K_{0}(G/B) from the representation ring of the split maximal torus T_{sc} such that the kernel ker(c) = I_{sc}^{W} is generated by augmented classes of fundamental representations. So, all relations in K_{0}(X) correspond to W -orbits of fundamental weights. If G is not simply-connected (as in the G^{sc}/μ_{2}-case), then the situation changes dramatically as by [6, Ex.5.4] we have

K_{0}(X) ≃ R(T)/(I_{sc}^{W} ∩ R(T))

and a finite set of generators of I_{sc}^{W} ∩R(T) is not known in general. Note that by definition we have inclusions of abelian groups I^{W} ⊆ I_{sc}^{W} ∩R(T) ⊆ I_{sc}^{W} which all coincide if taken with ℚ-coefficients. However, there are examples of semisimple groups (see [15, Ex.3.1] and [1]) where both quotients (I_{sc}^{W} ∩ R(T))/I^{W} and I_{sc}^{W} /(I_{sc}^{W} ∩ R(T)) are non-trivial. Our Theorem 3.4 provides a complete list of generators (Definition 3.2) of the ideal I_{sc}^{W} ∩ R(T) assuming the root system of G^{sc} satisfies the generalized flatness condition (see Definition 2.9). In Section 4 we show that this condition holds for types A and C. In the second part of the paper we study cohomological invariants of degree 3 of G. According to Garibaldi-Merkurjev-Serre [4, p.106], a degree d cohomological invariant is a natural transformation of functors

a: H^{1}( · , G) → H^{d}( · , ℚ/ℤ(d − 1))

The K-Theory and Cohomological Invariants 1119

on the category of field extensions over F, where the functor H^{1}( · , G) classifies G-torsors, H^{d}( · , ℚ/ℤ(d − 1)) is the Galois cohomology. Following Merkurjev [14], an invariant is called decomposable if it is given by a cup-product of invariants of smaller degrees; the factor group of (normalized) invariants modulo decomposable is called the group of indecomposable invariants. For d = 3 the latter (denoted by Inv_{ind}^{3}(G)) has been computed for all simple split groups in [14] and [2]; for some semi-simple groups of type A in [13] and [1]; for adjoint semisimple groups in [12]. Another key subgroup of semi-decomposable invariants introduced in [15] consists of invariants given by a cup-product of invariants up to some field extensions. For d = 3 it coincides with the group of decomposable invariants for all simple groups [15]. It was also shown that these groups are different for G = SO_{4} [15, Ex.3.1] and for some semisimple groups of type A (see [1]). The relationships between the subgroups I^{W} ⊆ I_{sc}^{W} ∩ R(T) ⊆ I_{sc}^{W} and the groups of cohomological invariants are explained in Section 5. In Sections 6-11, we compute the groups of decomposable, indecomposable and semi-decomposable invariants of degree 3 for new examples of semisimple groups (e.g. G^{sc}/μ_{2}, products of adjoint groups), hence, extending the results of [14], [2], [1], [15], [13]; to compute semi-decomposable invariants we use the generators of Definition 3.2. We show that

• The factor group Inv_{sd}^{3}(G) of semi-decomposable invariants of G modulo decomposable is nontrivial if and only if G is of classical type A, B, C, D. Moreover, we compute both groups Inv_{sd}^{3}(G) and Inv_{ind}^{3}(G) for an arbitrary product of simply-connected simple groups of the same Dynkin type modulo central subgroups of order 2 (see Corollaries 6.5, 7.2, 8.2, 9.2, and Proposition 11.2).

• If G is of type A, then both these groups can have an arbitrary order and contain any direct product of cyclic p-group (see Corollary 6.6). If G is of type B or C, then it is always a product of cyclic groups of order 2 (see Corollaries 7.2, 8.2).

• The group Inv_{sd}^{3}(G) is trivial for the simple group G = PGO_{8}, i.e., any semi-decomposable invariant is decomposable (Corollary 10.4).

Acknowledgements. S.B. was partially supported by National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1C1B2010037). The first author also would like to thank the university of Ottawa for the warm hospitality during his sabbatical in the fall of 2016. R.D. was partially supported by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada. R.D. and K.Z. were partially supported by the NSERC Discovery grant RGPIN-2015-04469, Canada.

2 Syzygies and divisibility for Laurent polynomials

Let Λ be a free abelian group of rank n with a fixed basis {x_{1}, …, x_{n}}. Let R be one of the rings ℤ or ℤ/mℤ, m ≥ 2. Consider the group ring R[Λ]. It consists of finite linear combinations ∑_{j} a_{j}e^{λ_{j}}, a_{j} ∈ R, λ_{j} ∈ Λ. We identify R[Λ] with the Laurent polynomial ring R[x_{1}^{±1}, …, x_{n}^{±1}] via e^{x_{i}} ↦ x_{i} and e^{−x_{i}} ↦ x_{i}^{−1}. By a polynomial we mean always a Laurent polynomial, i.e., an element of R[Λ]. We denote by Λ_{i} a free subgroup with the basis {x_{1}, …, x_{i}}, 1 ≤ i < n. Hence, R[Λ_{i}] = R[x_{1}^{±1}, …, x_{i}^{±1}].

Definition 2.1. Given f ∈ R[Λ], we can express it uniquely as

f = f_{k}x_{n}^{k} + f_{k−1}x_{n}^{k−1} + … + f_{m}x_{n}^{m}, where f_{i} ∈ R[Λ_{n−1}], k, m ∈ ℤ, k ≥ m.

The integer k is called the highest degree of f with respect to x_{n} and denoted hdeg_{n}(f). The integer m is called the lowest degree of f with respect to x_{n} and denoted ldeg_{n}(f). The difference k − m is called the degree of f with respect to x_{n} and denoted wdeg_{n}(f).

By definition, if wdeg_{n}(f) = 0, then f is a product of x_{n}^{k} and a polynomial in x_{1}, …, x_{n−1}.

Definition 2.2. Let f, p ∈ R[Λ] and let ldeg_{n}(f) ≥ d for some d ∈ ℤ. We say that it is possible to perform a divison of f by p bounded by d if there exist monomials q, r ∈ R[Λ] such that

1. f = pq + r.

2. Either r = 0 or (ldeg_{n}(r) ≥ d and hdeg_{n}(r) < d + wdeg_{n}(p)).

In this case q is called the quotient, and r is called the remainder.

Definition 2.3. We call p ∈ R[Λ] a divisor with respect to x_{n} if it satisfies the following condition: In the presentation of Definition 2.1

p = p_{k}x_{n}^{k} + … + p_{m}x_{n}^{m}, p_{i} ∈ R[Λ_{n−1}], k, m ∈ ℤ, k ≥ m,

the leading coefficient p_{k} is a monic monomial in x_{1}, …, x_{n−1}.

Lemma 2.4. Let f, p ∈ R[Λ] and let ldeg_{n}(f) ≥ d for some d ∈ ℤ. If p is a divisor with respect to x_{n}, then it is possible to perform a division of f by p bounded by d.

Proof. We proceed by induction on hdeg_{n}(f). If hdeg_{n}(f) < d + wdeg_{n}(p), then we set q = 0 and r = f. Suppose that hdeg_{n}(f) ≥ d + wdeg_{n}(p). Since p is a divisor, we can write it as

p = Y x_{n}^{k} + p′, where Y ∈ R[Λ_{n−1}] is a monic monomial and

p′ is either 0 or a polynomial with hdeg_{n}(p′) < hdeg_{n}(p) = k and ldeg_{n}(p′) = ldeg_{n}(p). Observe that Y is invertible in R[Λ_{n−1}]. We write f as f = gx_{n}^{m} + f′, where m = hdeg_{n}(f), g ∈ R[Λ_{n−1}], and f′ is either 0, or a polynomial with hdeg_{n}(f′) < m and ldeg_{n}(f′) = ldeg_{n}(f). Set q_{0} = gY^{−1}x_{n}^{m−k}. Then Y x_{n}^{k}q_{0} = gx_{n}^{m}. If both f′ and p′ are 0, then Y x_{n}^{k} = p and gx_{n}^{m} = f, so pq_{0} = f, and we are done. Consider the polynomial f′′ = f′ − q_{0}p′. We have hdeg_{n}(q_{0}) = ldeg_{n}(q_{0}) = m − k. Recall that either p′ = 0 or (hdeg_{n}(p′) < k and ldeg_{n}(p′) = ldeg_{n}(p)). So, either q_{0}p′ = 0, or (hdeg_{n}(q_{0}p′) < m and ldeg_{n}(q_{0}p′) = m − k + ldeg_{n}(p) = m − wdeg_{n}(p)). Recall also that either f′ = 0, or (hdeg_{n}(f′) < m and ldeg_{n}(f′) = ldeg_{n}(f)). So, if p′ and f′ are not both 0, then hdeg_{n}(f′′) < m. Also, if p′ and f′ are not both 0, then ldeg_{n}(f′′) ≥ min(m−wdeg_{n}(p), ldeg_{n}(f)). We know that m = hdeg_{n}(f) ≥ d + wdeg_{n}(p), so m − wdeg_{n}(p) ≥ d. Also, ldeg_{n}(f) ≥ d. So, ldeg_{n}(f′′) ≥ d, and we can apply the induction hypothesis. By induction, there exist polynomials q_{1} and r such that f′′ = pq_{1} + r, and (either r = 0 or (ldeg_{n}(r) ≥ d and hdeg_{n}(r) < d + wdeg_{n}(p))). Set q = q_{0} + q_{1}. Then

pq + r = pq_{0} + pq_{1} + r = (Y x_{n}^{k} + p′)q_{0} + f′′

= Y x_{n}^{k}q_{0} + p′q_{0} + f′ − q_{0}p′ = gx_{n}^{m} + f′ = f.

Definition 2.5. (cf. [3, §15.5]) Given a n-tuple of polynomials q⃗ = (q_{1}, …, q_{n}), a n-tuple of polynomials f⃗ = (f_{1}, …, f_{n}) is called a syzygy of q⃗ if ∑_{i} f_{i}q_{i} = 0. Observe that syzygies form a submodule of a free module of rank n over R[Λ]. An element of a submodule generated by

S_{ij} = (0, …, q_{j}, …, −q_{i}, …, 0),

where q_{j} is at the position i and −q_{i} is at the position j, i, j = 1, …, n and i ≠ j, is called a trivial syzygy of q⃗.

Lemma 2.6. Let q⃗ = (q_{i}), q_{i} ∈ ℤ[Λ] and let q̄ = (q̄_{i}), where q̄_{i} ∈ ℤ/dℤ[Λ] is the reduction modulo d, d ≥ 2. If f⃗′ = (f′_{i}) is a trivial syzygy of q̄, then there exists a trivial syzygy f⃗ = (f_{i}) of q⃗ such that its reduction modulo d coincides with f⃗′, i.e., a trivial syzygy can be always lifted to ℤ.

Proof. Let S̄_{ij} be the reduction modulo d of S_{ij}. We have f⃗′ = ∑_{i,j} g′_{ij}S̄_{ij} for some g′_{ij} ∈ ℤ/dℤ[Λ]. Let g_{ij} be liftings of g′_{ij} to ℤ[Λ]. Set f⃗ = ∑_{i,j} g_{ij}S_{ij}.

Definition 2.7. We say that a n-tuple of polynomials (q_{1}, …, q_{n}) satisfies the flatness condition if q_{i} ∈ R[Λ_{i}] for each i = 1, …, n, and q_{i} is a divisor with respect to x_{i}.

Lemma 2.8. If an n-tuple of polynomials r⃗ satisfies the flatness condition, then all syzygies of r⃗ are trivial.

Proof. First, consider the case where R is a domain (i.e., R = ℤ or ℤ/pℤ with p prime). We use induction on n. If n = 1, then the trivial syzygy 0 is the only syzygy. Let f⃗ = (f_{1}, …, f_{n}) be a syzygy of r⃗ = (r_{1}, …, r_{n}) with n ≥ 2. By Lemma 2.4, we can divide f_{i} = r_{n}g_{i} + h_{i} with bound d = min{ldeg_{n}(f_{i})}, where ldeg_{n}(h_{i}) ≥ d and hdeg_{n}(h_{i}) < d + wdeg_{n}(r_{n}) for 1 ≤ i ≤ n. Since (g_{1}r_{1}, …, g_{n−1}r_{n−1}, − ∑_{i=1}^{n−1} g_{i}r_{i}) is a trivial syzygy of r⃗, it suffices to show that (h_{1}, …, h_{n−1}, f_{n}+∑_{i=1}^{n−1} g_{i}r_{i}) is a trivial syzygy of r⃗. If ∑_{i=1}^{n−1} h_{i}r_{i} = (f_{n} + ∑_{i=1}^{n−1} g_{i}r_{i})r_{n} is nonzero, then by taking wdeg_{n} of both sides, we obtain

wdeg_{n}(r_{n}) > wdeg_{n}(∑_{i=1}^{n−1} h_{i}r_{i}) = wdeg((f_{n} + ∑_{i=1}^{n−1} g_{i}r_{i})r_{n}) ≥ wdeg(r_{n}),

a contradiction. Thus, we have f_{n} +∑_{i=1}^{n−1} g_{i}r_{i} = 0 and it remains to show that h⃗ = (h_{1}, …, h_{n−1}, 0) is a trivial syzygy of r⃗. Let e = d + wdeg_{n}(r_{n}) − 1. Write h_{i} = h_{id}x_{n}^{d} + ⋯ + h_{ie}x_{n}^{e} for all 1 ≤ i ≤ n − 1 and h⃗_{j} = (h_{1j}, …, h_{(n−1)j}, 0) for all d ≤ j ≤ e. Then, we have h⃗ = h⃗_{d}x_{n}^{d} + ⋯ + h⃗_{e}x_{n}^{e}. By induction, all syzygies h⃗_{j} are trivial, so is h⃗. Now we consider the case R = ℤ/mℤ. We proceed by induction on the number of prime factors in m. If m is a prime, it follows from the previous case.

