dim V = n = 2m. This is the main case of interest for generalization to central

simple algebras with involution, since central simple algebras of odd degree with

involution of the ¬rst kind are split (see (??)). Since dim V is even, the center of

C0 (V, q) is an ´tale quadratic F -algebra which we denote Z. Let ι be the nontrivial

e

automorphism of Z/F . In the Cli¬ord algebra C(V, q) we have

v · ζ = ι(ζ) · v for v ∈ V , ζ ∈ Z,

108 II. INVARIANTS OF INVOLUTIONS

hence

(9.2) ι(ζ) — (v — c1 ) = (v — c1 ) · ζ for v ∈ V , c1 ∈ C1 (V, q), ζ ∈ Z.

In view of this equation, we may consider V — C1 (V, q) as a right module over the

Z-algebra ι C0 (V, q)op —Z C0 (V, q): for v ∈ V , c1 ∈ C1 (V, q) and c0 , c0 ∈ C0 (V, q)

we set

(v — c1 ) · (ιcop — c0 ) = c0 — (v — c1 ) · c0 = v — c0 c1 c0 .

0

On the other hand V — C1 (V, q) also is a left module over End(V ); since the actions

of End(V ) and C0 (V, Q) commute, there is a natural homomorphism of F -algebras:

ν : ι C0 (V, q)op —Z C0 (V, q) ’ EndEnd(V ) V — C1 (V, q) = EndF C1 (V, q) .

This homomorphism is easily seen to be injective: this is obvious if Z is a ¬eld,

because then the tensor product on the left is a simple algebra. If Z F — F,

the only nontrivial ideals in the tensor product are generated by elements in Z.

However the restriction of ν to Z is injective, since the condition v · ζ = 0 for all

v ∈ V implies ζ = 0. Therefore, ν is injective.

The image of ν is determined as follows: through ν, the center Z of C0 (V, q) acts

on V — C1 (V, q) by End(V )-linear homomorphisms; the set V — C1 (V, q) therefore

has a structure of left End(V ) — Z-module (where the action of Z is through the

right action of C0 (V, q)): for f ∈ End(V ), ζ ∈ Z, v ∈ V and c1 ∈ C1 (V, q),

(f — ζ) · (v — c1 ) = f (v) — c1 ζ.

The map ν may then be considered as an isomorphism of Z-algebras:

∼

ν : ι C0 (V, q)op —Z C0 (V, q) ’ EndEnd(V )—Z V — C1 (V, q) = EndZ C1 (V, q) .

’

Equivalently, ν identi¬es ι C0 (V, q)op —Z C0 (V, q) with the centralizer of Z (= ν(Z))

in EndEnd(V ) V — C1 (V, q) .

9.B. De¬nition of the Cli¬ord bimodule. In order to de¬ne an analogue

of V — C1 (V, q) for a central simple algebra with quadratic pair (A, σ, f ), we ¬rst

de¬ne a canonical representation of the symmetric group S2n on A—n .

Representation of the symmetric group. As in §??, we write A for the

underlying vector space of the F -algebra A. For any integer n ≥ 2, we de¬ne a

generalized sandwich map

Sandn : A—n ’ HomF (A—n’1 , A)

by the condition:

Sandn (a1 — · · · — an )(b1 — · · · — bn’1 ) = a1 b1 a2 b2 · · · bn’1 an .

(Thus, Sand2 is the map denoted simply Sand in §??).

(9.3) Lemma. For any central simple F -algebra A, the map Sandn is an isomor-

phism of vector spaces.

Proof : Since it su¬ces to prove Sandn is an isomorphism after scalar extension, we

may assume that A = Mn (F ). It su¬ces to prove injectivity of Sandn , since A—n

and HomF (A—n’1 , A) have the same dimension. Let eij (i, j = 1, . . . , n) be the

matrix units of Mn (F ). Take any nonzero ± ∈ A—n and write

n n n n

±= ··· ci1 j1 ...in jn ei1 j1 — · · · — ein jn

i1 =1 j1 =1 in =1 jn =1

§9. THE CLIFFORD BIMODULE 109

with the ci1 j1 ...in jn ∈ F . Some coe¬cient of ±, say cp1 q1 ...pn qn is nonzero. Then,

Sandn (±)(eq1 p2 — eq2 p3 — · · · — eqi pi+1 — · · · — eqn’1 pn ) =

n n

ci1 q1 p2 q2 ...pn jn ei1 jn

i1 =1 jn =1

which is not zero, since its p1 qn -entry is not zero.

