PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 4, April 1998, Pages 1237–1244 S 0002-9939(98)04316-0

THE CHERN CHARACTER FOR CLASSICAL MATRIX GROUPS

JAY A. WOOD

(Communicated by Thomas Goodwillie)

To S. S. Chern

Abstract. We give explicit formulas for representations of classical matrix groups whose Chern characters have lowest order terms equal to standard characteristic classes. For SO(2r), the Euler class e does not arise in this way, but 2^{r−1}e does arise in this way.

A complex representation ρ of a compact Lie group G gives rise to a complex vector bundle ξ_{ρ} over the classifying space BG of G. In this way one obtains a ring homomorphism R(G) → K(BG) from the complex representation ring of G to the complex K-theory of BG. Composing with the Chern character yields a ring homomorphism,

R(G) → K(BG) ⟶^{ch} H^{∗∗}(BG; ℚ).

Our interest here is in determining which cohomology classes on BG occur as the lowest order terms of some Chern character ch(ξ_{ρ}). We will consider the classical matrix groups U(n), SU(n), Sp(n), and SO(n), and we will produce explicit representations ρ whose Chern characters ch(ξ_{ρ}) have lowest order terms equaling standard characteristic classes. In all cases considered here, the lowest order terms of the Chern characters ch(ξ_{ρ}) turn out to be integral classes. For U(n), SU(n), Sp(n), and SO(2r + 1), every homogeneous integral cohomology class modulo torsion on BG arises as the lowest order term of some Chern character ch(ξ_{ρ}); for SO(2r), the Euler class does not so arise, but 2^{r−1} times the Euler class does.

In the cases considered here where BG has integral cohomology which is free of torsion, namely, U(n), SU(n), and Sp(n), general results of Atiyah and Hirzebruch ([3], §2.5 and §4.8) imply that the lowest order terms of any Chern character ch(ξ_{ρ}) are integral classes and that any homogeneous integral class arises in this way. Thus our only contribution in these cases is the explicit form of certain representations which give rise to standard characteristic classes.

For SO(n), whose classifying space has 2-torsion, the general results of Atiyah and Hirzebruch do not apply directly. It may be possible to obtain some general results in the presence of torsion by using the Atiyah-Hirzebruch spectral sequence together with integrality results such as those of Adams [1] and Maunder [7]; we have

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors October 1, 1996.

1991 Mathematics Subject Classification. Primary 55R40.

The author was partially supported by NSA grants MDA904-94-H-2025 and MDA904-96-1-0067, and by Purdue University Calumet Scholarly Research Awards.

not pursued such general results here. Instead, we produce explicit representations which give rise to the Pontryagin classes and, when n = 2r, to 2^{r−1} times the Euler class. The explicit formulas we produce then allow us to show that the multiple 2^{r−1} is best possible.

We use, without comment, many facts from the representation and cohomology theories of these classical matrix groups. In particular, all computations really take place over a maximal torus T via identifications R(G) ≅ R(T)^{W} and (valid for these groups) H^{∗}(BG; ℤ)/torsion ≅ H^{∗}(BT; ℤ)^{W}. Relevant references are [5] and [6]. For notational convenience we will write ch(ρ) in place of ch(ξ_{ρ}).

We thank D. M. Davis, W. G. Dwyer, S. Gitler, R. J. Milgram, and S. Stolz for their helpful comments.

1. U(n)

Let ρ_{n} be the standard n-dimensional “birth-certificate” representation of U(n), and let Λ^{i} = Λ^{i}(ρ_{n}) be the ith exterior power of ρ_{n}. It is known that R(U(n)) = ℤ[Λ^{1}, Λ^{2}, …, Λ^{n}, Λ^{−n}]. Also known is that H^{∗}(B U(n); ℤ) = ℤ[c_{1}, …, c_{n}], where c_{i} ∈ H^{2i}(B U(n); ℤ) is the ith Chern class of ξ_{ρ_{n}}.