Write m = pl, l > 1, where p is a prime. Let f⃗ = (f_{i}) be a syzygy of r⃗ = (r_{i}) and let f̄ = (f̄_{i}) be the corresponding syzygy of r̄ = (r̄_{i}) over R̄ = ℤ/lℤ for 1 ≤ i ≤ n. By induction, we have

f̄ = ∑_{i,j} ḡ_{ij}S̄_{ij} for some ḡ_{ij} ∈ R̄[Λ].  (*)

Set

f⃗′ = f⃗ − ∑_{i,j} g_{ij}S_{ij}, where g_{ij} is a preimage of ḡ_{ij} in R[Λ].

By (*) we have f′_{i} = lf′′_{i} for some f′′_{i} ∈ R[Λ]. Since f⃗′ is a syzygy of r⃗, we have 0 = ∑_{i} f′_{i}r_{i} = l(∑_{i} f′′_{i}r_{i}) in R[Λ]. Thus, f⃗′′ = (f′′_{i}) is a syzygy of r⃗ modulo p. By the previous case, f⃗′′ is a trivial syzygy modulo p. So f⃗′′ = ∑_{i,j} g′_{ij}S_{ij} + ph⃗ for some n-tuple of polynomials h⃗ and preimages g′_{ij}. Then f⃗′ = lf⃗′′ = ∑_{ij} lg′_{ij}S_{ij} is a trivial syzygy of r⃗.

Definition 2.9. We say that a n-tuple of polynomials (q_{1}, …, q_{n}) satisfies a generalized flatness condition if there exists a matrix A ∈ GL_{n}(R[Λ]) such that the n-tuple (r_{1}, …, r_{n}) = (q_{1}, …, q_{n})A satisfies the flatness condition.

Lemma 2.10. Assume that a n-tuple of polynomials q⃗ = (q_{1}, …, q_{n}) satisfies the generalized flatness condition. Then all syzygies of q⃗ are trivial.

Proof. Let A be a matrix such that r⃗ = q⃗A satisfies the flatness condition. Let f⃗ = (f_{1}, …, f_{n}) be a syzygy of q⃗. Then (as a product of matrices)

0 = q⃗ · f⃗^{t} = q⃗A · A^{−1}f⃗^{t} = r⃗ · g⃗, where g⃗ = A^{−1}f⃗^{t}.

Hence, g⃗ is a syzygy of r⃗ and f⃗ = Ag⃗. By Lemma 2.8 it suffices to prove that if g⃗ = S_{ij} is a trivial syzygy of r⃗, then Ag⃗ is a trivial syzygy of q⃗. Let M_{ij}, i ≠ j denote a matrix where all entries are zeros except 1 at the position (i, j) and −1 at the position (j, i). The matrix M_{ij} is skew-symmetric. By definition, we have S_{ij} = M_{ij}(r⃗)^{t}. So all trivial syzygies of r⃗ are linear combinations with coefficients in R[Λ] of M_{ij}r^{t}. Similarly, all trivial syzygies of q⃗ are linear combinations of M_{ij}q^{t}. Then we obtain Ag⃗ = AM_{ij}r^{t} = AM_{ij}A^{t}q^{t}. Finally, since the matrix AM_{ij}A^{t} is skew-symmetric, it is a linear combination with coefficients in R[Λ] of matrices M_{i′j′} for various i′, j′.

3 The generators

Consider the weight lattice Λ of a semisimple root system corresponding to a group G. Let T^{∗} be a group of characters of a split maximal torus T of G. We assume that T^{∗} is of index 2 in Λ, i.e., Λ/T^{∗} = ℤ/2ℤ. Consider the ℤ/2ℤ-grading on Λ given by: a weight λ ∈ Λ has degree |λ| which is its class in the quotient Λ/T^{∗}. We denote by Λ^{(0)} = T^{∗} the subgroup of Λ of degree 0 and by Λ^{(1)} = Λ \ T^{∗} the subset of degree 1.

There is an induced grading on the group ring R[Λ] so that R[Λ] = R[Λ^{(0)}] ⊕ R[Λ^{(1)}]. Hence, we can uniquely express any f ∈ R[Λ] as a sum of its homogeneous components, i.e., f = f^{(0)} + f^{(1)}. We say that f ∈ R[Λ] is homogeneous of degree 0 or, equivalently, deg(f) = 0 (resp. f is of degree 1 or deg(f) = 1) if f ∈ R[Λ^{(0)}] (resp. f ∈ R[Λ^{(1)}]). Let {ω_{1}, …, ω_{n}} denote the set of fundamental weights (a ℤ-basis of Λ). Consider the orbit W (ω_{i}) of the fundamental weight ω_{i} by means of the Weyl group W. We denote by |i| the degree of ω_{i} with respect to the grading and by |W (ω_{i})| the number of elements in the orbit. Let

d = gcd_{ω_{i}∈Λ^{(1)}}(|W (ω_{i})|) = gcd_{|i|=1}(|W (ω_{i})|).

We set R = ℤ and we denote by bar the reduction modulo d, i.e., R̄ = ℤ/dℤ. We define

ρ(ω_{i}) = ∑_{λ∈W (ω_{i})} e^{λ} and ρ_{i} = ρ(ω_{i}) − |W (ω_{i})| ∈ R[Λ].

Since the Weyl group acts trivially on Λ/T^{∗}, we have deg(ρ(ω_{i})) = |i|. Reducing modulo d we obtain deg(ρ̄_{i}) = deg(\overline{ρ(ω_{i})}) = |i|. We will need the following

Lemma 3.1. Assume that (ρ̄_{1}, …, ρ̄_{n}) satisfies the generalized flatness condition with respect to some basis {x_{i}} of Λ. Assume that f_{i} ∈ R[Λ], i = 1, …, n are such that deg(∑_{i} f_{i}ρ_{i}) = 0. Then there exist polynomials g_{1}, …, g_{n} ∈ R[Λ] such that ∑_{i} f_{i}ρ_{i} = ∑_{i} g_{i}ρ_{i} and ḡ_{i}^{(1−|i|)} = 0 for each i.

Proof. Since deg(∑_{i} f_{i}ρ_{i}) = 0, we have ∑_{i} f̄_{i}^{(1−|i|)}ρ̄_{i} = 0. Hence, by Lemma 2.10 the n-tuple (f̄_{i}^{(1−|i|)}) is a trivial syzygy of (ρ̄_{i}). By Lemma 2.6 there exists a trivial syzygy (h_{i}), h_{i} ∈ R[Λ], of (ρ_{i}) such that h̄_{i} = f̄_{i}^{(1−|i|)}. Set g_{i} = f_{i} − h_{i}.

After a possible reindexing, we may assume that the first n′ fundamental weights {ω_{1}, …, ω_{n′}} have degree 1 and the remaining fundamental weights have degree 0. For 1 ≤ i ≤ n′ we set

d_{i} = gcd_{i≤j≤n′}(s_{j}), where s_{j} = |W (ω_{j})|.

So we have d = d_{1} | d_{2} | … | d_{n′} = s_{n′}. By a presentation of the gcd, there exist integers (denoted a_{i,j}, 1 ≤ i ≤ j ≤ n′) such that

d_{i} = a_{i,i}s_{i} + a_{i,i+1}s_{i+1} + … + a_{i,j}s_{j} + … + a_{i,n′}s_{n′}.

For 1 ≤ i < n′ we set

ρ̃_{i} = a_{i,i}ρ_{i} + a_{i,i+1}ρ_{i+1} + … + a_{i,n′}ρ_{n′}.

By definition, the coefficient of ρ̃_{i} at 1 = e^{0} is

−a_{i,i}s_{i} − a_{i,i+1}s_{i+1} − … − a_{i,n′}s_{n′} = −d_{i}.

Set ρ̃(ω_{i}) = ρ̃_{i} + d_{i} ∈ R[Λ^{(1)}].

Definition 3.2. Fix λ_{0} ∈ Λ^{(1)} and consider the following subsets of elements in R[T^{∗}]:

(1) \{h_{1,i} = e^{λ_{0}}(\frac{r_{i}}{s_{i}} ρ(ω_{i}) − \frac{r_{i}}{d_{i+1}} ρ̃(ω_{i+1})) | 1 ≤ i < n′, r_{i} = lcm(s_{i}, d_{i+1})\}.

(2) {h_{2,i} = ρ(ω_{i})ρ̃(ω_{1}) − ds_{i} | 1 ≤ i ≤ n′},

(3) {h_{3,i} = ρ_{i} | n′ < i ≤ n}.

Remark 3.3. The elements h_{1,i} will be extensively used (see (11) and (13)) in the computations of the group of semi-decomposable invariants.

Let I be the augmentation ideal of R[Λ], that is I is the kernel of the map R[Λ] → R given by e^{λ} ↦ λ. Let I_{sc}^{W} denote the ideal in R[Λ] generated by elements from R[Λ]^{W} ∩ I. By the Chevalley theorem I_{sc}^{W} is generated by the elements ρ_{i}, 1 ≤ i ≤ n, i.e., any f ∈ I_{sc}^{W} can be written as f = f_{1}ρ_{1} +…+f_{n}ρ_{n} for some f_{i} ∈ R[Λ]. Our main result is the following

Theorem 3.4. Assume that the n-tuple (ρ_{1}, …, ρ_{n}) satisfies the generalized flatness condition with respect to some basis {x_{i}} of the weight lattice Λ. Then the elements h_{k,i} of Definition 3.2 generate the ideal I_{sc}^{W} ∩R[T^{∗}] in R[T^{∗}].

Proof. Suppose that f_{1}ρ_{1} + … + f_{n}ρ_{n} ∈ R[T^{∗}] for some f_{i} ∈ R[Λ]. By Lemma 3.1 we may assume that d | f_{i}^{(1−|i|)} for each i. To prove the theorem we modify (f_{1}, …, f_{n}) in several steps. At each step, we subtract a linear combination of the elements h_{i,j} (with coefficients in R[T^{∗}]) from f_{1}ρ_{1} + … + f_{n}ρ_{n} so that the new polynomials f′_{1}, …, f′_{n} have fewer nonzero monomials. In the end they will all become 0, so that the original f_{1}ρ_{1} + … + f_{n}ρ_{n} will be replaced by a linear combination of h_{i,j}. Step 1. By definition we have for 1 ≤ i ≤ n′

h_{2,i} = ρ(ω_{i})ρ̃(ω_{1}) − ds_{i} = ρ(ω_{i})(ρ̃_{1} + d_{1}) − ds_{i}
= ρ(ω_{i})(a_{1,1}ρ_{1} + a_{1,2}ρ_{2} + … + a_{1,n′}ρ_{n′} + d) − ds_{i}
= a_{1,1}ρ(ω_{i})ρ_{1} + … + (a_{1,i}ρ(ω_{i}) + d)ρ_{i} + … + a_{1,n′}ρ(ω_{i})ρ_{n′}.

Since |i| = 1, d | f_{i}^{(0)}. Consider the difference

f′_{1}ρ_{1} + … + f′_{n}ρ_{n} = f_{1}ρ_{1} + … + f_{n}ρ_{n} − \frac{1}{d}f_{i}^{(0)}h_{2,i}.  (*)

Collecting the coefficients we obtain

The K-Theory and Cohomological Invariants 1125 • f′_{j} = f_{j} for all j > n′,

• f_{j}^{′(0)} = f_{j}^{(0)} for all j ≤ n′ and j ≠ i,

• f_{i}^{′(0)} = 0. Hence, applying (*) for each i, 1 ≤ i ≤ n′ we obtain new coefficients (f′_{1}, …, f′_{n}) such that f_{j}^{′(0)} = 0 for all j ≤ n′ and f′_{j} = f_{j} for all j > n′. (Observe that f_{j}^{′(1)} for j ≤ n′ does not necessarily coincides with f_{j}^{(1)}.) Step 2. We have for n′ < i ≤ n

ρ̃(ω_{1})h_{3,i} = ρ̃(ω_{1})ρ_{i} = (a_{1,1}ρ_{1} + a_{1,2}ρ_{2} + … + a_{1,n′}ρ_{n′} + d)ρ_{i}
= (a_{1,1}ρ_{i})ρ_{1} + … + (a_{1,n′}ρ_{i})ρ_{n′} + dρ_{i}.

Since |i| = 0, d | f_{i}^{(1)}. Consider the difference

f′_{1}ρ_{1} + … + f′_{n}ρ_{n} = f_{1}ρ_{1} + … + f_{n}ρ_{n} − \frac{1}{d}f_{i}^{(1)}ρ̃(ω_{1})h_{3,i}.  (**)

Collecting the coefficients we obtain • f′_{j} = f_{j} for all j > n′ and j ≠ i,

• f_{j}^{′(0)} = f_{j}^{(0)} for all j ≤ n′.

• f_{i}^{′(1)} = 0. Hence, applying (**) for each i, n′ < i ≤ n we obtain new coefficients (f′_{1}, …, f′_{n}) such that f_{j}^{′(1)} = 0 for all j > n′ and f_{j}^{′(0)} = 0 for all j ≤ n′. Step 3. As a result of step 2, we have f_{i} ∈ R[T^{∗}] for all i > n′. Subtracting

f_{1}ρ_{1} + … + f_{n}ρ_{n} − ∑_{i>n′} f_{i}h_{3,i}

we may assume that f_{i} = 0 for all i > n′. Step 4. Fix i, 1 ≤ i ≤ n′. If i > 1 we assume in addition that f_{1} = … = f_{i−1} = 0. So, we have f_{i}ρ_{i} + … + f_{n′}ρ_{n′} ∈ R[T^{∗}], where f_{j}^{(0)} = 0 for all i ≤ j ≤ n′ by previous steps. Hence, we can express it as f_{i}ρ_{i}+…+f_{n′}ρ_{n′} = f_{i}^{(1)}ρ(ω_{i})+…+f_{n′}^{(1)}ρ(ω_{n′})−(s_{i}f_{i}^{(1)}+…+s_{n′}f_{n′}^{(1)}) ∈ R[T^{∗}]. Since deg(ρ(ω_{j})) = 1 for i ≤ j ≤ n′, we obtain

s_{i}f_{i}^{(1)} = −s_{i+1}f_{i+1}^{(1)} − … − s_{n′}f_{n′}^{(1)}.