(9.4) Proposition. Let (A, σ) be a central simple F -algebra with involution of the

¬rst kind. If char F = 2, suppose further that σ is orthogonal. For all n ≥ 1 there is

a canonical representation ρn : S2n ’ GL(A—n ) of the symmetric group S2n which

is described in the split case as follows: for every nonsingular symmetric bilinear

space (V, b) and v1 , . . . , v2n ∈ V ,

ρn (π) •b (v1 — v2 ) — · · · — •b (v2n’1 — v2n ) =

•b vπ’1 (1) — vπ’1 (2) — · · · — •b (vπ’1 (2n’1) — vπ’1 (2n) )

∼

for all π ∈ S2n where •b : V — V ’ EndF (V ) is the standard identi¬cation (??).

’

Proof : We ¬rst de¬ne the image of the transpositions „ (i) = (i, i + 1) for i = 1,

. . . , 2n ’ 1.

If i is odd, i = 2 ’ 1, let

ρn „ (i) = IdA — · · · — IdA — σ — IdA — · · · — IdA ,

where σ lies in -th position. In the split case, σ corresponds to the twist under the

standard identi¬cation A = V — V ; therefore,

ρn „ (2 ’ 1) (v1 — · · · — v2 — v2 — · · · — v2n ) =

’1

v1 — · · · — v 2 — v 2 — · · · — v2n .

’1

If i is even, i = 2 , we de¬ne ρn „ (i) by the condition:

Sandn ρn „ (i) (u) (x) = Sandn (u) IdA — · · · — IdA — σ — IdA — · · · — IdA (x)

for u ∈ A—n and x ∈ A—n’1 where σ lies in -th position. The same computation

as in (??) shows that ρn „ (2 ) satis¬es the required condition in the split case.

In order to de¬ne ρn (π) for arbitrary π ∈ S2n , we use the fact that „ (1), . . . ,

„ (2n ’ 1) generate S2n : we ¬x some factorization

π = „1 —¦ · · · —¦ „ s where „1 , . . . , „s ∈ {„ (1), . . . , „ (2n ’ 1)}

and de¬ne ρn (π) = ρn („1 ) —¦ · · · —¦ ρn („s ). The map ρn (π) thus de¬ned meets the

requirement in the split case, hence ρn is a homomorphism in the split case. By

extending scalars to a splitting ¬eld, we see that ρn also is a homomorphism in the

general case. Therefore, the de¬nition of ρn (π) does not actually depend on the

factorization of π.

The de¬nition. Let (σ, f ) be a quadratic pair on a central simple F -algebra A.

For all n ≥ 1, let γn = ρn (1, 2, . . . , 2n)’1 ∈ GL(A—n ) where ρn is as in (??),

and let γ = •γn : T (A) ’ T (A) be the induced linear map. Thus, in the split case

(A, σ, f ) = EndF (V ), σq , fq , we have, under the standard identi¬cation A = V —V

of (??):

γ(v1 — · · · — v2n ) = γn (v1 — · · · — v2n ) = v2 — · · · — v2n — v1

110 II. INVARIANTS OF INVOLUTIONS

for v1 , . . . , v2n ∈ V .

Let also T+ (A) = •n≥1 A—n . The vector space T+ (A) carries a natural structure

of left and right module over the tensor algebra T (A). We de¬ne a new left module

structure — as follows: for u ∈ T (A) and v ∈ T+ (A) we set

u — v = γ ’1 u — γ(v) .

Thus, in the split case A = V — V , the product — avoids the ¬rst factor:

(u1 — · · · — u2i ) — (v1 — · · · — v2j ) = v1 — u1 — · · · — u2i — v2 — · · · — v2j

for u1 , . . . , u2i , v1 , . . . , v2j ∈ V . (Compare with the de¬nition of — in §??).

(9.5) De¬nition. The Cli¬ord bimodule of (A, σ, f ) is de¬ned as

T+ (A)

B(A, σ, f ) =

J1 (σ, f ) — T+ (A) + T+ (A) · J1 (σ, f )

where J1 (σ, f ) is the two-sided ideal of T (A) which appears in the de¬nition of the

Cli¬ord algebra C(A, σ, f ) (see (??)).

The map a ∈ A ’ a ∈ T+ (A) induces a canonical F -linear map

(9.6) b : A ’ B(A, σ, f ).

(9.7) Theorem. Let (A, σ, f ) be a central simple F -algebra with a quadratic pair.