Upon restriction to a maximal torus T of U(n), the representations Λ^{i} split as follows:

1 + Λ^{1}t + Λ^{2}t^{2} + ⋯ + Λ^{n}t^{n} = ∏_{i=1}^{n}(1 + α_{i}t), (1.1)

where α_{1}, …, α_{n} are the standard roots of T. It is well-known that H^{∗}(BT; ℤ) = ℤ[x_{1}, …, x_{n}], where x_{1}, …, x_{n} are classes of degree 2 with x_{i} = c_{1}(ξ_{α_{i}}). The work of Borel and Hirzebruch [4] (see [6], p. 282) tells us that the Chern character for bundles ξ_{ρ} over BT arises by replacing α_{i} by e^{x_{i}}. For example, we have our first proposition. (We shall use the notation hot to mean higher order terms.)

Proposition 1.1. On U(n), let ρ = Λ^{n} − Λ^{n−1} + Λ^{n−2} − ⋯ + (−1)^{n} · 1. Then ch(ρ) = c_{n} + hot.

Proof. Setting t = −1 in (1.1), we have ρ = ∏(α_{i} − 1). Then ch(ρ) = ∏(e^{x_{i}} − 1) = ∏(x_{i} + hot) = c_{n} + hot.

Here we have made use of some standard facts concerning Chern classes. In the identification H^{∗}(B U(n); ℤ) ≅ H^{∗}(BT; ℤ)^{W}, c_{i} equals the ith elementary symmetric polynomial in x_{1}, …, x_{n}. In particular, c_{n} = ∏ x_{i}.

Next, we need to understand how the representations Λ^{i} = Λ^{i}(ρ_{n}) of U(n) behave upon restriction to U(n − 1). This amounts to setting α_{n} = 1 in (1.1). A short calculation leads to the next lemma.

Lemma 1.2. The restriction of the representation Λ^{i}(ρ_{n}) to U(n − 1) equals

Λ^{i}(ρ_{n−1}) + Λ^{i−1}(ρ_{n−1}).

Theorem 1.3. For any k, 1 ≤ k ≤ n, let d = n−k, and set ρ_{k,n} to be the following representation of U(n):

ρ_{k,n} = ∑_{j=0}^{k}(−1)^{j}\binom{d + j}{d}Λ^{k−j}
= Λ^{k} − (d + 1)Λ^{k−1} + · · · .

Then for any m, k ≤ m ≤ n, the restriction of ρ_{k,n} to U(m) equals ρ_{k,m}. In particular, ρ_{k,n} restricted to U(k) is ρ_{k,k} = Λ^{k} − Λ^{k−1} + Λ^{k−2} − ⋯ + (−1)^{k} · 1.

Proof. The general result will follow by induction once we know that ρ_{k,n} restricts to ρ_{k,n−1} on U(n − 1). This is immediate from Lemma 1.2.

Corollary 1.4. Over U(n), we have ch(ρ_{k,n}) = c_{k} + hot.

Proof. Restriction from U(n) to U(k) induces a projection

H^{∗}(B U(n); ℤ) ⟶^{i^{∗}} H^{∗}(B U(k); ℤ),

whose kernel is the ideal generated by c_{k+1}, …, c_{n}. Since the Chern character is natural, we have

i^{∗} ch(ρ_{k,n}) = ch(ρ_{k,k}) = c_{k} + hot,

by Proposition 1.1. But all the non-zero elements of ker i^{∗} have higher order than c_{k}, hence the result.

Remark 1.5. It turns out that ρ_{k,n} = γ^{k}(ρ_{n} − n), where ρ_{n} is the birth-certificate representation of U(n), and γ^{k} is the K-theory operation due to Grothendieck (see [6], pp. 162–3). In factored form, ρ_{k,n} is the kth elementary symmetric polynomial in α_{1} − 1, …, α_{n} − 1.

Corollary 1.6. Any non-zero homogeneous element of H^{∗}(B U(n); ℤ) arises as the lowest order term of ch(ρ) for some ρ ∈ R(U(n)).

Proof. Any homogeneous element y of H^{∗}(B U(n); ℤ) is expressible as a sum of monomials in the Chern classes c_{k}. Since the Chern character is a ring homomorphism, the corresponding sum of monomials in the ρ_{k,n} will produce y as its lowest order term.