The right hand side of this equation is divisible by r_{i}, hence, \frac{r_{i}}{s_{i}} | f_{i}^{(1)}.

Documenta Mathematica 22 (2017) 1117–1148

By definition, we have

h_{1,i} = e^{λ_{0}}(\frac{r_{i}}{s_{i}} ρ(ω_{i}) − \frac{r_{i}}{d_{i+1}} ρ̃(ω_{i+1})) = e^{λ_{0}}(\frac{r_{i}}{s_{i}} ρ_{i} − \frac{r_{i}}{d_{i+1}} ρ̃_{i+1})
= e^{λ_{0}}(\frac{r_{i}}{s_{i}} ρ_{i} − \frac{r_{i}}{d_{i+1}} (a_{i+1,i+1}ρ_{i+1} + a_{i+1,i+2}ρ_{i+2} + … + a_{i+1,n′}ρ_{n′}))

Consider the difference

f′_{i}ρ_{i} + … + f′_{n′}ρ_{n′} = f_{i}ρ_{i} + … + f_{n′}ρ_{n′} − \frac{s_{i}}{r_{i}} f_{i}^{(1)}e^{−λ_{0}}h_{1,i}.  (***)

Collecting the coefficients we obtain f′_{i} = 0 while keeping f_{j}^{′(0)} = f_{j}^{(0)} = 0 for all i < j ≤ n′. Hence, applying (***) inductively starting with i = 1, we obtain that f_{i} = 0 for all 1 ≤ i ≤ n′.

4 The generalized flatness condition

In the present section we prove that the n-tuple of W -orbits (ρ_{1}, …, ρ_{n}) in R[Λ] satisfies the generalized flatness condition when Λ is a weight lattice for a semi-simple root system of type A and C. Observe that it is enough to prove the generalized flatness condition for each simple component.

4.1 Type A

Let Λ̃ = ℤ^{n+1} with a standard basis e_{1}, …, e_{n+1}. The weight lattice of type A is then given by Λ = Λ̃/(e_{1} + … + e_{n+1}). We denote the class of e_{i} in Λ by ē_{i}. The basis of Λ is given by the fundamental weights ω_{i} = ē_{1} + … + ē_{i}, i = 1, …, n. The Weyl group (the symmetric group S_{n+1}) acts by permutations of {e_{1}, …, e_{n+1}}. Let x_{i} = e^{ω_{i}} in ℤ[Λ] and let y_{i} = e^{e_{i}} in ℤ[Λ̃]. Consider the induced map ϕ: ℤ[Λ̃] → ℤ[Λ] given by ϕ(y_{1}) = x_{1}, ϕ(y_{i}) = eē_{i} = x_{i}x_{i−1}^{−1}, 1 < i ≤ n and ϕ(y_{n+1}) = x_{n}^{−1}. The image of the elementary symmetric function σ_{i} = σ_{i}(y_{1}, …, y_{n+1}) gives the W -orbit ρ(ω_{i}). Let g_{i} be (the complete sum symmetric function) the sum of all monomials of total degree i in variables y_{1}, …, y_{n+2−i}. We have the following analogue of the Newton relation (see [16, Relation (2)]) for i > 0

∑_{j=0}^{i}(−1)^{j}g_{j}(y_{1}, …, y_{n+2−j})σ_{i−j}(y_{1}, …, y_{n+1−j}) = 0 (here g_{0} = σ_{0} = 1) (1)

which implies that the ideal I_{σ} = (σ_{1}, …, σ_{n+1}) in ℤ[y_{1}, …, y_{n+1}] coincides with the ideal I_{g} = (g_{1}, …, g_{n+1}). Consider the involution τ of ℤ[y_{1}, …, y_{n+1}] given by y_{i} ↦ 1 − y_{i}. We get

(σ_{1} − s_{1}, …, σ_{n+1} − s_{n+1}) = τ(I_{σ}) = τ(I_{g}) = (g̃_{1}, …, g̃_{n+1}),

where s_{i} = σ_{i}(1, …, 1) = |W (ω_{i})| and g̃_{i} is the (non-homogeneous) polynomial in variables y_{1}, …, y_{n+2−i} of degree i such that its coefficient at y_{n+2−i}^{i} is ±1. Taking its images in ℤ[Λ] we obtain

(ρ_{1}, …, ρ_{n}) = ϕ(τ(I_{σ})) = ϕ(τ(I_{g})) = (r_{1}, …, r_{n+1}),

where ρ_{i} = ρ(ω_{i}) − s_{i} and r_{i} = ϕ(g̃_{n+2−i}). We claim that r_{n+1} can be written as a linear combination of r_{1}, …, r_{n}. Indeed, taking the sum of relations (1) for all i we obtain

1 = ∑_{i=0}^{n+1}(g_{i} ∏_{j=1}^{n+1−i}(1 − y_{j})).

After applying τ we obtain 1 = ∑_{i=0}^{n+1} g̃_{i}y_{1} … y_{n+1−i}. Since y_{1} … y_{n+1} = 1, after taking its image in ℤ[Λ] we obtain the desired linear combination.

4.2 Type C

Consider the weight lattice Λ of type C. It is generated by the standard vectors {e_{1}, …, e_{n}} with fundamental weights ω_{i} = e_{1} + … + e_{i}. The Weyl group W acts on the standard vectors by permutations and changing signs. Consider the embedding ϕ: ℤ[y_{1}, …, y_{n}] ↪ ℤ[Λ] given by ϕ(y_{i}) = e^{e_{i}} + e^{−e_{i}}. The image of the elementary symmetric function σ_{i} = σ_{i}(y_{1}, …, y_{n}) gives the W -orbit ρ(ω_{i}). As in type A let g_{i} be the sum of all monomials of total degree i in variables y_{1}, …, y_{n+1−i}. Then (σ_{1}, …, σ_{n}) = (g_{1}, …, g_{n}) as ideals in ℤ[y_{1}, …, y_{n}]. Applying the involution τ we obtain (ρ_{1}, …, ρ_{n}) = (r_{1}, …, r_{n}), where ρ_{i} = ρ(ω_{i}) − s_{i}, r_{i} = ϕ(τ(g_{n+1−i}) and s_{i} = |W (ω_{i})|. The n-tuple (r_{1}, …, r_{n}) satisfies the flatness condition and there is an invertible transformation matrix between ρ_{i} and r_{i}.

5 Characters and invariants

In the present section we introduce some notation and recall basic definitions for the group of characters, characteristic classes and invariants which will be used in the subsequent sections. We follow [14], [15] and [1].

5.1 Characters.

Let H and H′ be simply connected simple split groups of the same Dynkin type D over a field F. Assume that there is a central diagonal subgroup μ_{k} in H × H′. The quotient G = (H × H′)/μ_{k} will be called a group of index k of type D. We denote by T_{sc} the split maximal torus of H × H′, by T the split maximal torus of G and by T_{ad} the split maximal torus of the product of the adjoint forms H_{ad} × H′_{ad}. Then there is an exact sequence for the groups of characters

0 → T^{∗}/T_{ad}^{∗} → T_{sc}^{∗}/T_{ad}^{∗} → ℤ/kℤ → 0

which can be used to describe T^{∗}. Indeed, the quotient T_{sc}^{∗}/T_{ad}^{∗} is the group of characters of the center Z(G) and the map T_{sc}^{∗}/T_{ad}^{∗} → ℤ/kℤ is induced by the diagonal embedding μ_{k} → Z(G). Moreover, T_{sc}^{∗}/T_{ad}^{∗} is the product of groups of characters of the centers of H and H′, hence,

T_{sc}^{∗}/T_{ad}^{∗} = Λ_{w}/Λ_{r} ⊕ Λ′_{w}/Λ′_{r} → ℤ/kℤ, is given by taking the sum, where Λ_{w} (resp. Λ′_{w}) is the weight lattice and Λ_{r} (resp. Λ′_{r}) is the root lattice of H (resp. of H′).

5.2 Invariant forms.

Let W = W_{H} × W_{H′} be the Weyl group of H × H′. It naturally acts on T_{sc}^{∗} = Λ_{w} ⊕ Λ′_{w}. Consider the group of W -invariant quadratic forms. It is a direct sum of cyclic groups

S^{2}(T_{sc}^{∗})^{W} = S^{2}(Λ_{w})^{W_{H}} ⊕ S^{2}(Λ′_{w})^{W_{H′}} = ℤq ⊕ ℤq′,

where q and q′ are generators given by normalized Killing forms. So any form ϕ ∈ S^{2}(T_{sc}^{∗})^{W} can be written uniquely as ϕ = dq + d′q′, d, d′ ∈ ℤ. The list of Killing forms for all types can be found in [14, §4]. Let {ω_{i}} and {ω′_{i}} denote the fundamental weights of H and H′, i.e., the ℤbases of Λ_{w} and Λ′_{w}. Choose a ℤ-basis {x_{i}} of T^{∗}. Expressing each ω_{i} and ω′_{i} in terms of x_{j}’s and substituting into ϕ allows us to explicitly describe the subgroup

Q(G) = S^{2}(T^{∗})^{W} = S^{2}(T^{∗}) ∩ S^{2}(T_{sc}^{∗})^{W} .

5.3 Characteristic map

Consider the group ring ℤ[T_{sc}^{∗}] that is the representation ring R(T_{sc}) of T_{sc}. Each element of ℤ[T_{sc}^{∗}] can be written as a finite linear combination ∑_{i} a_{i}e^{λ_{i}}, a_{i} ∈ ℤ, λ_{i} ∈ T_{sc}^{∗}. Fix a basis of T_{sc}^{∗} consisting of fundamental weights {ω_{i}} and {ω′_{j}}. Following [14, §3c] and [5] we define a ℤ[W]-module homomorphism

c_{2} : ℤ[T_{sc}^{∗}] → S^{≤2}(T_{sc}^{∗}) → S^{2}(T_{sc}^{∗})

by sending 1 ↦ 1, e^{−ω_{i}} ↦ 1 − ω_{i} and e^{ω_{i}} ↦ 1 + ω_{i} + ω_{i}^{2} (resp. for ω′_{j}) and then taking the degree 2 homogeneous component. Let I_{sc} denote the augmentation ideal in ℤ[T_{sc}^{∗}], i.e., the kernel of the trace map ℤ[T_{sc}^{∗}] → ℤ, e^{λ} ↦ 1. Then the image c_{2}(I_{sc}^{3}) = 0, so c_{2} can be restricted to I_{sc}^{2}[5]. Observe that the filtration by powers of the ideal I_{sc} can be viewed as a γ-filtration on K_{0}(BT); its image in K_{0}(G/B) via the characteristic map c: ℤ[T_{sc}^{∗}] → K_{0}(G/B) gives the Grothendieck γ-filtration on K_{0}(G/B) (e.g., see [19]).

5.4 Invariants

Given λ ∈ T^{∗} denote by ρ(λ) = ∑_{χ∈W (λ)} e^{χ}, where W (λ) is the W -orbit of λ. If restricted to invariants, the map c_{2} defines a group homomorphism

c_{2} : ℤ[T^{∗}]^{W} → S^{2}(T^{∗})^{W}

with image generated by forms c_{2}(ρ(λ)) = −\frac{1}{2} ∑_{χ∈W (λ)} χ^{2} for all λ ∈ T^{∗} [14, §3c]. It was shown in [14] that the image c_{2}(ℤ[T^{∗}]^{W}) can be identified with the group of degree 3 decomposable invariants Dec(G) and the quotient Q(G)/ Dec(G) with the group of indecomposable invariants denoted by Inv_{ind}^{3}(G). By definition for any two semisimple groups G_{1}, G_{2} we have

Q(G_{1} × G_{2}) = Q(G_{1}) × Q(G_{2}) and Dec(G_{1} × G_{2}) = Dec(G_{1}) × Dec(G_{2}). (2)

Let I_{sc}^{W} denote an ideal in ℤ[T_{sc}^{∗}] generated by W -invariants from the augmentation ideal I_{sc}, namely, I_{sc}^{W} = (I_{sc} ∩ ℤ[T_{sc}^{∗}]^{W}). The main result of [15] says that the image c_{2}(ℤ[T^{∗}] ∩ I_{sc}^{W}) in S^{2}(T^{∗})^{W} coincides with the subgroup of semi-decomposable invariants Sdec(G) and that Dec(H) = Sdec(H) if H is a simple group. Observe that we have

Dec(G) ⊆ Sdec(G) ⊆ Q(G).  (3)

We denote by Inv_{sd}^{3}(G) the quotient Sdec(G)/ Dec(G).

6 Type A

In the present section we consider semisimple groups of type A. The following lemma gives a simple geometric proof for the coincidence between the normalized invariants and semi-decomposable invariants (c.f. [1]):

Lemma 6.1. Let G = (∏_{i=1}^{m} SL_{n_{i}})/μ_{k}, m, n_{i}, k ≥ 2, where μ ≃ μ_{k} is a diagonal (central) subgroup. Then, Q(G) = Sdec(G), i.e., each degree 3 indecomposable invariant of G is semi-decomposable.