(1) The F -vector space B(A, σ, f ) carries a natural C(A, σ, f )-bimodule structure

where action on the left is through —, and a natural left A-module structure.

(2) In the split case (A, σ, f ) = EndF (V ), σq , fq , the standard identi¬cation

∼

•q : V — V ’ EndF (V )

’

induces a standard identi¬cation of Cli¬ord bimodules

∼

V —F C1 (V, q) ’ B(A, σ, f ).

’

(3) The canonical map b : A ’ B(A, σ, f ) is an injective homomorphism of left 13

A-modules.

(4) dimF B(A, σ, f ) = 2(deg A)’1 deg A.

Proof : By extending scalars to split A, it is easy to verify that

J2 (σ, f ) — T+ (A) ⊆ T+ (A) · J1 (σ, f ) and T+ (A) · J2 (σ, f ) ⊆ J1 (σ, f ) — T+ (A).

Therefore, the actions of T (A) on T+ (A) on the left through — and on the right

through the usual product induce a C(A, σ, f )-bimodule structure on B(A, σ, f ).

We de¬ne on T+ (A) a left A-module structure by using the multiplication map

A ’ A which carries a—b to ab. Explicitly, for a ∈ A and u = u1 —· · ·—ui ∈ A—i ,

—2

we set

a · u = au1 — u2 — · · · — ui .

Thus, in the split case (A, σ, f ) = (EndF (V ), σq , fq ), we have, under the standard

identi¬cation A = V — V :

a · (v1 — · · · — v2i ) = a(v1 ) — v2 — · · · — v2i .

It is then clear that the left action of A on T+ (A) commutes with the left and right

actions of T (A). Therefore, the subspace J1 (σ, f ) — T+ (A) + T+ (A) · J1 (σ, f ) is

13 Therefore, the image of a ∈ A under b will be written ab .

§9. THE CLIFFORD BIMODULE 111

preserved under the action of A, and it follows that B(A, σ, f ) inherits this action

from T+ (A).

∼

In the split case, the standard identi¬cation •’1 : A ’ V — V induces a sur-

’

q

jective linear map from B(A, σ, f ) onto V — C1 (V, q). Using an orthogonal decom-

position of (V, q) into 1- or 2-dimensional subspaces, one can show that

dimF B(A, σ, f ) ¤ dimF V dimF C1 (V, q).

Therefore, the induced map is an isomorphism. This proves (??) and (??), and (??)

follows by dimension count. Statement (??) is clear in the split case (see (??)), and

the theorem follows.

As was observed in the preceding section, there is no signi¬cant loss if we

restrict our attention to the case where the degree of A is even, since A is split if

its degree is odd. Until the end of this subsection, we assume deg A = n = 2m.

According to (??), the center Z of C(A, σ, f ) is then a quadratic ´tale F -algebra.

e

Let ι be the non-trivial automorphism of Z/F . By extending scalars to split the

algebra A, we derive from (??):

(9.8) x · ζ = ι(ζ) — x for x ∈ B(A, σ, f ), ζ ∈ Z.

Therefore, we may consider B(A, σ, f ) as a right module over ι C(A, σ, f )op —Z

C(A, σ, f ): for c, c ∈ C(A, σ, f ) and x ∈ B(A, σ, f ), we set

x · (ιcop — c ) = c — x · c .

Thus, B(A, σ, f ) is an A-ι C(A, σ, f )op —Z C(A, σ, f )-bimodule, and there is a natural

homomorphism of F -algebras:

ν : ι C(A, σ, f )op —Z C(A, σ, f ) ’ EndA B(A, σ, f ).

By comparing with the split case, we see that the map ν is injective, and that its

image is the centralizer of Z (= ν(Z)) in EndA B(A, σ, f ). Endowing B(A, σ, f )

with a left A —F Z-module structure (where the action of Z is through ν), we may

thus view ν as an isomorphism

∼

ν : ι C(A, σ, f )op —Z C(A, σ, f ) ’ EndA—Z B(A, σ, f );

(9.9) ’

ι op

it is de¬ned by xν( c —c )

= c — x · c for c, c ∈ C(A, σ, f ) and x ∈ B(A, σ, f ).

The canonical involution. We now use the involution σ on A to de¬ne an

involutorial A-module endomorphism ω of B(A, σ, f ). As in §??, σ denotes the

involution of C(A, σ, f ) induced by σ, and „ is the involution on C(V, q) which is

the identity on V .

(9.10) Proposition. The A-module B(A, σ, f ) is endowed with a canonical endo-

morphism14 ω such that for c1 , c2 ∈ C(A, σ, f ), x ∈ B(A, σ, f ) and a ∈ A:

(c1 — x · c2 )ω = σ(c2 ) — xω · σ(c1 ) (ab )ω = ab ,

and

where b : A ’ B(A, σ, f ) is the canonical map. Moreover, in the split case

(A, σ, f ) = (EndF (V ), σq , fq )

we have ω = IdV — „ under the standard identi¬cations A = V — V , B(A, σ, f ) =

V — C1 (V, q).

14 Since B(A, σ, f ) is a left A-module, ω will be written to the right of its arguments.

112 II. INVARIANTS OF INVOLUTIONS

Proof : Let ω = γ ’1 —¦σ : T+ (A) ’ T+ (A) where σ is the involution on T (A) induced

by σ. Thus, in the split case A = V — V :

ω(v1 — · · · — v2n ) = v1 — v2n — v2n’1 — · · · — v3 — v2 .

By extending scalars to a splitting ¬eld of A, it is easy to check that for a ∈ A, u 1 ,

u2 ∈ T (A) and v ∈ T+ (A),

ω(u1 — v · u2 ) = σ(u2 ) — ω(v) · σ(u1 ), ω(a · v) = a · ω(v) and ω(a) = a.

It follows from the ¬rst equation that

ω J1 (σ, f ) — T+ (A) = T+ (A) · σ J1 (σ, f ) ⊆ T+ (A) · J1 (σ, f )

and

ω T+ (A) · J1 (σ, f ) = σ J1 (σ, f ) — T+ (A) ⊆ J1 (σ, f ) — T+ (A),

hence ω induces an involutorial F -linear operator ω on B(A, σ, f ) which satis¬es

the required conditions.

We thus have ω ∈ EndA B(A, σ, f ). Moreover, it follows from the ¬rst property

of ω in the proposition above and from (??) that for x ∈ B(A, σ, f ) and ζ ∈ Z,

(x · ζ)ω = σ(ζ) — xω = xω · ι —¦ σ(ζ) .

The restriction of σ to Z is determined in (??): σ is of the ¬rst kind if m is even

and of the second kind if m is odd. Therefore, ω is Z-linear if m is odd, hence it

belongs to the image of ι C(A, σ, f )op —Z C(A, σ, f ) in EndA B(A, σ, f ) under the

natural monomorphism ν. By contrast, when m is even, ω is only ι-semilinear. In

this case, we de¬ne an F -algebra

ι

C(A, σ, f ) —Z C(A, σ, f ) • ι C(A, σ, f ) —Z C(A, σ, f ) · z

E(A, σ, f ) =

where multiplication is de¬ned by the following equations:

z(ιc — c ) = (ιc — c)z z 2 = 1.

for c, c ∈ C(A, σ, f ),

We also de¬ne a map ν : E(A, σ, f ) ’ EndA B(A, σ, f ) by

ω

(ιc1 —c2 +ιc3 —c4 ·z)

xν = σ(c1 ) — x · c2 + σ(c3 ) — x · c4

for x ∈ B(A, σ, f ) and c1 , c2 , c3 , c4 ∈ C(A, σ, f ). The fact that ν is a well-de¬ned

F -algebra homomorphism follows from the properties of ω in (??), and from the

hypothesis that deg A is divisible by 4 which ensures that σ is an involution of the

¬rst kind: see (??).

(9.11) Proposition. If deg A ≡ 0 mod 4, the map ν is an isomorphism of F -

algebras.

Proof : Suppose ν (u + vz) = 0 for some u, v ∈ ι C(A, σ, f ) —Z C(A, σ, f ). Then

ν (u) = ’ν (vz); but ν (u) is Z-linear while ν (vz) is ι-semilinear. Therefore,

ν (u) = ν (vz) = 0. It then follows that u = v = 0 since the natural map ν is

injective. This proves injectivity of ν . Surjectivity follows by dimension count:

since dimF B(A, σ, f ) = 2deg A’1 deg A, we have rdimA B(A, σ, f ) = 22m’1 hence

deg EndA B(A, σ, f ) = 22m’1 by (??). On the other hand,

dimF E(A, σ, f ) = 2 dimF ι C(A, σ, f ) —Z C(A, σ, f )

2

= 22(2m’1) .

= dimF C(A, σ, f )

§9. THE CLIFFORD BIMODULE 113

9.C. The fundamental relations. In this section, A denotes a central simple

F -algebra of even degree n = 2m with a quadratic pair (σ, f ). The fundamental

relations between the Brauer class [A] of A and the Brauer class of the Cli¬ord

algebra C(A, σ, f ) are the following:

(9.12) Theorem. Let Z be the center of the Cli¬ord algebra C(A, σ, f ).

(1) If deg A ≡ 0 mod 4 (i.e., if m is even), then

2

(9.13) C(A, σ, f ) =1 in Br(Z).

(9.14) NZ/F C(A, σ, f ) = [A] in Br(F ).

(2) If deg A ≡ 2 mod 4 (i.e., if m is odd ), then

2

(9.15) C(A, σ, f ) = [AZ ] in Br(Z).

(9.16) NZ/F C(A, σ, f ) = 1 in Br(F ).

(If Z = F — F , the norm NZ/F is de¬ned by NF —F/F (C1 — C2 ) = C1 —F C2 : see

the end of §??).

Proof : Equations (??) and (??) follow by (??) from the fact that the canonical

involution σ on C(A, σ, f ) is of the ¬rst kind when deg A ≡ 0 mod 4 and of the

second kind when deg A ≡ 2 mod 4.

To prove equations (??) and (??), recall the natural isomorphism (??):

∼

ν : ι C(A, σ, f )op —Z C(A, σ, f ) ’ EndA—Z B(A, σ, f ).

’

By (??), it follows that AZ = A —F Z is Brauer-equivalent to ι C(A, σ, f )op —Z

C(A, σ, f ). If m is odd, the canonical involution σ is of the second kind; it therefore

de¬nes an isomorphism of Z-algebras:

ι

C(A, σ, f )op C(A, σ, f ),

and (??) follows. Note that the arguments above apply also in the case where

Z F —F ; then C(A, σ, f ) = C + —C ’ for some central simple F -algebras C + , C ’ ,

and there is a corresponding decomposition of B(A, σ, f ) which follows from its Z-

module structure:

B(A, σ) = B+ — B’ .

Then EndA—Z B(A, σ, f ) = EndA B+ —EndA B’ and ι C(A, σ, f )op = C ’op —C +op ,

and the isomorphism ν can be considered as

∼

ν : (C + —F C ’op ) — (C ’ —F C +op ) ’ (EndA B+ ) — (EndA B’ ).

’

Therefore, A is Brauer-equivalent to C + —F C ’op and to C ’ — C +op . Since σ is

of the second kind when m is odd, we have in this case C ’op C + , hence A is

Brauer-equivalent to C +—2 and to C ’—2 , proving (??).

Similarly, (??) is a consequence of (??), as we proceed to show. Let

B (A, σ) = B(A, σ)ω

denote the F -subspace of ω-invariant elements in B(A, σ). For the rest of this

section, we assume deg A ≡ 0 mod 4. As observed before (??), the involution ω is

ι-semilinear, hence the multiplication map

B (A, σ) —F Z ’ B(A, σ)

is an isomorphism of Z-modules. Relation (??) follows from (??) and the following

claim:

114 II. INVARIANTS OF INVOLUTIONS

Claim. The isomorphism ν of (??) restricts to an F -algebra isomorphism

∼

NZ/F C(A, σ, f ) ’ EndA B (A, σ, f ).

’

To prove the claim, observe that the subalgebra

EndA B (A, σ, f ) ‚ EndA B(A, σ, f )

is the centralizer of Z and ω, hence it is also the F -subalgebra of elements which

commute with ω in EndA—Z B(A, σ, f ). The isomorphism ν identi¬es this algebra

with the algebra of switch-invariant elements in ι C(A, σ, f ) —Z C(A, σ, f ), i.e., with

NZ/F C(A, σ, f ).

§10. The Discriminant Algebra

A notion of discriminant may be de¬ned for hermitian spaces on the same model

as for symmetric bilinear spaces. If (V, h) is a hermitian space over a ¬eld K, with

respect to a nontrivial automorphism ι of K (of order 2), the determinant of the

Gram matrix of h with respect to an arbitrary basis (e1 , . . . , en ) lies in the sub¬eld

F ‚ K elementwise ¬xed under ι and is an invariant of h modulo the norms of K/F .

We may therefore de¬ne the determinant by

· N (K/F ) ∈ F — /N (K/F )

det h = det h(ei , ej ) 1¤i,j¤n

where N (K/F ) = NK/F (K — ) is the group of norms of K/F . The discriminant is

the signed determinant:

disc h = (’1)n(n’1)/2 det h(ei , ej ) ∈ F — /N (K/F ).

1¤i,j¤n

If δ ∈ F — is a representative of disc h, the quaternion algebra K • Kz where

multiplication is de¬ned by zx = ι(x)z for x ∈ K and z 2 = δ does not depend on

the choice of the representative δ. We denote it by (K, disc h)F ; thus

√

(±, δ)F if K = F ( ±) (char F = 2),

(K, disc h)F =

[±, δ)F if K = F „˜’1 (±) (char F = 2).

Our aim in this section is to generalize this construction, associating a central

simple algebra D(B, „ ) to every central simple algebra with involution of unitary

type (B, „ ) of even degree, in such a way that D EndK (V ), σh is Brauer-equivalent

to (K, disc h)F (see (??)). In view of this relation with the discriminant, the algebra

D(B, „ ) is called the discriminant algebra, a term suggested by A. Wadsworth. This

algebra is endowed with a canonical involution of the ¬rst kind.

As preparation for the de¬nition of the discriminant algebra, we introduce in the

¬rst four sections various constructions related to exterior powers of vector spaces.

For every central simple algebra A over an arbitrary ¬eld F , and for every positive

integer k ¤ deg A, we de¬ne a central simple F -algebra »k A which is Brauer-

k

equivalent to A—k and such that in the split case »k EndF (V ) = EndF ( V ). In

the second section, we show that when the algebra A has even degree n = 2m, the

algebra »m A carries a canonical involution γ of the ¬rst kind. In the split case

A = EndF (V ), the involution γ on »m A = EndF ( m V ) is the adjoint involution

m m n

with respect to the exterior product § : V— V’ V F . The third

section is more speci¬cally concerned with the case where char F = 2: in this case,

we extend the canonical involution γ on »m A into a canonical quadratic pair (γ, f ),

when m ≥ 2. Finally, in §??, we show how an involution on A induces an involution

on »k A for all k ¤ deg A.

§10. THE DISCRIMINANT ALGEBRA 115

10.A. The »-powers of a central simple algebra. Let A be a central sim-

ple algebra of degree n over an arbitrary ¬eld F . Just as for the Cli¬ord bimodule,

the de¬nition of »k A uses a canonical representation of the symmetric group, based

on Goldman elements.

(10.1) Proposition. For all k ≥ 1, there is a canonical homomorphism g k : Sk ’

—

A—k from the symmetric group Sk to the group of invertible elements in A—k ,

such that in the split case A = EndF (V ) we have under the identi¬cation A—k =

EndF (V —k ):

gk (π)(v1 — · · · — vk ) = vπ’1 (1) — · · · — vπ’1 (k)

for all π ∈ Sk and v1 , . . . , vk ∈ V .

Proof : We ¬rst de¬ne the image of the transpositions „ (i) = (i, i + 1) for i = 1,

. . . , k ’ 1, by setting

gk „ (i) = 1 — · · · — 1 — g — 1 — · · · — 1

i’1 k’i’1

where g ∈ A — A is the Goldman element de¬ned in (??). From (??), it follows

that in the split case

gk „ (i) (v1 — · · · — vk ) = v1 — · · · — vi+1 — vi — · · · — vk ,

as required.

In order to de¬ne gk (π) for arbitrary π ∈ Sk , we ¬x some factorization

π = „1 —¦ · · · —¦ „ s where „1 , . . . , „s ∈ {„ (1), . . . , „ (k ’ 1)}

and set gk (π) = gk („1 ) · · · gk („s ). Then, in the split case

gk (π)(v1 — · · · — vk ) = vπ’1 (1) — · · · — vπ’1 (k) ,

and it follows that gk is a homomorphism in the split case. It then follows by scalar

extension to a splitting ¬eld that gk is also a homomorphism in the general case.

Therefore, the de¬nition of gk does not depend on the factorization of π.

(10.2) Corollary. For all π ∈ Sk and a1 , . . . , ak ∈ A,

gk (π) · (a1 — · · · — ak ) = (aπ’1 (1) — · · · — aπ’1 (k) ) · gk (π).

Proof : The equation follows by scalar extension to a splitting ¬eld of A from the

description of gk (π) in the split case.

For all k ≥ 2, de¬ne

sgn(π)gk (π) ∈ A—k ,

sk =

π∈Sk

where sgn(π) = ±1 is the sign of π.

(10.3) Lemma. The reduced dimension of the left ideal A—k sk is given by

n