2. SU(n)

Since SU(n) ⊂ U(n), the representations ρ_{k,n} = Λ^{k} − (n − k + 1)Λ^{k−1} + ⋯ of U(n) restrict to SU(n) and have ch(ρ_{k,n}) = c_{k} + hot. It turns out that we can do slightly better than this, achieving c_{k} via a representation involving only 1, Λ^{1}, …, Λ^{k−1}.

Since Λ^{n} = 1 over SU(n), we have R(SU(n)) = ℤ[Λ^{1}, Λ^{2}, …, Λ^{n−1}]. Also, H^{∗}(B SU(n); ℤ) = ℤ[c_{2}, …, c_{n}], i.e., c_{1} = 0.

Over U(n), the representation ρ_{k,n} arose by finding representations over U(n) which would restrict to ρ_{k,k} = Λ^{k} − Λ^{k−1} + Λ^{k−2} − ⋯ + (−1)^{k} · 1 over U(k). For example, to find ρ_{k,k+1}, Lemma 1.2 requires ρ_{k,k+1} to start with Λ^{k}. Restricting back to U(k), this Λ^{k} will contribute Λ^{k} + Λ^{k−1}. Then ρ_{k,k+1} needs Λ^{k} − 2Λ^{k−1} to account for all the Λ^{k−1}’s over U(k). This helps explain the origins of the ρ_{k,n}.

Over SU(n), we still have ρ_{n,n} = Λ^{n} − Λ^{n−1} + Λ^{n−2} − ⋯ + (−1)^{n} · 1, but now Λ^{n} = 1. Thus we can write ρ_{n,n} = −Λ^{n−1} + Λ^{n−2} − ⋯ + (−1)^{n} · 1 + 1. As in the previous paragraph, we can lift ρ_{n,n} to ρ′_{n,n+1}, but now the formulas are different.

Theorem 2.1. For any k, 1 ≤ k ≤ n, let d = n−k, and set ρ′_{k,n} to be the following representation of SU(n):

ρ′_{k,n} = 1 + ∑_{j=0}^{k−1}(−1)^{j+1}\binom{d + j}{d}Λ^{k−1−j}
= −Λ^{k−1} + (d + 1)Λ^{k−2} − · · · .

Then for any m, k ≤ m ≤ n, the restriction of ρ′_{k,n} to SU(m) equals ρ′_{k,m}. In particular, ρ′_{k,n} restricted to SU(k) is ρ′_{k,k} = ρ_{k,k} = −Λ^{k−1}+Λ^{k−2}−⋯+(−1)^{k}·1+1.

Corollary 2.2. Over SU(n), we have ch(ρ′_{k,n}) = c_{k} + hot. Any non-zero homogeneous element of H^{∗}(B SU(n); ℤ) arises as the lowest order term of ch(ρ) for some ρ ∈ R(SU(n)).

3. Sp(n)

The birth-certificate representation of Sp(n) acts on ℍ^{n}. Restricting to the underlying complex vector space ℂ^{2n} ≅ ℍ^{n}, we obtain the standard complex representation ρ_{2n} of Sp(n). Let Λ^{i} = Λ^{i}(ρ_{2n}) be the ith exterior power of ρ_{2n}, 1 ≤ i ≤ 2n. The representation ρ_{2n} preserves a non-degenerate symplectic form, and this implies isomorphisms Λ^{i} ≅ Λ^{2n−i}. While the Λ^{i} are not irreducible, they do account for all of R(Sp(n)):

R(Sp(n)) = ℤ[Λ^{1}, Λ^{2}, …, Λ^{n}].

Now R(Sp(n)) ≅ R(T)^{W}, with T a maximal torus in Sp(n), and R(T) = ℤ[α_{1}, α_{1}^{−1}, …, α_{n}, α_{n}^{−1}]. In this notation we have the splitting (over T)

1 + Λ^{1}t + ⋯ + Λ^{2n}t^{2n} = ∏_{i=1}^{n}(1 + α_{i}t)(1 + α_{i}^{−1}t). (3.1)

Setting t = −1 and using Λ^{i} ≅ Λ^{2n−i}, we have

Λ^{n} − 2Λ^{n−1} + 2Λ^{n−2} − ⋯ + (−1)^{n} · 2 = ∏_{i=1}^{n}(α_{i} + α_{i}^{−1} − 2). (3.2)