Proof. We follow arguments used in [1, §5]. Assume

μ = {(λ_{1}, …, λ_{m}) ∈ μ_{n_{1}} × ⋯ × μ_{n_{m}} | λ_{1}^{k} = ⋯ = λ_{m}^{k} = 1, λ_{1} = ⋯ = λ_{m}}

is the diagonal subgroup. The corresponding versal flag variety X over the function field F′ of the classifying space of G can be replaced by the product of Severi-Brauer varieties SB(A_{1})×⋯×SB(A_{m}), where A_{i} are central simple F′- algebras of degree n_{i} for 1 ≤ i ≤ m such that k[A_{i}] = 0 and [A_{1}] = ⋯ = [A_{m}] in the Brauer group Br(F′). Let B be the common underlying division algebra of A_{1}, …, A_{m} such that the index and the exponent of B are all equal to k. Then, we obtain CH^{2}(X)_{tors} = CH^{2}(SB(B))_{tors}. By [10, Cor.4] we have CH^{2}(SB(B))_{tors} = 0, thus by the main theorem of [15] Q(G) coincides with Sdec(G).

Remark 6.2. Using arguments in [1] one can compute the quotient

Inv_{ind}^{3}(G) = Q(G)/ Sdec(G) = CH^{2}(X)_{tors}

for an arbitrary semisimple group G of type A. For instance, when m = 2, the same arguments in the proof work if we replace the diagonal subgroup μ ≃ μ_{2} by a central subgroup μ_{2} × 1 ⊆ μ_{n_{1}} × μ_{n_{2}} or 1 × μ_{2} ⊆ μ_{n_{1}} × μ_{n_{2}}.

The following proposition deals with groups of p-primary index for any prime p, which in turn computes the p-primary component of Inv_{ind}^{3}((SL_{m} × SL_{n})/μ_{k}) for any diagonal (central) subgroup μ_{k}.

Proposition 6.3. Let G = (SL_{m} × SL_{n})/μ, m, n ≥ 2, where μ ≃ μ_{k} is a p-primary diagonal (central) subgroup. Then,

Q(G) = \{dq + d′q′ | ( \frac{k−1}{k})(md + nd′) ≡ 0 mod 2k\}

and

Dec(G) ≃ \begin{cases}kℤq ⊕ kℤq′ & if p ≠ 2 or p = 2, min\{v_{2}(m), v_{2}(n)\} > v_{2}(k), \\ kℤ(q − q′) ⊕ kℤ(q + q′) & if p = 2, v_{2}(m) = v_{2}(n) = v_{2}(k), \\ kℤq ⊕ 2kℤq′ & if p = 2, v_{2}(m) > v_{2}(k) = v_{2}(n), \end{cases}

where q (resp. q′) is the normalized Killing form of SL_{m} (resp. SL_{n}).

Proof. Let G = (SL_{m} × SL_{n})/μ_{k}, m, n, k ≥ 2, k | gcd(m, n), where μ_{k} is a diagonal subgroup. Then, by 5.1 the character group of the split maximal torus T of G is given by

T^{∗} = \{ ∑_{i=1}^{m−1} a_{i}ω_{i} + ∑_{j=1}^{n−1} a′_{j}ω′_{j} | ∑_{r=0}^{\frac{m−k}{k}} ∑_{i=1}^{k−1} ia_{i+rk} + ∑_{s=0}^{\frac{n−k}{k}} ∑_{i=1}^{k−1} ia′_{i+sk} ≡ 0 mod k\}. (4)

Following 5.2 the group of W -invariant quadratic forms S^{2}(T_{sc}^{∗})^{W} is generated by the normalized Killing forms

q = ∑_{i=1}^{m−1} ω_{i}^{2} − ∑_{i=1}^{m−2} ω_{i}ω_{i+1} and q′ = ∑_{j=1}^{n−1} ω_{j}^{′2} − ∑_{j=1}^{n−2} ω′_{j}ω′_{j+1}.

Consider the ℤ-basis {x_{1}, …, x_{m−1}, x′_{1}, …, x′_{n−1}} of the character group T^{∗} given by

x_{i+rk} = ω_{i+rk} + iω′_{n−1}, x_{k+rk} = ω_{k+rk} and x′_{i+sk} = ω′_{i+sk} + iω′_{n−1}, x′_{k+sk} = ω′_{k+sk},

where 1 ≤ i ≤ k − 1, 0 ≤ r ≤ \frac{m−k}{k}, 0 ≤ s ≤ \frac{n−k}{k} (for convenience, we set x_{j} = x′_{j} = 0 for any j > n − 1 or j > m − 1). In this basis a form ϕ ∈ S^{2}(T_{sc}^{∗})^{W} can be written as

ϕ = (\frac{md(k−1)+nd′(k−1)}{2k^{2}})x_{n−1}^{′2} + ψ, d, d′ ∈ ℤ

where ψ is a quadratic form with integer coefficients. Hence, we obtain

Q(G) = \{dq + d′q′ | (\frac{k−1}{k})(md + nd′) ≡ 0 mod 2k\}.  (5)

From now on we assume that k is p-primary. We claim that Dec(G) ⊆ kℤq ⊕ kℤq′. To show this we extend the arguments in [4, p.136]. We use the standard presentation of the root system of type A, namely, that Λ_{w} (resp. Λ′_{w}) consists of vectors in the standard basis {e_{1}, .., e_{m}} (resp. {e′_{1}, .., e′_{n}}) whose sum of coordinates is zero. Choose a character χ ∈ T^{∗}. Assume χ has l (resp. l′) distinct coordinates in some order b_{1} > ⋯ > b_{l} (resp. b′_{1} > … > b′_{l′}) which repeat r_{1}, .., r_{l} (resp. s_{1}, .., s_{l′}) times with respect to the basis {e_{i}} (resp. {e′_{j}}). Then, for the orbit ρ(χ) of χ under the action of W we obtain

c_{2}(ρ(χ)) = \frac{n!}{s_{1}!···s_{l′}!} · [r, b] · q + \frac{m!}{r_{1}!···r_{l}!} · [s, b′] · q′, where  (6)

[r, b] = \frac{(m−2)!}{r_{1}!···r_{l}!} (m(∑_{i=1}^{l} r_{i}b_{i}^{2}) − (∑_{i=1}^{l} r_{i}b_{i})^{2}), [s, b′] = \frac{(n−2)!}{s_{1}!···s_{l}!} (n(∑_{j=1}^{l′} s_{j}b_{j}^{′2}) − (∑_{j=1}^{l′} s_{j}b′_{j})^{2}).

Observe that by (4) we have ∑ r_{i}b_{i} + ∑ s_{j}b′_{j} ≡ 0 mod k. Let c = min{v_{p}(r_{i})} and d = min{v_{p}(s_{j})}. If v_{p}(k) ≤ d, then v_{p}(k) ≤ v_{p}(∑ r_{i}b_{i}). So we obtain v_{p}(k) ≤ v_{p}(gcd(m, ∑ r_{i}b_{i})). Hence, by [4, p.137, Lemma 11.4] the coefficient [r, b] is divisible by k. Similarly, if v_{2}(k) ≤ c, then [s, b′] is divisible by k. Hence, we may assume that c, d ≤ v_{p}(k). By [4, p.137, Lemma 11.3], we have

v_{p}(\frac{n!}{s_{1}!···s_{l′}!}) ≥ v_{p}(k) − d ≥ 0 and v_{p}(\frac{m!}{r_{1}!···r_{l}!}) ≥ v_{p}(k) − c ≥ 0,

which implies that

v_{p}(gcd(m, ∑ r_{i}b_{i})) ≥ d and v_{p}(gcd(n, ∑ s_{j}b′_{j})) ≥ c.

Therefore, again by [4, p.137, Lemma 11.4] we see that each coefficient of q and q′ in (6) is divisible by k, which proves the claim. Finally, we compute the group Dec(G) case by case. As

c_{2}(ρ(kω_{1})) = −k^{2}q, c_{2}(ρ(2ω_{1} − ω_{2})) = −2mq  (7)

and similarly, c_{2}(ρ(kω′_{1})) = −k^{2}q′, c_{2}(ρ(2ω′_{1} − ω′_{2})) = −2mq′, we have

gcd(2, p)kq ∈ Dec(G) if v_{p}(m) = v_{p}(k)  (8)

and similarly, gcd(2, p)kq′ ∈ Dec(G) if v_{p}(n) = v_{p}(k). Moreover, we get

c_{2}(ρ(ω_{k})) = (kk′)q with gcd(k′, p) = 1 if v_{p}(m) > v_{p}(k)  (9)

and similarly, c_{2}(ρ(ω_{k})) = (kk′′)q with gcd(k′′, p) = 1 if v_{p}(n) > v_{p}(k) (see also [2, Thm.4.1]). Thus if p ≠ 2, then by (7) and (9) we obtain

gcd(k^{2}, kk′)q = kq ∈ Dec(G), gcd(k^{2}, kk′′)q = kq′ ∈ Dec(G).

Therefore, by (8) and the claim above, Dec(G) = kℤq ⊕ kℤq′ if p ≠ 2. Now assume that p = 2. If v_{2}(m) > v_{2}(k) and v_{2}(n) > v_{2}(k), then by (7), (9) and the claim above, we have Dec(G) = kℤq ⊕ kℤq′. If v_{2}(m) = v_{2}(n) = v_{2}(k), then c_{2}(ρ(ω_{k/2} + ω′_{k/2})) ≡ −kq − kq′ mod 2k. Hence, by (8), k(q − q′), k(q + q′) ∈ Dec(G). Since Dec(SL_{m} /μ_{k}) = 2kq if v_{2}(m) = v_{2}(k), and Dec(SL_{n} /μ_{k}) = 2kq′ if v_{2}(n) = v_{2}(k) ([2, Thm.4.1]), it follows from the claim above that Dec(G) = kℤ(q − q′) ⊕ kℤ(q + q′). Similarly, if v_{2}(m) > v_{2}(k) = v_{2}(n), then by (7) and (9) we have kq ∈ Dec(G). Since Dec(SL_{n} /μ_{k}) = 2kq′, we get kq′, k(q ± q′) ∉ Dec(G). Therefore, by the claim we obtain Dec(G) = kℤq ⊕ 2kℤq′.

Remark 6.4. This proposition and its proof generalize [14, Theorem 4.4] and [2, Theorem 4.1] for split simple groups G′ = SL_{m} /μ_{k} of type A. Indeed, following the proof of Proposition 6.3 one can easily check that Q(G′) = Q(G)|_{d′=0}. For instance, in order to obtain Q(PGL_{m}) we simply set d′ = 0 and k = m in (5), then we get Q(PGL_{m}) = 2 gcd(2, m)ℤq [14, Theorem 4.4]. Similarly, we can obtain Dec(PGL_{m}) in the same way. To compute the indecomposable groups for G′ = SL_{2m} /μ_{2}, we set d′ = 0, k = 2. Then it follows by Proposition 6.3 that Q(G′) = {dq | md ≡ 0 mod 4} and

Dec(G′) = \begin{cases}2ℤq & if v_{2}(m) > 0, \\ 4ℤq & if v_{2}(m) = 0. \end{cases}

Hence, we obtain [2, Theorem 4.1], that is

Inv_{ind}^{3}(G′) ≃ \begin{cases}(ℤ/2ℤ)q & if v_{2}(m) ≥ 2, \\ 0 & otherwise. \end{cases}

Finally observe that together with Lemma 6.1, properties (2) and (3) it computes the group Inv_{sd}^{3}(G′′), where G′′ = (SL_{2m} /μ_{2}) × SL_{2n}, n, m ≥ 1.

The following corollary generalizes [15, Example 3.1] to groups of type A (see also [1]). Similarly, by using Lemma 6.1 and Proposition 6.3, one can compute both groups Inv_{ind}^{3}(G) and Inv_{sd}^{3}(G) for any p-primary diagonal subgroup μ_{k}.

Corollary 6.5. Let G = (∏_{i=1}^{m} SL_{2n_{i}})/μ_{2}, n_{i} ≥ 1, where μ_{2} is the diagonal subgroup. Then

Inv_{sd}^{3}(G) = Inv_{ind}^{3}(G) = \begin{cases}(ℤ/2ℤ)^{⊕m} & if n_{i} ≡ 0 mod 4 ∀i, \\ (ℤ/2ℤ)^{⊕m−1} & otherwise. \end{cases}

Proof. Let G = (SL_{2m} × SL_{2n})/μ_{2}. Then, it follows from Proposition 6.3 that

Inv_{ind}^{3}(G) ≃ \begin{cases}(ℤ/2ℤ)q ⊕ (ℤ/2ℤ)q′ & m ≡ n ≡ 0 mod 4, \\ (ℤ/2ℤ)q & m ≡ 0, n ≡ 2 mod 4 or m ≡ 0 mod 4, n is odd, \\ (ℤ/2ℤ)(q − q′) & m ≡ n ≡ 2 mod 4 or both m, n are odd, \\ (ℤ/2ℤ)(q − 2q′) & m ≡ 2 mod 4, n is odd. \end{cases}

Hence, the result follows by Lemma 6.1. Applying the same arguments for three and more groups completes the proof.

In the following we show that the both indecomposable group Inv_{ind}^{3}(G) and the semi-decomposable group Inv_{sd}^{3}(G) can have an arbitrary order (c.f. [1]). In particular, the order can be arbitrarily large.

Corollary 6.6. For an arbitrary integer k ≥ 2 there exists a semisimple group G of type A such that

| Inv_{ind}^{3}(G)| = | Inv_{sd}^{3}(G)| = k.

Moreover, for any homocyclic p-group C there exists a semisimple group H of type A such that Inv_{ind}^{3}(H) = Inv_{sd}^{3}(H) = C.

Proof. Let p^{r} (r ≥ 1) be a prime factor of k and let n = gcd(2, p)p^{r}. We denote G[p^{r}] = (SL_{n} × SL_{n})/μ_{p^{r}}, where μ_{p^{r}} is the diagonal subgroup. By Proposition 6.3 we have

Inv_{ind}^{3}(G[p^{r}]) = Inv_{sd}^{3}(G[p^{r}]) = ℤ/p^{r}ℤ.

Set G = (SL_{gcd(2,k)k} × SL_{gcd(2,k)k})/μ_{k}. Then, the same argument as in [15, §3b] shows that Inv_{ind}^{3}(G[p^{r}]) is a p-primary component of Inv_{ind}^{3}(G) and the first statement follows from Lemma 6.1. Let C = (ℤ/p^{r}ℤ)^{⊕m} be a homocyclic p-group of rank m for some prime p. It suffices to consider the case m ≥ 2. Let H = (SL_{n})^{m}/μ_{p^{r}}, where n = gcd(2, p)p^{2r} and μ_{p^{r}} is the diagonal subgroup. Then, the arguments used in the proof of Proposition 6.3 yield

Q(G) = {∑_{i=1}^{m} d_{i}q_{i} | (n/p^{r}) ∑_{i=1}^{m} d_{i} ≡ 0 mod gcd(2, p)p^{r}},

where q_{i} is the corresponding normalized Killing form of SL_{n}. Similarly, we have Dec(G) = ⨁_{i=1}^{m} p^{r}ℤq_{i}. Then the second statement follows by Remark 6.2.

7 Type B

In the present section we show that any semi-decomposable invariant of semisimple groups of type B is decomposable, except in the case of a product of groups of type B_{2} = C_{2} modulo the diagonal subgroup μ_{2}. We first consider the index 2 case.

Proposition 7.1. Let G = (Spin_{2m+1} × Spin_{2n+1})/μ_{2}, m, n ≥ 2, where μ_{2} is the diagonal subgroup. Then, we have Inv_{ind}^{3}(G) = ℤ/2ℤ and

Inv_{sd}^{3}(G) = \begin{cases}ℤ/2ℤ & if m = n = 2, \\ 0 & otherwise, \end{cases}

i.e., each semi-decomposable invariant is decomposable unless m = n = 2.

Proof. Following 5.1 the character group of the split maximal torus T of G is given by

T^{∗} = {∑_{i=1}^{m} a_{i}ω_{i} + ∑_{i=1}^{n} a′_{i}ω′_{i} | a_{m} ≡ a′_{n} mod 2}.

Following 5.2 the group S^{2}(T_{sc}^{∗})^{W} is generated by the normalized Killing forms

q = ∑_{i=1}^{m−1}(ω_{i}^{2} − ω_{i}ω_{i+1}) + 2ω_{m}^{2} − ω_{m−1}ω_{m}, q′ = ∑_{i=1}^{n−1}(ω_{i}^{′2} − ω′_{i}ω′_{i+1}) + 2ω_{m}^{′2} − ω′_{m−1}ω′_{m}.

Choose a ℤ-basis {ω_{1}, …, ω_{m−1}, ω′_{1}, …, ω_{m−1}, v_{1}, v′_{1}} of T^{∗} where v_{1} = ω_{m}+ω′_{n} and v′_{1} = ω_{m} − ω′_{n}. For any ϕ ∈ S^{2}(T^{∗})^{W} there exist d, d′ ∈ ℤ such that ϕ = dq + d′q′, thus in this basis we have

ϕ = \frac{1}{2}(d + d′)(v_{1}^{2} + v_{1}^{′2}) + ψ

for some quadratic form ψ with integer coefficients. Hence, we obtain

Q(G) = {dq + d′q′ | d + d′ ≡ 0 mod 2} = ℤ(q − q′) ⊕ ℤ(q + q′).  (10)

We claim that Dec(G) = 2ℤq⊕2ℤq′. The result for the group of indecomposable invariants then follows immediately. To prove the claim, since q = \frac{1}{2}(∑_{i=1}^{m} e_{i}^{2}) and q′ = \frac{1}{2}(∑_{j=1}^{n} e_{j}^{′2}) in terms of the standard basis of T_{sc}^{∗} = ℤ^{m} ⊕ ℤ^{n}, we conclude that c_{2}(ρ(ω_{1})) = 2q and c_{2}(ρ(ω′_{1})) = 2q′ are contained in Dec(G). On the other hand, as Dec(G) is generated by c_{2}(ρ(λ)) for all λ ∈ T^{∗} and the Weyl group of G contains normal subgroups (ℤ/2ℤ)^{m} and (ℤ/2ℤ)^{n} generated by sign switching, we see that the coefficient at each e_{i} in the expansion of c_{2}(ρ(λ)) is divisible by 2 (c.f. [4, Lemma 14.2]). We now compute the group Sdec(G). Assume that m = n = 2. Consider the element

y = e^{ω_{2}}z ∈ ℤ[T^{∗}] ∩ I_{sc}^{W} with z = ρ̄(ω_{2}) − ρ̄(ω′_{2}),  (11)

where ρ̄(ω_{i}) denotes the augmented orbit ρ(ω_{i}) − |W (ω_{i})|. As (e^{ω_{2}} − 1)z ∈ I_{sc}^{3}, we see that c_{2}(y) = c_{2}(z). Since c_{2}(ρ̄(ω_{2})) = q and c_{2}(ρ̄(ω′_{2})) = q′, we conclude that q − q′ ∈ Sdec(G). Therefore, Inv_{sd}^{3}(G) = ℤ/2ℤ. Assume that m, n ≥ 3. We will show that q − q′ which is a generator of Inv_{ind}^{3}(G) does not belong to Sdec(G). Let x ∈ ℤ[T^{∗}]∩I_{sc}^{W}. Similar to [15, §3c] write

x = ∑_{i=1}^{m}(d_{i} + δ_{i})ρ̄(ω_{i}) + ∑_{j=1}^{n}(d′_{j} + δ′_{j})ρ̄(ω′_{j})

for some d_{i}, d′_{j} ∈ ℤ and δ_{i}, δ′_{j} ∈ I_{sc}. As c_{2}(I_{sc}^{3}) = 0, we have

c_{2}(x) = ∑_{i=1}^{m} d_{i}c_{2}(ρ̄(ω_{i})) + ∑_{j=1}^{n} d′_{j}c_{2}(ρ̄(ω′_{j})).

On the other hand, we have

c_{2}(ρ̄(ω_{i})) = 2m_{i}q and c_{2}(ρ̄(ω′_{j})) = 2m′_{j}q′

for all 1 ≤ i ≤ m and 1 ≤ j ≤ n and for some m_{i}, m′_{j} ∈ ℤ. Hence, c_{2}(x) ≡ 0 mod 2, thus q − q′ ∉ Sdec(G). Similarly, if m = 2 and n ≥ 3, then c_{2}(x) ≡ (d_{1} + d_{2})q mod 2, thus q − q′ ∉ Sdec(G), which completes the proof.

The previous proposition yields the following. Combining these results we obtain both indecomposable and semi-decomposable subgroups for an arbitrary split semisimple group of type B modulo central subgroups μ_{2}.

Corollary 7.2. (1) Let G = ∏_{i=1}^{m} SO_{2n_{i}+1}, n_{i} ≥ 2. Then,

Inv_{ind}^{3}(G) = Inv_{sd}^{3}(G) = 0.

(2) Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ_{2}, n_{i} ≥ 2, m ≥ 2, where μ_{2} is the diagonal subgroup. Then,

Inv_{ind}^{3}(G) = (ℤ/2ℤ)^{⊕m−1} and Inv_{sd}^{3}(G) = (ℤ/2ℤ)^{⊕k−1},

where k is the number of n_{i}’s such that n_{i} = 2.

(3) Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1}) × (∏_{i=1}^{m′} SO_{2n′_{i}+1}), n_{i}, n′_{i} ≥ 2. Then,

Inv_{ind}^{3}(G) = (ℤ/2ℤ)^{⊕k} and Inv_{sd}^{3}(G) = 0,

where k is the number of n_{i}’s such that n_{i} ≥ 3.

Proof. (1) We set d′ = 0 in (10). Then, we obtain Q(SO_{2m+1}) = 2ℤq ([14, §4b]). It immediately follows from the proof of Proposition 7.1 that we also have Dec(SO_{2m+1}) = 2ℤq.

(2) This follows by the same argument as in Proposition 7.1.

(3) As Dec(Spin_{2n+1}) = Sdec(Spin_{2n+1}) for any n ≥ 2, the same argument as in (1) shows that Sdec(G) = Dec(G). By [4, Theorem 13.4], Q(Spin_{2n+1}) = 2 Dec(Spin_{2n+1}) for any n ≥ 3 and Inv_{ind}^{3}(Spin_{5}) = 0, thus the same argument as in (1) proves the result for the indecomposable group.

8 Type C

In the present section we compute the groups of indecomposable and semidecomposable invariants for semisimple groups of type C. In particular, we show that for groups G = (∏_{i=1}^{m} SP_{2n_{i}})/μ_{2}, where m ≥ 2, n_{i} ≢ 0 mod 4 for all i = 1, …, m, and μ_{2} is the diagonal subgroup, any indecomposable invariant is semi-decomposable. We consider the index 2 case, which generalizes the example [15, Example 3.1] (the case n = m = 1) to groups of type C.

Proposition 8.1. Let G = (SP_{2m} × SP_{2n})/μ_{2}, m, n ≥ 1 where μ_{2} is the diagonal subgroup. Then, we have

Inv_{ind}^{3}(G) = \begin{cases}ℤ/2ℤ ⊕ ℤ/2ℤ & if m ≡ n ≡ 0 mod 4, \\ ℤ/2ℤ & otherwise, \end{cases}

and

Inv_{sd}^{3}(G) = \begin{cases}ℤ/2ℤ & if m ≡ n ≡ 0 or m ≢ 0 ≢ n mod 4, \\ 0 & otherwise. \end{cases}

In particular, if both n and m are not divisible by 4, then each indecomposable invariant is semi-decomposable.

Proof. Let {e_{1}, ⋯, e_{m}, e′_{1}, ⋯, e′_{n}} be a standard basis of T_{sc}^{∗} = ℤ^{m}⊕ℤ^{n}. Then T^{∗} consists of all linear combinations of standard basis elements with even sums of coefficients. Consider the ℤ-basis {x_{1}, …, x_{m}, x′_{1}, …, x′_{n}} of T^{∗} given by

x_{1} = e_{1} + e′_{1}, x′_{1} = e_{1} − e′_{1}, x_{i} = e_{1} − e_{i}, x′_{j} = e′_{1} − e′_{j}, i, j, > 1,

The standard basis can be expressed in terms of this basis over ℚ as

e_{i} = e_{1} − x_{i}, e′_{j} = e′_{1} − x′_{j}, where e_{1} = \frac{1}{2}(x_{1} + x′_{1}) and e′_{1} = \frac{1}{2}(x_{1} − x′_{1}).

The group S^{2}(T_{sc}^{∗})^{W} is generated by q = ∑_{i=1}^{m} e_{i}^{2} and q′ = ∑_{j=1}^{n} e_{j}^{′2}. Therefore, for any ϕ ∈ S^{2}(T_{sc}^{∗})^{W} there exist d, d′ ∈ ℤ such that ϕ = dq + d′q′ =

d( \frac{1}{4} (x_{1} + x′_{1})^{2} + ∑_{i>1}( \frac{1}{2} (x_{1} + x′_{1}) − x_{i})^{2}) + d′( \frac{1}{4} (x_{1} − x′_{1})^{2} + ∑_{j>1}( \frac{1}{2} (x_{1} − x′_{1}) − x′_{j})^{2}).

The form ϕ has integer coefficients at x_{1}x′_{1}, x_{i}x′_{j}, x_{i}^{2}, x_{j}^{′2}, i, j > 1 and it has coefficient \frac{1}{4}(dm + d′n) at x_{1}^{2} and at x_{1}^{′2}. Hence,

Q(G) = {dq + d′q′ | dm + d′n ≡ 0 mod 4}.  (12)

Consider the subgroup Dec(G) of decomposable invariants of G. As in the proof of [4, Lemma 14.2], since the Weyl group of G contains normal subgroups (ℤ/2ℤ)^{m} and (ℤ/2ℤ)^{n} generated by sign switching, we conclude that the coefficient at each e_{i} in the expansion of q_{χ} is divisible by 2, hence, Dec(G) ⊆ 2ℤq ⊕ 2ℤq′. Since c_{2}(ρ̄(2e_{1})) = 4q and c_{2}(ρ̄(2e′_{1})) = 4q′, we have 4ℤq ⊕ 4ℤq′ ⊆ Dec(G). Assume n ≡ m ≡ 0 mod 2. Since c_{2}(ρ̄(x_{2})) = 2(m − 1)q and c_{2}(ρ̄(x′_{2})) = 2(n−1)q′, we obtain Dec(G) = 2ℤq ⊕2ℤq′ and Q(G) = \{dq +d′q′ | d\frac{m}{2} +d′ \frac{n}{2} ≡ 0 mod 2}. Hence,

Inv_{ind}^{3}(G) ≃ \begin{cases}(ℤ/2ℤ)q ⊕ (ℤ/2ℤ)q′ & if n ≡ m ≡ 0 mod 4, \\ (ℤ/2ℤ)q & if m ≡ 0 ≢ n mod 4, \\ (ℤ/2ℤ)q′ & if m ≢ 0 ≡ n mod 4, \\ (ℤ/2ℤ)(q + q′) & if m ≢ 0 ≢ n mod 4. \end{cases}

Assume both n and m are odd. If n ≡ −m mod 4, then Q(G) = {dq + d′q′ | d ≡ d′ mod 4} ≃ ℤ/4ℤ(q + q′). Since c_{2}(ρ̄(x_{1})) = 2nq + 2mq′, Dec(G) = 2ℤ(q + q′) ⊕ 2ℤ(q − q′) and, therefore, Inv_{ind}^{3}(G) ≃ (ℤ/2ℤ)(q + q′). Similarly, if n ≡ m mod 4, then Inv_{ind}^{3}(G) ≃ (ℤ/2ℤ)(q − q′). Finally, assume n is odd and m is even. If m ≡ 0 mod 4, then Q(G) = ℤq⊕4ℤq′. Since c_{2}(ρ̄(x_{2})) = 2(m−1)q, Dec(G) = 2ℤq⊕4ℤq′, hence, Inv_{ind}^{3}(G) ≃ (ℤ/2ℤ)q. If m ≢ 0 mod 4, then d′ is even and

Q(G) = \{dq + d′q′ | d + \frac{d′}{2} ≡ 0 mod 2\} = \{(¯0, ¯0), (¯2, ¯0), (¯1, ¯2), (−¯1, ¯2)\},

where (d̄, d̄′) denotes dq + d′q′ modulo 4. Since Dec(G) = 2ℤq + 4ℤq′, we have Inv_{ind}^{3}(G) ≃ (ℤ/2ℤ)(q + 2q′).