H^{∗}(B Sp(n); ℤ) is known to be torsion-free, and H^{∗}(B Sp(n); ℤ) ≅ H^{∗}(BT; ℤ)^{W}. If we write H^{∗}(BT; ℤ) ≅ ℤ[x_{1}, …, x_{n}], with deg x_{i} = 2, then H^{∗}(B Sp(n); ℤ) ≅ ℤ[q_{1}, q_{2}, …, q_{n}], with q_{i} ∈ H^{4i}(B Sp(n); ℤ). The element q_{i} equals the ith elementary symmetric polynomial in x_{1}^{2}, …, x_{n}^{2}, and q_{i} is called the ith symplectic Pontryagin class.

Proposition 3.1. Over Sp(n), denote ρ_{n,n} = Λ^{n} −2Λ^{n−1} +2Λ^{n−2} −⋯+(−1)^{n} ·2. Then ch(ρ_{n,n}) = q_{n} + hot.

Proof. From (3.2) we have ch(ρ_{n,n}) = ∏(e^{x_{i}} + e^{−x_{i}} − 2) = ∏(x_{i}^{2} + hot) = q_{n} + hot.

Next we determine how the Λ^{i} = Λ^{i}(ρ_{2n}) behave under restriction to Sp(n − 1). This amounts to setting α_{n} = α_{n}^{−1} = 1 in (3.1). It then follows that Λ^{i}(ρ_{2n}) restricts over Sp(n − 1) to Λ^{i}(ρ_{2n−2}) + 2Λ^{i−1}(ρ_{2n−2}) + Λ^{i−2}(ρ_{2n−2}). For what follows, set

\binom{n}{k}′ = \binom{n}{k} + \binom{n − 1}{k}. (3.3)

Theorem 3.2. For any k, 1 ≤ k ≤ n, let d = n−k, and set ρ_{k,n} to be the following representation of Sp(n):

ρ_{k,n} = ∑_{j=0}^{k}(−1)^{j+1}\binom{2d + j}{2d}′Λ^{k−j}
= Λ^{k} − (2d + 2)Λ^{k−1} + · · · .

Then for any m, k ≤ m ≤ n, the restriction of ρ_{k,n} to Sp(m) equals ρ_{k,m}. In particular, ρ_{k,n} restricted to Sp(k) is ρ_{k,k} = Λ^{k} − 2Λ^{k−1} + 2Λ^{k−2} − ⋯ + (−1)^{k} · 2.

Proof. This is a straight-forward verification.

Corollary 3.3. Over Sp(n), we have ch(ρ_{k,n}) = q_{k} + hot. Any non-zero homogeneous element of H^{∗}(B Sp(n); ℤ) arises as the lowest order term of ch(ρ) for some ρ ∈ R(Sp(n)).

Proof. Since the restriction map

H^{∗}(B Sp(n); ℤ) = ℤ[q_{1}, …, q_{n}] → H^{∗}(B Sp(k); ℤ) = ℤ[q_{1}, …, q_{k}]

has as kernel the ideal generated by q_{k+1}, …, q_{n}, the same argument as in Corollary 1.4 applies.

4. SO(n): Pontryagin classes

The standard complex representation ρ_{n} of SO(n) on ℂ^{n} is the complexification of the birth-certificate representation of SO(n) on ℝ^{n}. Let Λ^{i} = Λ^{i}(ρ_{n}) be the ith exterior power of ρ_{n}. Because ρ_{n} preserves the symmetric form coming from ℝ^{n}, we have isomorphisms Λ^{i} ≅ Λ^{n−i} ; Λ^{i} is irreducible for i < n/2.

If n = 2r + 1 is odd, this is all that happens:

R(SO(2r + 1)) = ℤ[Λ^{1}, Λ^{2}, …, Λ^{r}],

with

1 + Λ^{1}t + ⋯ + Λ^{2r+1}t^{2r+1} = (1 + t) ∏_{i=1}^{r}(1 + α_{i}t)(1 + α_{i}^{−1}t) (4.1)

being the splitting over a maximal torus T (of rank r).

If n = 2r is even, we have the splitting

1 + Λ^{1}t + ⋯ + Λ^{2r}t^{2r} = ∏_{i=1}^{r}(1 + α_{i}t)(1 + α_{i}^{−1}t) (4.2)

over a maximal torus T. The middle dimensional Λ^{r} = Λ^{r}(ρ_{2r}) decomposes as Λ^{r} = Λ_{+}^{r} + Λ_{−}^{r} into self-dual and anti-self-dual parts. The representations Λ^{1}, …, Λ^{r−1}, Λ_{+}^{r}, Λ_{−}^{r} generate the representation ring R(SO(2r)), subject to a single relation (which we will not need to know explicitly).

Turning now to cohomology, H^{∗}(B SO(n); ℤ) has 2-torsion and

H^{∗}(B SO(n); ℤ)/torsion ≅ H^{∗}(BT; ℤ)^{W} .

Writing H^{∗}(BT; ℤ) = ℤ[x_{1}, …, x_{r}], with deg x_{i} = 2, we have

H^{∗}(B SO(2r + 1); ℤ)/torsion ≅ ℤ[p_{1}, p_{2}, …, p_{r}],
H^{∗}(B SO(2r); ℤ)/torsion ≅ ℤ[p_{1}, p_{2}, …, p_{r−1}, e],

where the ith Pontryagin class p_{i} ∈ H^{4i}(B SO(n); ℤ)/torsion is the ith elementary symmetric polynomial in x_{1}^{2}, …, x_{r}^{2}, and e = x_{1}x_{2} ⋯ x_{r} is the Euler class (when n = 2r). Of course, e^{2} = p_{r}. In this section we will show that every Pontryagin class arises as the lowest order term of some ch(ρ); the Euler class will be treated in the next section.

Let ρ_{r,2r} = Λ^{r} − 2Λ^{r−1} + 2Λ^{r−2} − ⋯ + (−1)^{r} · 2, a representation of SO(2r). As usual, it arises by setting t = −1 in (4.2), so that

ρ_{r,2r} = ∏_{i=1}^{r}(α_{i} + α_{i}^{−1} − 2).

Just as for Sp(n), we have

Proposition 4.1. Over SO(2r), let ρ_{r,2r} = Λ^{r} − 2Λ^{r−1} + 2Λ^{r−2} − ⋯ + (−1)^{r} · 2. Then ch(ρ_{r,2r}) = p_{r} + hot.

Using (4.1) and (4.2) we see that Λ^{i}(ρ_{n+1}) of SO(n + 1) restricts to Λ^{i}(ρ_{n}) + Λ^{i−1}(ρ_{n}) on SO(n). Remember the notation from (3.3).

Theorem 4.2. For any k, 1 ≤ k ≤ n/2, let d = n − 2k, and set ρ_{k,n} to be the following representation of SO(n):

ρ_{k,n} = ∑_{j=0}^{k}(−1)^{j}\binom{d + j}{d}′Λ^{k−j}
= Λ^{k} − (d + 2)Λ^{k−1} + · · · .

Then for any m, 2k ≤ m ≤ n, the restriction of ρ_{k,n} to SO(m) equals ρ_{k,m}. In particular, ρ_{k,n} restricted to SO(2k) is ρ_{k,2k} = Λ^{k} −2Λ^{k−1} +2Λ^{k−2} −⋯+(−1)^{k} ·2.

Corollary 4.3. Over SO(n), we have ch(ρ_{k,n}) = p_{k} + hot. Any non-zero homogeneous element of H^{∗}(B SO(2r + 1); ℤ)/torsion arises as the lowest order term of ch(ρ) for some ρ ∈ R(SO(2r + 1)). The same is true for SO(2r) provided the element is expressible entirely in terms of Pontryagin classes (including p_{r} = e^{2}; no odd powers of e are allowed).

5. SO(2r): Euler class

Recall that on SO(2r), Λ^{r} = Λ_{+}^{r} + Λ_{−}^{r}. Since ρ_{r,2r} = Λ^{r} − 2Λ^{r−1} + 2Λ^{r−2} − ⋯ + (−1)^{r} · 2 has ch(ρ_{r,2r}) = p_{r} + hot, it makes sense to split ρ_{r,2r} as ρ_{r,2r} = μ_{+} + μ_{−} where

μ_{±} = Λ_{±}^{r} − Λ^{r−1} + Λ^{r−2} − ⋯ + (−1)^{r} · 1.

Theorem 5.1. Over SO(2r) we have ch(μ_{±}) = ±2^{r−1}e+hot. No smaller multiple of the Euler class e arises as the lowest order term of ch(ρ) for any ρ ∈ R(SO(2r)).

Proof. We follow a computation of Atiyah and Hirzebruch, [2], §4.1.

The representations Λ_{±}^{r}, when pulled back to Spin(2r), are related to the spinor representations Δ^{±} by

Δ^{+} ⊗ Δ^{+} = Λ_{+}^{r} + Λ^{r−2} + Λ^{r−4} + ⋯,
Δ^{−} ⊗ Δ^{−} = Λ_{−}^{r} + Λ^{r−2} + Λ^{r−4} + · · · .

Observe that Λ_{+}^{r} − Λ_{−}^{r} = Δ^{+} ⊗ Δ^{+} − Δ^{−} ⊗ Δ^{−} = (Δ^{+} + Δ^{−})(Δ^{+} − Δ^{−}).

Over a maximal torus for Spin(2r) we have the splittings

Δ^{±} = ∑_{ϵ_{1}···ϵ_{r}=±1} α_{1}^{ϵ_{1}/2} ⋯ α_{r}^{ϵ_{r}/2},

where ϵ_{i} = ±1. Then

ch(Δ^{±}) = ∑_{ϵ_{1}···ϵ_{r}=±1} e^{1/2(ϵ_{1}x_{1}+···+ϵ_{r}x_{r})}.

A straight-forward computation shows that

ch(Δ^{+} + Δ^{−}) = ∏_{i=1}^{r} (e^{x_{i}/2} + e^{−x_{i}/2})

and

ch(Δ^{+} − Δ^{−}) = ∏_{i=1}^{r} (e^{x_{i}/2} − e^{−x_{i}/2}) .

Then ch(Λ_{+}^{r} − Λ_{−}^{r}) equals their product, i.e.,

ch(Λ_{+}^{r} − Λ_{−}^{r}) = ∏_{i=1}^{r} (e^{x_{i}} − e^{−x_{i}}) = 2^{r}e + hot.

Since μ_{+} + μ_{−} = ρ_{r,2r} and μ_{+} − μ_{−} = Λ_{+}^{r} − Λ_{−}^{r}, we can solve for ch(μ_{±}):

ch(μ_{±}) = \frac{1}{2} ch(ρ_{r,2r}) ± \frac{1}{2} ch(Λ_{+}^{r} − Λ_{−}^{r}).

Now, ch(ρ_{r,2r}) = p_{r} + hot, while ch(Λ_{+}^{r} − Λ_{−}^{r}) = 2^{r}e + hot. But p_{r} has degree 4r while e has degree 2r, so ch(μ_{±}) = ±2^{r−1}e + hot.

As for the sharpness of this result, we first observe that the inclusion

R(SO(2r + 1)) → R(SO(2r))

induced by restriction makes R(SO(2r)) a free module over R(SO(2r+1)), generated by 1 and Λ_{+}^{r} (see [5], VI, (6.6)). The fact that μ_{+} has the form

μ_{+} = Λ_{+}^{r} − Λ^{r−1} + Λ^{r−2} + ⋯ + (−1)^{r} · 1

allows one to use 1 and μ_{+} as the free module generators. In addition, the form of ρ_{k,2r+1} = Λ^{k} − ⋯ implies that

R(SO(2r + 1)) = ℤ[Λ^{1}, …, Λ^{r}] = ℤ[ρ_{1,2r+1}, …, ρ_{r,2r+1}].

The sharpness now follows from Corollary 4.3 and the first part of this theorem.

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[5] Th. Bröcker and T. tom Dieck, Representations of compact Lie groups, Grad. Texts Math., vol. 98, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1985. MR 86i:22023 [6] D. Husemoller, Fibre bundles, 2nd ed., Grad. Texts Math., vol. 20, Springer-Verlag, New York,

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Department of Mathematics, Computer Science & Statistics, Purdue University Calumet, Hammond, Indiana 46323-2094

   E-mail address: wood@calumet.purdue.edu