As for semi-decomposable invariants, consider the element (cf. with h_{1,i} of Definition 3.2)

z = \frac{n}{gcd(m,n)} ρ̄(ω_{1}) − \frac{m}{gcd(m,n)} ρ̄(ω′_{1}).  (13)

By definition, we have y = e^{e_{1}}z ∈ ℤ[T^{∗}] ∩ I_{sc}^{W} and we obtain

c_{2}(y) = c_{2}((1 + (e^{e_{1}} − 1))z) = c_{2}(z) = \frac{n}{gcd(m,n)}q − \frac{m}{gcd(m,n)}q′

where the second equality holds since (e^{e_{1}} − 1)z ∈ I_{sc}^{3}. The element c_{2}(y) ∈ Sdec(G) coincides with the generator of Inv_{ind}^{3}(G) = ℤ/2ℤ except if n ≡ m ≡ 0 mod 4. So Inv_{sd}^{3}(G) = Inv_{ind}^{3}(G) = ℤ/2ℤ except if n ≡ m ≡ 0 mod 4. Assume that m ≡ n ≡ 0 mod 4. Then {q, q′} are generators of the group of indecomposable invariants. Consider an arbitrary element x ∈ ℤ[T^{∗}] ∩ I_{sc}^{W} and the ring homomorphism

ϕ: ℤ[T_{sc}^{∗}] → ℤ[T_{sc}^{∗}/T^{∗}] = ℤ[t]/(t^{2} − 2t)

given by ϕ(1−e^{−w_{odd}}) = ϕ(1−e^{−w′_{odd}}) = t and ϕ(1−e^{−w_{even}}) = ϕ(1−e^{−w′_{even}}) = 0. Write

x = ∑_{i=1}^{m}(d_{i} + δ_{i})ρ̄(ω_{i}) + ∑_{j=1}^{n}(d′_{i} + δ′_{i})ρ̄(ω′_{i}), d_{i}, d′_{j} ∈ ℤ, δ_{i}, δ′_{j} ∈ I_{sc}.

Since ker ϕ ⊃ ℤ[T^{∗}] ∩ I_{sc}^{W}, we obtain

0 = ϕ(x) = ∑_{odd i} 2^{i}\binom{m}{i}(d_{i} + 2s_{i})t + ∑_{odd j} 2^{j} \binom{n}{j}(d′_{j} + 2s′_{j})t.

Observe that if 2^{r} | m and i is odd, then 2^{r} | \binom{m}{i}. Dividing by the 2-primary part 2^{r} of the greatest common divisor of all the coefficients we obtain

(m/2^{r−1})d_{1} + (n/2^{r−1})d′_{1} ≡ 0 mod 2,

where 2^{r−1} = g.c.d.(v_{2}(n), v_{2}(m)) is the g.c.d. of the 2-primary parts. Therefore,

\begin{cases}d_{1} + d′_{1} ≡ 0 mod 2 & if v_{2}(m) = v_{2}(n) \\ d_{1} ≡ 0 mod 2 & if v_{2}(m) < v_{2}(n). \end{cases}  (14)

We then have

c_{2}(x) = (∑_{i=1}^{m} 2^{i−1}\binom{m − 1}{i − 1}d_{i})q + (∑_{j=1}^{n} 2^{j−1}\binom{n − 1}{j − 1}d′_{j})q′.

So c_{2}(x) ≡ d_{1}q + d′_{1}q′ mod Dec(G), where d_{1} and d′_{1} satisfy (14). Since

c_{2}(y) = \begin{cases}q + q′ mod Dec(G) & if v_{2}(n) = v_{2}(m) \\ q′ mod Dec(G) & if v_{2}(m) < v_{2}(n) \end{cases}

we conclude that c_{2}(y) is also a generator of Inv_{sd}^{3}(G) ≃ ℤ/2ℤ.

We now present a generalization of the previous proposition, which in turn determine both indecomposable and semi-decomposable subgroups for an arbitrary split semisimple group of type C modulo central subgroups μ_{2}.

Corollary 8.2. (1) Let G = ∏_{i=1}^{m} PGSp_{2n_{i}}, m, n_{i} ≥ 1. Then,

Inv_{ind}^{3}(G) ≃ (ℤ/2ℤ)^{⊕k},

where k is the number of n_{i}’s which are divisible by 4, and Sdec(G) = Dec(G), i.e., each semi-decomposable invariant is decomposable.

(2) Let G = (∏_{i=1}^{m} SP_{2n_{i}})/μ_{2}, m, n_{i} ≥ 1, where μ_{2} is the diagonal subgroup. Then,

Inv_{ind}^{3}(G) = \begin{cases}(ℤ/2ℤ)^{⊕m} & if ∀n_{i} ≡ 0 mod 4, \\ (ℤ/2ℤ)^{⊕m−1} & otherwise, \end{cases}

and

Inv_{sd}^{3}(G) = \begin{cases}(ℤ/2ℤ)^{⊕m−1} & if ∀n_{i} ≡ 0 or ∀n_{i} ≢ 0 mod 4, \\ (ℤ/2ℤ)^{⊕m−2} & otherwise. \end{cases}

(3) Let G = (∏_{i=1}^{m} PGSp_{2n_{i}}) × (∏_{i=1}^{m′} SP_{2n′_{i}}), n_{i}, n′_{i} ≥ 1. Then,

Inv_{ind}^{3}(G) = (ℤ/2ℤ)^{⊕k} and Sdec(G) = Dec(G),

where k is the number of n_{i}’s which are divisible by 4.

Proof. (1) Let G_{1} = PGSp_{2m} and G_{2} = PGSp_{2n}, m, n ≥ 1. It suffices to consider the case G = G_{1} ×G_{2} since the same arguments can be easily adapted to prove the case of three and more groups. We simply set d′ = 0 (resp. d = 0) in 12. Then, we have Q(G_{1}) = 4/ gcd(4, m)ℤq (resp. Q(G_{2}) = 4/ gcd(4, n)ℤq′). Similarly, by the proof of Proposition 8.1 we get Dec(G_{1}) = 4/ gcd(2, m)ℤq (resp. Dec(G_{2}) = 4/ gcd(2, n)ℤq′) ([14, §4b]). By (2) the answer for Inv_{ind}^{3}(G) then follows. As for semi-decomposable invariants, by (3) and Dec(G) = Q(G) for n ≢ 0 mod 4, it suffices to consider the case n ≡ m ≡ 0 mod 4. We follow arguments used in [15, §3c]. Let x ∈ ℤ[T^{∗}] ∩ I_{sc}^{W} be an arbitrary element. Write

x = ∑_{i=1}^{m}(d_{i} + δ_{i})ρ̄(ω_{i}) + ∑_{j=1}^{n}(d′_{j} + δ′_{j})ρ̄(ω′_{j})

for some d_{i}, d′_{j} ∈ ℤ and δ_{i}, δ′_{j} ∈ I_{sc}. Consider the ring homomorphism induced by the quotient map T_{sc}^{∗} → T_{sc}^{∗}/T^{∗}

ϕ: ℤ[T_{sc}^{∗}] ⟶ ℤ[T_{sc}^{∗}/T^{∗}] = ℤ[Λ_{w}/Λ_{r}] ⊗ ℤ[Λ′_{w}/Λ′_{r}] = ℤ[t, t′]/(t^{2} − 2t, t^{′2} − 2t′).

It is given by

ϕ(1 − e^{−ω_{odd}}) = t, ϕ(1 − e^{−ω′_{odd}}) = t′ and ϕ(1 − e^{−ω_{even}}) = ϕ(1 − e^{−ω′_{even}}) = 0.

Since x ∈ ℤ[T^{∗}], ϕ(x) = 0. Moreover, we have ϕ(I_{sc}) = (t, t′) and ϕ(ρ̄(ω_{i})) = |W (ω_{i})| · t, ϕ(ρ̄(ω′_{i})) = |W (ω′_{i})| · t′. Combining these facts we obtain

0 = ∑_{i=1}^{m}(d_{i} + ϕ(δ_{i}))ϕ(ρ̄(ω_{i})) + ∑_{j=1}^{n}(d′_{j} + ϕ(δ′_{j}))ϕ(ρ̄(ω′_{j}))

= ∑_{odd i} |W (ω_{i})| · (d_{i} + s_{i}t + s′_{i}t′)t + ∑_{odd j} |W (ω′_{j})| · (d′_{j} + r_{j}t + r′_{j}t′)t′,

for some s_{i}, s′_{i}, r_{j}, r′_{j} ∈ ℤ. Since |W (ω_{i})| = 2^{i}\binom{m}{i}, collecting the coefficients at t and t′, we get

d_{1} + 2s_{1} ≡ d′_{1} + 2r′_{1} ≡ 0 mod 2.

Hence, both d_{1} and d′_{1} are even. We now compute c_{2}(x). Since c_{2}(I_{sc}^{3}) = 0 and c_{2}(ρ̄(ω_{i})) = 2^{i−1}\binom{m−1}{i−1}q, we obtain

c_{2}(x) = ∑_{i=1}^{m} d_{i}c_{2}(ρ̄(ω_{i})) + ∑_{j=1}^{n} d′_{j}c_{2}(ρ̄(ω′_{j})) = 2sq + 2rq′, for some r, s ∈ ℤ.

Therefore, Sdec(G_{1}) ⊆ 2ℤq = Dec(G_{1}) and Sdec(G_{2}) ⊆ 2ℤq′ = Dec(G_{2}).

(2) This immediately follows from the same argument as in Proposition 8.1.

(3) As Dec(SP_{2n}) = Sdec(SP_{2n}) = Q(SP_{2n}) for any n ≥ 1 ([4, Theorem 14.3]), the same argument as in (1) shows that Sdec(G) = Dec(G) and the result for the indecomposable subgroup.

9 Type D

In this section we calculate the groups of indecomposable and semidecomposable invariants for an arbitrary product of simply-connected simple groups of type D modulo the (diagonal) central subgroups. We first consider the groups of index 2 and 4.

Proposition 9.1. Let G = (Spin_{2m} × Spin_{2n})/μ, where m, n ≥ 4 and m + n is even, and μ is a diagonal subgroup of G. Then

Inv_{ind}^{3}(G) = \begin{cases}ℤ/4ℤ & if μ ≃ μ_{4}, \\ ℤ/2ℤ & if μ ≃ μ_{2}, \end{cases} and Inv_{sd}^{3}(G) = \begin{cases}ℤ/2ℤ & if μ ≃ μ_{4}, \\ 0 & if μ ≃ μ_{2}. \end{cases}

Proof. Observe that there is a unique diagonal subgroup μ ≃ μ_{4} in the case where m and n are odd and there are two different diagonal subgroups μ ≃ μ_{2} : μ ⊆ μ_{4}^{2} if both m and n are odd and μ ⊆ μ_{2}^{4} otherwise. First, assume that μ ≃ μ_{4}. Then, by 5.1 the character group of the split maximal torus T of G is given by

\{ ∑_{i=1}^{m} a_{i}ω_{i} + ∑_{j=1}^{n} a′_{i}ω′_{i} | a_{m−1} + 3a_{m} + 2 ∑_{i=1}^{\frac{m−1}{2}} a_{2i−1} ≡ 3a′_{n−1} + a′_{n} + 2 ∑_{j=1}^{\frac{n−1}{2}} a′_{2j−1} mod 4\}.

(15)

Write ∑_{i=1}^{m} a_{i}ω_{i} +∑_{j=1}^{n} a′_{i}ω′_{i} = ∑_{i=1}^{m} b_{i}e_{i} +∑_{j=1}^{n} b′_{i}e′_{i} in terms of the standard basis vectors {e_{1}, ⋯, e_{m}, e′_{1}, ⋯, e′_{n}} of T_{sc}^{∗} = ℤ^{m} ⊕ ℤ^{n}. Then, the relation in (15) is equivalent to

2(b_{m−2} +b_{m−1} +b_{m} + ∑_{i=1}^{\frac{m−3}{2}} b_{2i−1} −b_{2i}) ≡ 2(b′_{n−2} +b′_{n−1} −b′_{n} + ∑_{j=1}^{\frac{n−3}{2}} b_{2j−1} −b_{2j}) mod 4.

(16)

Using (15) we choose the following basis {x_{1}, …, x_{m}, x′_{1}, …, x′_{n}} of T^{∗}:

\begin{matrix}x_{2i−1} = ω_{2i−1} + 2ω′_{n}, x_{2k} = ω_{2k}, x_{m−1} = ω_{m−1} + ω′_{n}, x_{m} = ω_{m} + 3ω′_{n}, \\ x′_{2j−1} = ω′_{2j−1} + 2ω′_{n}, x′_{2l} = ω′_{2l}, x′_{n−1} = ω′_{n−1} + ω′_{n}, x′_{n} = 4ω′_{n} \end{matrix} (17)

for 1 ≤ i ≤ (m − 1)/2, 1 ≤ k ≤ (m − 3)/2, 1 ≤ j ≤ (n − 1)/2, and 1 ≤ l ≤ (n − 3)/2. Let ψ be a quadratic form on (17) with integer coefficients. As the group S^{2}(T_{sc}^{∗})^{W} is generated by the normalized Killing forms

\begin{matrix}q := ∑_{i=1}^{m} ω_{i}^{2} − 2 ∑_{i=1}^{m−2} ω_{i}ω_{i+1} − 2ω_{m−2}ω_{m} = (∑_{i=1}^{m} e_{i}^{2})/2, \\ q′ := ∑_{j=1}^{n} ω_{j}^{′2} − 2 ∑_{j=1}^{n−2} ω′_{j}ω′_{j+1} − 2ω′_{n−2}ω′_{n} = (∑_{j=1}^{n} e_{j}^{′2})/2, \end{matrix}

from the equation ϕ = dq + d′q′ we get

ϕ = (\frac{md+nd′}{8} )x_{n}^{′2} + ψ,

where ψ is a quadratic form on (17) with integer coefficients. Therefore,

Q(G) = {dq + d′q′ | md + nd′ ≡ 0 mod 8}.  (18)

We show that Dec(G) = 4ℤ(q − q′) + 4ℤ(q + q′). First, by (15) we see that all elements c_{2}(ρ(2ω_{1})) = −8q, c_{2}(ρ(2ω′_{1})) = −8q′, c_{2}(ρ(ω_{1} + ω′_{1})) = −4nq − 4mq′ are contained in Dec(G), thus 4(q −q′), 4(q +q′) ∈ Dec(G). On the other hand, since Dec(G) is generated by c_{2}(ρ(λ)) for all λ ∈ T^{∗} and W contains normal subgroups (ℤ/2ℤ)^{m−1} and (ℤ/2ℤ)^{n−1}, we see that the coefficient at each e_{i} in the expansion of c_{2}(ρ(λ)) is divisible by 2, thus, Dec(G) ⊆ 4ℤq ⊕ 4ℤq′ (note that by (18) 4q, 4q′ ∉ Dec(G)). Hence,

Inv_{ind}^{3}(G) = ℤ/4ℤ( \frac{n}{gcd(m,n)} q − \frac{m}{gcd(m,n)} q′).

Now we show that Sdec(G) = 2ℤ(\frac{n}{gcd(m,n)}q − \frac{m}{gcd(m,n)}q′). First of all, as

c_{2}(ρ̄(ω_{i})) = 2m_{i}q and c_{2}(ρ̄(ω′_{j})) = 2m′_{j}q′  (19)

for all 1 ≤ i ≤ m and 1 ≤ j ≤ n and for some m_{i}, m′_{j} ∈ ℤ, by the same argument as in the proof of Proposition 7.1 we obtain \frac{n}{gcd(m,n)}q − \frac{m}{gcd(m,n)}q′ ∉ Sdec(G). On the other hand, consider an element

z = \frac{n}{gcd(m,n)} ρ̄(ω_{1}) − \frac{m}{gcd(m,n)} ρ̄(ω′_{1}).

Then, by (16) we obtain y := e^{e_{1}}z ∈ ℤ[T^{∗}] ∩ I_{sc}^{W}. Therefore, as in the proof of Proposition 8.1, we have c_{2}(y) = 2(\frac{n}{gcd(m,n)}q − \frac{m}{gcd(m,n)}q′) ∈ Sdec(G). Hence, the result for Inv_{sd}^{3}(G) follows. Now we assume that μ ≃ μ_{2}, so that μ ⊆ μ_{4}^{2} if m and n are odd and μ ⊆ μ_{2}^{4} otherwise. In both cases, the corresponding character group of T is given by

T^{∗} = { ∑_{i=1}^{m} a_{i}ω_{i} + ∑_{j=1}^{n} a′_{i}ω′_{i} | a_{m−1} + a_{m} ≡ a′_{n−1} + a′_{n} mod 2}.

By applying the same argument in the above μ ≃ μ_{4} case, we obtain

Q(G) = {dq + d′q′ | d + d′ ≡ 0 mod 2}.

Since c_{2}(ρ(ω_{1})) = −2q and c_{2}(ρ(ω′_{1})) = −2q′, we have 2q, 2q′ ∈ Dec(G). Moreover, by (19), we get q − q′ ∉ Sdec(G), thus Sdec(G) = Dec(G) = 2ℤq ⊕ 2ℤq′ and Inv_{ind}^{3}(G) = ℤ/2ℤ(q − q′).

We obtain the following generalization of the previous proposition. Together with Remark 9.3, they determine the groups of indecomposable and semidecomposable invariants for an arbitrary product of simply-connected simple groups of type D modulo the central subgroups μ_{2}.

Corollary 9.2. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ, n_{i} ≥ 4, m ≥ 1, where either all n_{i} are even or odd, and μ is a diagonal subgroup. Then,

Inv_{ind}^{3}(G) = \begin{cases}(ℤ/4ℤ)^{m−1} & if μ ≃ μ_{4}, \\ (ℤ/2ℤ)^{m−1} & if μ ≃ μ_{2}, \end{cases} and Inv_{sd}^{3}(G) = \begin{cases}(ℤ/2ℤ)^{m−1} & if μ ≃ μ_{4}, \\ 0 & if μ ≃ μ_{2}. \end{cases}

Proof. By the same argument as in Proposition 9.1, we obtain

Q(G) = \begin{cases}\{∑_{i=1}^{m} d_{i}q_{i} | ∑ n_{i}d_{i} ≡ 0 mod 8\} & if μ ≃ μ_{4}, \\ \{∑_{i=1}^{m} d_{i}q_{i} | ∑ d_{i} ≡ 0 mod 2\} & if μ ≃ μ_{2}, \end{cases}

where q_{1}, …, q_{m} are the corresponding normalized Killing forms of Spin_{2n_{i}}. Similarly,

Dec(G) = \begin{cases}⨁_{i=2}^{m} 4ℤ(q_{1} − q_{i}) ⊕ 4ℤ(q_{1} + q_{2}) & if μ ≃ μ_{4}, \\ ⨁_{i=1}^{m} 2ℤq_{i} & if μ ≃ μ_{2}, \end{cases}

thus, the factor groups follow. Following Proposition 9.1, we see that

Sdec(G)/ Dec(G) = ⨁_{i=2}^{m} ℤ/2ℤ(\frac{n_{i}}{gcd(n_{1},n_{i})}q_{1} − \frac{n_{1}}{gcd(n_{1},n_{i})}q_{i})

if μ ≃ μ_{4} and q_{1} − q_{i} ∉ Sdec(G) for all 2 ≤ i ≤ m if μ ≃ μ_{2}, which completes the proof.

Remark 9.3. (1) Let G′ = G × (∏_{i=1}^{m′} Spin_{2n′_{i}}), where G is the group from Corollary 9.2. Then, similar to the proof of Proposition 9.1 one can show that Inv_{sd}^{3}(G′) = Inv_{sd}^{3}(G). Moreover, by [4, Theorem 15.4] Inv_{ind}^{3}(G′) = Inv_{ind}^{3}(G) ⊕ (ℤ/2ℤ)^{⊕m′}.

(2) Let G′ = G × (∏_{i=1}^{m′} Spin_{2n′_{i}}), where G is either SO_{2m} or HSpin_{2m}. If G = SO_{2m}, then by [4, §15] Dec(G) = Dec(Spin_{2m}), thus Dec(G′) = Sdec(G′). Similarly, if G = HSpin_{2m}, then it follows from [2, §5] and [15, §3d] that Dec(G′) = Sdec(G′). One can also easily compute the indecomposable groups.

10 The PGO_{8}-case

In the present section we use the techniques developed in section 3 to give a direct proof of the main result of [15, Appendix]. In this section, G = PGO_{8} that is an adjoint group of Dynkin type D_{4}. The weight lattice of type D_{4} can be constructed as follows. We first take a ℚvector space with basis e_{1}, …, e_{4}. Then Λ has the following ℤ-basis consisting of fundamental weights:

ω_{1} = e_{1}, ω_{2} = e_{1} + e_{2}, ω_{3} = (e_{1} + e_{2} + e_{3} − e_{4})/2, ω_{4} = (e_{1} + e_{2} + e_{3} + e_{4})/2.

So the coordinates of elements of Λ are either all integers or half-integers. The group T^{∗} consists of all points such that all coordinates are integers, and the sum of coordinates is divisible by 2. We have Λ/T^{∗} = ℤ/2ℤ ⊕ ℤ/2ℤ with elements ¯0 = (0, 0), ω̄_{1} = (0, 1), ω̄_{3} = (1, 0), ω̄_{4} = (1, 1). The quotient map Λ → Λ/T^{∗} induces a grading on Λ and, hence, on ℤ[Λ]. We denote by Λ^{(0,0)}, Λ^{(0,1)}, Λ^{(1,0)}, Λ^{(1,1)} the respective homogeneous components. Each polynomial f ∈ ℤ[Λ] can be split into a sum of its homogeneous components, which we will denote by f^{(i,j)}. Denote the orbits in ℤ[Λ] by ρ(ω_{1}), …, ρ(ω_{4}) and the augmented orbits by ρ_{1}, …, ρ_{4} as in section 3.

Lemma 10.1. Let f_{1}, …, f_{4} ∈ ℤ[Λ] be such that f_{1}ρ_{1} + … + f_{4}ρ_{4} ∈ ℤ[T^{∗}]. Then, for each element i ∈ Λ/T^{∗} except for i = (0, 1) the sum of coefficients of f_{1}^{(i)} is even.

Proof. Consider a subgroup

Λ′ = {x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}e_{4} ∈ T^{∗} | x_{2} + x_{3} + x_{4} and x_{1} is even}.

We have Λ/Λ′ = ℤ/2ℤ ⊕ ℤ/4ℤ with a generator ω̄_{1} of order 2 and ω̄_{4} of order 4. Set R = ℤ/4ℤ. Consider a natural map ℤ[Λ] → R[Λ/Λ′] given by f ↦ f̄. Since Λ′ ⊂ T^{∗}, R[Λ/Λ′] is also a Λ/T^{∗}-graded algebra, and this map preserves the grading. By definition, the sum of coefficients of f modulo 4 is the sum of coefficients of f̄. So, it is sufficient to prove that for each element i ∈ Λ/T^{∗} except for i = (0, 1), the sum of coefficients of f̄_{1}^{(i)} = f̄_{1}^{(i)} is even.

Since ρ̄_{1} = 2\overline{e^{e_{1}}} + 2\overline{e^{e_{2}}}, ρ̄_{2} = ρ̄_{3} = ρ̄_{4} = 0 in R[Λ/Λ′] and f̄_{1}ρ̄_{1} + … + f̄_{4}ρ̄_{4} ∈ R[T^{∗}/Λ′], we obtain that f̄_{1}ρ̄_{1} ∈ R[T^{∗}/Λ′]. Since ρ̄_{1} ∈ ℤ[Λ^{(0,1)}], for each i ∈ Λ/T^{∗}, i ≠ 0, we have

0 = (f̄_{1}ρ̄_{1})^{(i)} = f̄_{1}^{(i−(0,1))}ρ̄_{1} = 2\overline{e^{e_{1}}}(1 + \overline{e^{e_{1}+e_{2}}})f̄_{1}^{(i−(0,1))}.

Therefore, for each j ∈ Λ/T^{∗}, j ≠ (0, 1), all coefficients of (1 + \overline{e^{e_{1}+e_{2}}})f̄_{1}^{(j)} are divisible by 2. Observe that the classes of 0 and of ω_{2} in Λ/Λ′ are all elements of Λ/Λ′ that belong to T^{∗}/Λ′. So, if λ ∈ Λ/Λ′, and f ∈ R[Λ/Λ′], then the coefficient of (1+\overline{e^{e_{1}+e_{2}}})f at e^{λ} is the sum of coefficients of f in front of e^{λ′} for all λ′ ∈ Λ/Λ′ such that λ = λ′ mod T^{∗} (there are exactly two such λ′, one of them is λ, the other is λ + e_{1} + e_{2}). If f is a homogeneous polynomial of degree j ∈ Λ/T^{∗}, and λ ∈ Λ/Λ′ is mapped to j by the natural projection Λ/Λ′ → Λ/T^{∗}, then the coefficient of (1 + \overline{e^{e_{1}+e_{2}}})f in front of e^{λ} is the sum of all coefficients of f. Therefore, the sum of all coefficients of f̄_{1}^{(j)} is divisible by 2 for each j ∈ Λ/T^{∗}, j ≠ (0, 1).

Lemma 10.2. Let f_{1}, …, f_{4} ∈ ℤ[Λ] be such that f_{1}ρ_{1} + … + f_{4}ρ_{4} ∈ ℤ[T^{∗}]. Then,

(1) for each element i ∈ Λ/T^{∗} except for i = (1, 0) the sum of coefficients of f_{3}^{(i)} is even;

(2) for each element i ∈ Λ/T^{∗} except for i = (1, 1) the sum of coefficients of f_{4}^{(i)} is even.

Proof. (1) Consider an automorphism ψ of Λ (induced by an outer automorphism of PGO_{8}) that interchanges ω_{1} and ω_{3} and keeps ω_{2} and ω_{4} invariant. It preserves T^{∗}. So, it also acts on Λ/T^{∗} = ℤ/2 ⊕ ℤ/2 by interchanging (0, 1) and (1, 0) (and keeping (0, 0) and (1, 1)). By definition ψ maps the graded components of a polynomial f ∈ ℤ[T^{∗}] to graded components of ψ(f), more precisely, ψ(f^{(i)}) = ψ(f)^{ψ(i)}. In particular, ψ(f)^{(0,1)} = ψ(f^{(1,0)}). Since ψ interchanges ρ_{1} and ρ_{3}, and keeps ρ_{2} and ρ_{4} unchanged,

ψ(f_{1}ρ_{1} + … + f_{4}ρ_{4}) = ψ(f_{1})ρ_{3} + ψ(f_{2})ρ_{2} + ψ(f_{3})ρ_{1} + ψ(f_{4})ρ_{4}.

Finally, observe that the sum of coefficients of ψ(f) is the same as the sum of coefficients of f. We then apply Lemma 10.1.

(2) The proof is completely similar to the proof of the previous case, the only difference is that now ψ should interchange ω_{1} and ω_{4} and keep ω_{2} and ω_{3} unchanged.

Proposition 10.3. Let f_{1}, …, f_{4} ∈ ℤ[Λ] be such that f_{1}ρ_{1}+…+f_{4}ρ_{4} ∈ ℤ[T^{∗}]. Then the sum of all coefficients of f_{i} is even for i = 1, 3, 4.

Proof. Set ω_{(0,1)} = ω_{1}, ω_{(1,0)} = ω_{3}, ω_{(1,1)} = ω_{4}, and ω_{(0,0)} = ω_{2}. Then the class of ω_{(i,j)} in Λ/T^{∗} is precisely (i, j). Set R = ℤ/16ℤ and consider the natural map ℤ[Λ] → R[Λ/T^{∗}] given by f ↦ f̄. Since f_{1}ρ_{1}+…+f_{4}ρ_{4} ∈ ℤ[T^{∗}], ρ̄_{i} = 8\overline{e^{ω_{i}}} −8, i = 1, 3, 4 and ρ̄_{2} = 24\overline{e^{ω_{2}}} −24 = 0,

f̄_{1}ρ̄_{1} + f̄_{3}ρ̄_{3} + f̄_{4}ρ̄_{4} is a constant in R[Λ/T^{∗}].

For i = 1, 3, 4 and (j, k) ∈ Λ/T^{∗}, denote by c_{i}^{(j,k)} the sum of coefficients of f_{i}^{(j,k)} modulo 16. Then f̄_{i}^{(j,k)} = c_{i}^{(j,k)}\overline{e^{ω_{(j,k)}}}. By Lemma 10.1 and 10.2, all numbers c_{i}^{(j,k)} are even, except for, possibly, c_{1}^{(0,1)}, c_{3}^{(1,0)}, and c_{4}^{(1,1)}. Observe that if c_{i}^{(j,k)} is even, then c_{i}^{(j,k)}\overline{e^{ω_{(j,k)}}} ρ̄_{i} = 0 in R[Λ/T^{∗}] since ρ̄_{i} is divisible by 8. Therefore, both expressions

c_{1}^{(0,1)}\overline{e^{ω_{1}}} ρ̄_{1} + c_{3}^{(1,0)}\overline{e^{ω_{3}}} ρ̄_{3} + c_{4}^{(1,1)}\overline{e^{ω_{4}}} ρ̄_{4} and
8c_{1}^{(0,1)}\overline{e^{ω_{1}}} + 8c_{3}^{(1,0)}\overline{e^{ω_{3}}} + 8c_{4}^{(1,1)}\overline{e^{ω_{4}}} + 8(c_{1}^{(0,1)} + c_{3}^{(1,0)} + c_{4}^{(1,1)})

are constants in R[Λ/T^{∗}]. So, the coefficients c_{1}^{(0,1)}, c_{3}^{(1,0)}, and c_{4}^{(1,1)} are even which means that for all i = 1, 3, 4 and (j, k) ∈ Λ/T^{∗}, c_{i}^{(j,k)} is even. But then the sum of all coefficients of f_{i} is even for i = 1, 3, 4.

We now give a direct proof of the result obtained in [15, Appendix] using a computer algorithm

Corollary 10.4. If G = PGO_{8}, then any semi-decomposable invariant of G is decomposable.

Proof. Let x ∈ ℤ[T^{∗}] ∩ I_{sc}^{W}. Similar to [15, §3c] we write

x = ∑_{i=1}^{4}(d_{i} + δ_{i})ρ̄(ω_{i}) for some d_{i} ∈ ℤ, δ_{i} ∈ I_{sc}.

Then c_{2}(x) = ∑_{i=1}^{4} d_{i}c_{2}(ρ̄_{i}). By Proposition 10.3 we have d_{1} ≡ d_{3} ≡ d_{4} ≡ 0 mod 2. Since c_{2}(ρ̄_{i}) = 2q for i = 1, 3, 4 and c_{2}(ρ̄_{2}) = 12q by [15, §3d], we obtain that c_{2}(x) ∈ 4ℤq = Dec(G).

11 Type E

We now treat the exceptional cases. In the following we show that any semidecomposable invariant is decomposable for semisimple groups of type E_{6} and E_{7}.

Lemma 11.1. Let G be a split semisimple group of type E_{6} or E_{7}. Then, Sdec(G) = Dec(G), i.e., each semi-decomposable invariant is decomposable.

Proof. We denote by E_{6}^{ad} (resp. E_{7}^{ad}) a split simple adjoint group of type E_{6} (resp. E_{7}) and by E_{6}^{sc} (resp. E_{7}^{sc}) a split simple simply connected group of type E_{6} (resp. E_{7}). We first consider the case where G is a semisimple group of type E_{6}, i.e., G = (E_{6}^{sc} × ⋯ × E_{6}^{sc})/μ (n copies of E_{6}^{sc}) for some central subgroup μ. Let q_{1}, ⋯, q_{n} be the corresponding normalized Killing forms for each copy of E_{6}^{sc} in G. Since Dec(E_{6}^{sc}) = Dec(E_{6}^{ad}) = 6q_{i} by [14, §4b], we have Dec(E_{6}^{sc} × ⋯ × E_{6}^{sc}) = Dec(E_{6}^{ad} × ⋯ × E_{6}^{ad}) = 6ℤq_{1} ⊕ ⋯ ⊕ 6ℤq_{n}. As Dec(E_{6}^{ad} × E_{6}^{ad}) ⊆ Dec(G) ⊆ Dec(E_{6}^{sc} × E_{6}^{sc}), we conclude that Dec(G) = 6ℤq_{1} ⊕ ⋯ ⊕ 6ℤq_{n}. Now we show Sdec(G) ⊆ Dec(G). Similar to the proof of Proposition 7.1, we consider an arbitrary element x ∈ I_{sc}^{W}:

x = ∑_{j=1}^{n} ∑_{i=1}^{6}(d_{ij} + δ_{ij})ρ̄(ω_{ij})

for some d_{ij} ∈ ℤ and δ_{ij} ∈ I_{sc}, where {ω_{1j}, …, ω_{6j}} is the fundamental weights of each copy of E_{6}^{sc}. Since c_{2}(I_{sc}^{3}) = 0, we obtain

c_{2}(x) = ∑_{j=1}^{n} ∑_{i=1}^{6} d_{ij}c_{2}(ρ̄(ω_{ij})).  (20)

By [11, §2], each element c_{2}(ρ̄(ω_{ij})) in (20) iz contained in 6ℤq_{j}. Hence, Sdec(G) ⊆ Dec(G), so the equality holds. Let G = (E_{7}^{sc} × ⋯ × E_{7}^{sc})/μ (n copies of E_{7}^{sc}) for some central subgroup μ. Then, the same argument together with Dec(E_{6}^{sc}) = Dec(E_{6}^{ad}) = 12q_{i} ([14, §4b]) shows that Dec(G) = Sdec(G) = 12ℤq_{1} ⊕ ⋯ ⊕ 12ℤq_{n}.

We determine the indecomposable groups for an arbitrary product of split simply-connected simple groups of type E_{6} (resp. E_{7}) modulo the diagonal subgroups μ_{3} (resp. μ_{2}).

Proposition 11.2. (1) Let G = (E_{6}^{sc} × ⋯ × E_{6}^{sc})/μ_{3} with n (≥ 2) copies of a split simple simply connected group E_{6}^{sc} of type E_{6} and the diagonal subgroup μ_{3}. Then

Inv_{ind}^{3}(G) = ℤ/2ℤ ⊕ (ℤ/6ℤ)^{⊕n−1} and Sdec(G) = Dec(G).

(2) Let G = (E_{7}^{sc} × ⋯ × E_{7}^{sc})/μ_{2} with n (≥ 2) copies of a split simple simply connected group E_{7}^{sc} of type E_{7} and the diagonal subgroup μ_{2}. Then

Inv_{ind}^{3}(G) = ℤ/3ℤ ⊕ (ℤ/12ℤ)^{⊕n−1} and Sdec(G) = Dec(G).

Proof. By Lemma 11.1, it suffices to compute the indecomposable groups.

(1) Assume that n = 2. Then, by 5.1 the character group of the split maximal torus T of G is given by

T^{∗} = {∑_{i=1}^{6} a_{i}ω_{i} + ∑_{i=1}^{6} a′_{i}ω′_{i} | a_{1} + a′_{1} + a_{5} + a′_{5} ≡ a_{3} + a′_{3} + a_{6} + a′_{6} mod 3}. (21)

Choose a basis {x_{1}, … x_{6}, x′_{1}, …, x′_{6}} of (21) as follows

\begin{matrix}x_{1} = ω_{1} + ω′_{6}, x_{2} = ω_{2}, x_{3} = ω_{3} + 2ω′_{6}, x_{4} = ω_{4}, x_{5} = ω_{5} + ω′_{6}, x_{6} = ω_{6} + 2ω′_{6}, \\ x′_{1} = ω′_{1} + ω′_{6}, x′_{2} = ω′_{2}, x′_{3} = ω′_{3} + 2ω′_{6}, x′_{4} = ω′_{4}, x′_{5} = ω′_{5} + ω′_{6}, x′_{6} = 3ω′_{6}. \end{matrix}

Let ϕ be a quadratic form on x_{i}, x′_{i} over ℤ. Since the group S^{2}(T_{sc}^{∗})^{W} is generated by the normalized Killing forms

\begin{matrix}q := ω_{1}^{2} − ω_{1}ω_{3} + ω_{2}^{2} − ω_{2}ω_{4} + ω_{3}^{2} − ω_{3}ω_{4} + ω_{4}^{2} − ω_{4}ω_{5} + ω_{5}^{2} − ω_{5}ω_{6} + ω_{6}^{2} & and \\ q′ := ω_{1}^{′2} − ω′_{1}ω′_{3} + ω_{2}^{′2} − ω′_{2}ω′_{4} + ω_{3}^{′2} − ω′_{3}ω′_{4} + ω_{4}^{′2} − ω′_{4}ω′_{5} + ω_{5}^{′2} − ω′_{5}ω′_{6} + ω_{6}^{′2}, \end{matrix} ; (22)

from the equation ϕ = dq + d′q′, we obtain

ϕ = (\frac{2d+2d′}{3} )x_{6}^{′2} + ψ,

where ψ is a quadratic form with integer coefficients. Therefore,

Q(G) = {dq + d′q′ | d + d′ ≡ 0 mod 3}.  (23)

Therefore, by (23) and the proof of the previous lemma, we obtain

Inv_{ind}^{3}(G) = ℤ/2ℤ(3q + 3q′) ⊕ ℤ/6ℤ(q + 2q′).

For n ≥ 3, the same argument shows that

Q(G) = {d_{1}q + ⋯ d_{n}q_{n} | ∑_{i=1}^{n} d_{i} ≡ 0 mod 3},

where q_{1}, …, q_{n} are the corresponding normalized Killing forms. Hence, the result for the indecomposable group follows from Lemma 11.1.

(2) Assume that n = 2. Then, by 5.1 the character group of the split maximal torus T of G is given by

T^{∗} = {∑_{i=1}^{7} a_{i}ω_{i} + ∑_{i=1}^{7} a′_{i}ω′_{i} | a_{2} + a_{5} + a_{7} ≡ a′_{2} + a′_{5} + a′_{7} mod 2}.

Since the group S^{2}(T_{sc}^{∗})^{W} is generated by the normalized Killing forms

q := q_{6} − ω_{6}ω_{7} + ω_{7}^{2} and q′ := q′_{6} − ω′_{6}ω′_{7} + ω_{7}^{′2},

where q_{6} and q′_{6} are the normalized Killing forms of E_{6} in (22), the same argument as in (1) shows

Q(G) = {dq + d′q′ | d + d′ ≡ 0 mod 4}.

Hence, by Lemma 11.1 Inv_{ind}^{3}(G) = ℤ/3ℤ(4q′) ⊕ ℤ/12ℤ(q − q′). For n ≥ 3, the same argument shows that Q(G) = {d_{1}q+⋯ d_{n}q_{n} | ∑_{i=1}^{n} d_{i} ≡ 0 mod 4}, where q_{1}, …, q_{n} are the corresponding normalized Killing forms, which comptues the indecomposable group together with Lemma 11.1.

Remark 11.3. (1) Let G = (E_{6}^{ad}×⋯×E_{6}^{ad})×(E_{6}^{sc}×⋯×E_{6}^{sc}), n(≥ 0) copies of E_{6}^{ad} and m(≥ 0) copies of E_{6}^{sc}. It follows by (2), (23) and the proof of Lemma 11.1 that Inv_{ind}^{3}(G) = (ℤ/2ℤ)^{⊕n} ⊕ (ℤ/6ℤ)^{⊕m}. Similarly, for the same group G replacing E_{6} by E_{7}, we have Inv_{ind}^{3}(G) = (ℤ/3ℤ)^{⊕n} ⊕ (ℤ/12ℤ)^{⊕m}.

(2) Note that the center μ_{3} ×μ_{3} of E_{6}^{sc} ×E_{6}^{sc} contains two nontrivial (≠ μ_{3} ×1, 1 × μ_{3}) central subgroups which is isomorphic to μ_{3}: a diagonal subgroup and a non-diagonal subgroup. Assume that μ is non-diagonal. Then, the character group of T becomes

T^{∗} = {∑_{i=1}^{6} a_{i}ω_{i} + ∑_{i=1}^{6} a′_{i}ω′_{i} | a_{1} + a′_{3} + a_{5} + a′_{6} ≡ a′_{1} + a_{3} + a′_{5} + a_{6} mod 3}.

In this case, we have the same Q(G) as in (23), thus have the same indecomposable group.

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Sanghoon Baek

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea Rostislav Devyatov Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada

Kirill Zainoulline

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada