This article was downloaded by: [Northeastern University]On: 03 December 2014, At: 13:04Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

Alternating Forms of Higher DegreeM. Rafiq Omar aa Department of Mathematics , University of the Western Cape , Cape Town, South AfricaPublished online: 31 Aug 2006.

To cite this article: M. Rafiq Omar (2003) Alternating Forms of Higher Degree, Communications in Algebra, 31:4, 1705-1724,DOI: 10.1081/AGB-120018504

To link to this article: http://dx.doi.org/10.1081/AGB-120018504

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Alternating Forms of Higher Degree

M. Rafiq Omar*

Department of Mathematics, University of the Western Cape,Cape Town, South Africa

ABSTRACT

We define hyperbolic alternating forms of higher degree and deter-mine their isometry groups. If (H(V ), f ) is the hyperbolic alternatingspace of degree dþ 1 on a vector space V, we show that its isometrygroup is GL(V )fflF, where F is a co-Schur functor.

Key Words: Alternating form; Isometry group.

I. INTRODUCTION

In contrast to the symmetric case, not a great deal is known abouthigher degree (i.e., degree > 2) alternating forms. Trilinear alternatingforms over algebraically closed fields have been classified up to dimension8 for characteristic 0 (Gurevich, 1964, pp. 390–395), and up to dimension

*Correspondence: M. Rafiq Omar, Department of Mathematics, University ofthe Western Cape, Cape Town, South Africa; E-mail: romar@uwc.ac.za.

COMMUNICATIONS IN ALGEBRA�

Vol. 31, No. 4, pp. 1705–1724, 2003

1705

DOI: 10.1081/AGB-120018504 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

7 for arbitrary characteristic (Cohen, Helminck, 1988, Theorem 2.1, p. 3).Cohen and Helminck (ibid.) have also computed their isometry groupsover certain fields. The classification of alternating forms of degreegreater than 3 is known in only the simplest cases (Gurevich, 1964,p. 391).

The isometry group of a bilinear alternating form, known as its sym-plectic group, may be described explicitly in matrix terms (Scharlau, 1985,Lemma 7.4, p. 263), or geometrically (Artin, 1957, pp. 139–140). Abilinear alternating form is nondegenerate iff its rank is even iff it is hyper-bolic. We shall extend the notion of hyperbolic to alternating forms ofhigher degree, and compute their isometry groups. Our main result is:

Theorem 1. Let (H(V ), f ) be the hyperbolic alternating space of degreedþ 1 on a vector space V. Then its isometry group G(H(V ), f ) is thesemi-direct product of GL(V ) and Km(V

�), the co-Schur functor associatedwith the Young diagram

(with d blocks in the first column).

We summarize the contents of this paper.In Sec. 2 we introduce the basic definitions and notation and

sketch some background on alternating forms of higher degree. InSec. 3 we define the hyperbolic alternating forms of higher degree,in close analogy to the symmetric case treated in Keet (1993). InSec. 4 we prove the first part of Theorem 1, viz. that G(H(V ), f )¼GL(V ) ffl F, where F is the space of forms of degree dþ 1 on Vsatisfying two symmetry conditions: (i) alternation in the first d vari-ables; and (ii) a (signed) Jacobi identity (Proposition 6). In the sym-metric case, the notion of singularity (Keet, 1993, Equation (4),p. 426) affords a considerable simplification of the computation. Sincethe concept of singularity makes no sense in the alternating case, weuse instead the notion of the ith domain of an alternating form, takenfrom Cohen, Helminck (1988). In Sec. 5 we complete the proof ofTheorem 1 by showing that the space of forms F occurring as the sec-ond factor in the isometry group is the co-Schur functor Km(V

�),in the notation of Akin et al. (1982) (Proposition 7).

1706 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

II. PRELIMINARIES

In this paper we assume that d is a positive integer �2, F is a fieldwhose characteristic does not divide d!, and V is a vector space of dimen-sion n over F. We denote the dual of V by V �, its d-fold Cartesian pro-duct V� � � � �V by Vd, and the dth tensor power (or grade d componentof the tensor algebra T(V )) by T d(V ).

Let y be a multilinear form of degree d on V, y :Vd!F, withv1, . . . , vn a basis for V and x1, . . . , xn the dual basis for V �. Denotethe space of degree d multilinear forms on V by MultdV. The groupGL(V ) acts (on the left) on MultdV by s � y¼ y � s�1, i.e.,s � y(v1, . . . , vd)¼ y(s�1v1, . . . , s

�1vd). This defines an action of the sym-metric group Sd on MultdV : p � y(v1, . . . , vd)¼ y(vp1, . . . , vpd). (This actionis via a representation of Sd in Vd.)

We say y is skew-symmetric if p � y¼ E(p)y for all p2Sd. (E(p) denotesthe sign of p.) Since any p2Sd is a product of transpositions of adjacentintegers, it is sufficient that y(v1, . . . , vi, viþ1, . . . , vd)¼� y(v1, . . . , viþ1,vi, . . . , vd) for all i¼ 1, . . . , d� 1. We say y is alternating if y(v1, . . . , vd)¼ 00 whenever vi¼ viþ1 for some i¼ 1, . . . , d� 1. Clearly we have: y alternat-ing implies y skew-symmetric; and, if char F 6¼ 2 (as we assume), theconverse also holds. We shall use the above conditions interchangeably,and usually refer to forms satisfying them as alternating.

Denote the vector space of alternating forms of degree d on V byAltdV, which is isomorphic to ^d(V )�. We have several identifications:the grade d component ^d(V �) of the Hopf algebra ^(V �) can be identi-fied with the space Fdhx1, . . . , xni of homogeneous anticommutative poly-nomials of degree d, and the space T d(V �)alt of alternating tensors inT d(V �) is isomorphic to AltdV. From these identifications we get a basisfor AltdV consisting of the n

d

� �elements xk1^ � � � ^ xkd, one for each

strictly increasing list k : d! n (where d¼f1, . . . , dg).We describe the antisymmetrization map ^d(V �)!T d(V �)alt, as well

as its inverse, the multilinearization map, both in Hopf algebra terms andexplicitly. The multilinearization map is given in terms of the comultipli-cation on the Hopf algebra ^(V �), while the antisymmetrization mapcomes from the multiplication.

The multilinearization map T d(V �)alt!^d(V �) is easy to describe. Itis the canonical projection T d(V �)!^d(V �), or, since ^1V � ¼V �, it isthe component V � � � � � �V � !^d(V �) of d-fold multiplication in theHopf algebra ^(V �) (Akin et al., 1982, Sec. I.2, p. 213). Explicitly, thismap can be described as follows:

Let t¼Pa(t j a1 . . . ad )(xa1� � � � � xad)2T d(V �), where a : d! n. (The

xa1� � � � � xad form a basis for T d(V �).) If t2T d(V �)alt (i.e., p � t¼ E(p)t

Alternating Forms of Higher Degree 1707

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

for all p2Sd), then (tja1 . . . ad )¼ E(p)(tjap1 . . . apd ) for all p2Sd.Hence t 7!P

ri¼1

Pp2Sd

(tjai1 . . . aid )xai1^ � � �^xaid¼P

ri¼1 d!(tjai1 . . . aid )

xai1^ � � � ^ xaid¼ d!P

a(tja1 . . . ad )xa1^ � � � ^ xad2^d(V �), where the sumis taken over all a such that 1 a1 < � � � < ad n.

The antisymmetrization map ^d(V �)!T d(V �)alt maps x1^ � � � ^ xdto

PsE(s)xs1� � � � � xsd. It may be viewed as the component ^d(V �)!

V � � � � � �V � of d-fold comultiplication in the Hopf algebra ^V �, andit is a split monomorphism over F (Akin et al., 1982, Sec. I.2, p. 213).If we take an arbitrary element p2^d(V �), where p¼P

a( pja1 . . . ad )xa1^ � � � ^ xad, then its antisymmetrization is

Pa( pja1 . . . ad ) [(1=d!)P

s2SdE(s)xas1� � � � � xasd], where the sum is taken over all a : d! n such

that 1 a1 < � � � < ad n. (Such a give a basis for ^d(V �).)Following Harrison (1975, p. 124), we define an alternating space

of degree d to be a pair (V, y), where V is a finite-dimensional vectorspace and y is an alternating multilinear degree d form on V, i.e.,y :Vd!F. Two alternating spaces (V, y) and (W, f) are called iso-morphic (or isometric) if there exists a vector space isomorphisms :V!W such that y(v1, . . . , vd)¼f(sv1, . . . , svd)¼ s�1 �f(v1, . . . , vd)for all v1, . . . , vd2V, i.e., W¼ sV and f¼ s � y. The antisymmetrizationmap gives an obvious bijective correspondence between alternatingspaces (V, y) of degree d over F and elements of ^d(V �). Since anti-symmetrization preserves the actions of GL(V ) on ^d(V �) and onAltdV, isomorphic alternating spaces (V, y), (sV, s � y) of degree d cor-respond to GL(V )-equivalent elements of ^d(V �) (or equivalent anti-commutative polynomials).

Let (V, y) be an alternating space of degree d, and suppose v2V.Define an alternating multilinear form y(v) of degree d� 1 on V by

yðvÞðv1; . . . ; vd�1Þ ¼ yðv; v1; . . . ; vd�1Þ for all v1; . . . ; vd�1 2 V :

We call (V, y(v)) (or just y(v)) the derivative of (V, y) (or just y) in the

direction of v. It is easy to check that, if (V, y)ffis (W, f), then (V, y(v))ffis(W,f(s�v)). Choose abasis e1, . . . , en andadual basisx1, . . . , xn, and supposev¼ a1e1þ � � � þ anen. If f is the anticommutative polynomial correspondingto (V, y) (via the given bases), then (V, y(v)) corresponds to(1=d )

Pni¼1ai(@f=@xi), where the derivatives of f are as defined by Berezin

(1987, pp. 74–75).We call (V, y) (or simply y) nondegenerate if y(v)¼ 0 implies v¼ 0,

i.e., if y(v, v1, . . . , vd�1)¼ 0 for all v1, . . . , vd�12V implies v¼ 0. Other-wise, we call (V, y) degenerate. This condition is clearly preserved byisomorphism.

1708 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

Let f be a multilinear form of degree d on a vector space V. If s is anautomorphism of V, we call s an isometry if s � f¼ f, and the set of allsuch s is the isometry group of f, denoted G(V, f ), or just G( f ).

III. HYPERBOLIC ALTERNATING SPACES

Let F be a field in which d! 6¼ 0 and let V be a finite-dimensional vec-tor space over F. Let Altd�1V denote the space of alternating (d� 1)-multilinear forms on V and put H(V )¼V�Altd�1V. Take d� 2. Definethe hyperbolic degree d alternating space on V, (H(V ), C), by putting(we use a to denote that a is omitted)

C½ðx1; y1Þ; . . . ; ðxd ; ydÞ� ¼Xdi¼1

ð�1Þi�1yiðx1; . . . ; bxixi; . . . ; xdÞ:For d¼ 2 this gives the usual hyperbolic bilinear alternating space

(V�V �, C), with C[(x, f ), ( y, g)]¼ f( y)� g(x) (Scharlau, 1985, p. 239).

Remark. The hyperbolic degree d symmetric space on V is H(V )¼V�Symd�1V, with C[(x1, y1), . . . , (xd, yd)]¼

Pdi¼1yi(x1, . . . , bxixi, . . . , xd).

(See Keet, 1993, p. 425)We show that C is (i) alternating (ii) d-multilinear (iii) nondegene-

rate. First we prove the following:

Lemma 1. Suppose f(v1, . . . , vd)¼ 0 for all alternating forms f of degree don a vector space V, and for all v2, . . . , vd in V. Then v1¼ 0.

Proof. Let e1, . . . , en be basis for V. We may assume that n� d. Putv1¼

Pni¼1 aiei. Then we have

Xni¼1

ai f ðei; v2; . . . ; vdÞ ¼ 0 for all v2; . . . ; vd and all f : ð1Þ

Fix j, 1 j d. Choose (v2, . . . , vd)¼ (e1, . . . , bejej, . . . , ed) and f¼ ej� ^ e1

� ^� � � ^ be�je�j ^ � � � ^ ed

� in (1). Then aj¼ 0. This is true for all j¼ 1, . . . , d, hencea1¼ � � � ¼ ad¼ 0. Now fix j, dþ 1 j n.

Choosing (v2, . . . , vd)¼ (e1, . . . , ed�1) and f¼ ej� ^ e1

� ^ � � � ^ ed�1� in (1),

we obtain aj¼ 0. This is true for all j¼ dþ 1, . . . , n, hence adþ1¼ � � � ¼an¼ 0. Thus v1¼ 0, as required. &

Alternating Forms of Higher Degree 1709

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

Remark. In the case of symmetric forms, this Lemma follows easily byinvoking the existence of nondegenerate symmetric forms of arbitrarydegree in arbitrary dimension. But there is no nondegenerate alternatingform of degree n� 1 on a vector space of dimension n. (MacLane,Birkhoff, 1967, Proposition 15, p. 568.)

Proposition 1. C is (i) alternating (ii) d-multilinear (iii) nondegenerate.

Proof. (i) By Sec. 2 it suffices to show that C¼ 0 if two adjacent argu-ments are equal. Suppose that (xi, yi)¼ (xiþ1, yiþ1)¼ (x, y). Then

C½ðx1; y1Þ; . . . ; ðx; yÞ; ðx; yÞ; . . . ; ðxd ; ydÞ�¼ y1ðx2; . . . ; x; x; . . . ; xdÞ þ � � �þ ð�1Þi�1yðx1; . . . ; xi�1; x; xiþ2; . . . ; xdÞþ ð�1Þiyðx1; . . . ; xi�1; x; xiþ2; . . . ; xdÞ þ � � �þ ð�1Þd�1ydðx1; . . . ; x; x; . . . ; xd�1Þ ¼ 0;

since all terms are zero except the ith and (iþ 1)th, which cancel oneanother.

(ii)C[(x1, y1), . . . , a(xj, yj)þ b(xj0, yj0), . . . , (xd,yd)]

¼C½ðx1;y1Þ; . . . ;ðaxjþbx0j;ayj þby0jÞ; . . . ;ðxd ;ydÞ�¼X

i 6¼jð�1Þi�1yiðx1; . . . ; bxixi; . . . ;axjþbx0j; . . . ;xdÞ

þð�1Þj�1ðayjþby0jÞðx1; . . . ; bxjxj; . . . ;xdÞ¼ a

Xi 6¼j

ð�1Þi�1yiðxi; . . . ; bxixi; . . . ;xj; . . . ;xdÞþb

Xi 6¼j

ð�1Þi�1yiðx1; . . . ; bxixi; . . . ;x0j ; . . . ;xdÞþð�1Þj�1ayjðx1; . . . ; bxjxj; . . . ;xdÞþð�1Þj�1by0jðx1; . . . ; bxjxj; . . . ;xdÞ

Combining the first and third, and second and fourth terms,respectively, this equals aC[(x1, y1), . . . , (xj, yj), . . . , (xd, yd)]þ bC[(x1,y1),. . . , (xj

0, yj0), . . . , (xd, yd)].

(iii) Suppose that C[(v, y), (v2, y2), . . . , (vd, yd)]¼ 0, i.e.,

yðv2;v3;...;vdÞ�y2ðv;v3;...;vdÞþ���þð�1Þd�1ydðv;v2;...;vd�1Þ¼0

for all vi;yi;2 id: ð2Þ

1710 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

We need to show v¼ 0, y¼ 0. If v2¼ 0 in (2), then y2(v, v3, . . . , vd)¼ 0for all v3, . . . , vd, and all y2. By Lemma 1, v¼ 0.

If we choose y2¼ � � � ¼ yd¼ 0, we obtain y(v2, . . . , vd)¼ 0 for allv2, . . . , vd, hence y¼ 0. &

We now show that hyperbolic alternating space is cofinal for alternat-ing spaces:

Proposition 2. Every alternating space (V, f ) of degree d can be isometri-cally embedded in some hyperbolic alternating space of degree d.

Proof. Define f : (V, f )! (V�Altd�1V, C) by f(v)¼ (v, f (v)), where f (v)

denotes the derivative of f with respect to v. It is obvious that f is linear(observe that f (vþw)¼ f (v)þ f (w)), and that it is injective. It remains toshow that f is an isometry, i.e., that C[f(v1), . . . ,f(vd)]¼ f(v1, . . . , vd)for all v1, . . . , vd. Now

C½fðv1Þ; . . . ;fðvdÞ� ¼ C½ðv1; f ðv1ÞÞ; . . . ; ðvd ; f ðvd ÞÞ�

¼Xdi¼1

ð�1Þi�1f ðviÞðv1; . . . ; bvivi; . . . ; vdÞ¼

Xdi¼1

ð�1Þi�1f ðvi; v1; . . . ; bvivi; . . . ; vdÞ¼ d f ðv1; . . . ; vdÞ:

Then (1=d )f is the required isometry. &

Remarks. 1. All nondegenerate bilinear alternating forms are hyper-bolic. For d > 2, there exist (simply from dimension considerations)nondegenerate alternating forms which are not hyperbolic.

2. Of the canonical trilinear alternating forms discussed by Cohen andHelminck (1988, Table 1, p. 4), f1 (dimension 3) and f4 (dimension 6) arehyperbolic. The other dimension 6 form, f3, as well as all the others, arenot hyperbolic.

IV. THE ISOMETRY GROUP

We now determine the isometry group of an alternating hyperbolicspace of arbitrary degree and dimension. Recall (Sec. 3) the degreedþ 1 alternating hyperbolic, which consists of the space H¼V�AltdV

Alternating Forms of Higher Degree 1711

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

together with the alternating form f defined by f [(v1, y1), . . . ,(vdþ1, ydþ1)]¼

Pdþ1i¼1 (� 1)i�1yi(v1, . . . , bvivi, . . . , vdþ1). We want to determine

the isometry group G(H, f ). The case dimV¼ d is easily dispensed with:

Proposition 3. If dimV¼ d, then G(H, f )¼SLdþ1(F ).

Proof. dimH¼ nþ�nd

�¼ dþ

�dd

�¼ dþ 1. If y is an arbitrary

alternating form of degree dþ 1 on a vector space of dimensiondþ 1 and s is an isometry then y(v1^� � � ^ vdþ1)¼ y(sv1^ � � � ^ svdþ1)¼(det s)y(v1^ � � � ^ vdþ1), essentially because the determinant is the onlyalternating form of degree dþ 1 in dþ 1 variables. Hence det s¼ 1,i.e., s2SLdþ1(F ). &

In order to determine the isometry group G(H, f ) (dimV > d ),we write an isometry as a matrix M¼

�A BC D

�, where A :V!V,

B :AltdV!V, C :V!AltdV and D :AltdV!AltdV are linear maps.The major task is to show that M(AltdV ) AltdV. This implies thatB¼ 0 and describing A, C and D is then easy.

There is in general a great deal of analogy with the case of a hyper-bolic symmetric form of higher degree, but proving the invariance ofAltdV requires a quite different approach. In the symmetric case, thespace SymdV equals the singular set of the hyperbolic form (Keet, 1993,Equation (4), p. 426), which is easily seen to be invariant underisometry (Keet, 1993, Proposition 6(ii), p. 424); in the alternating case,we cannot define the singular set, so proving the invariance of AltdV isdone as follows:

Let x2V. Recall that the derivative of f with respect to x is thedegree d� 1 alternating form f (x) given by

f ðxÞðx1; . . . ; xd�1Þ ¼ f ðx; x1; . . . ; xd�1Þ for all x1; . . . ; xd�1 2 V :

The kernel of f is

ker f ¼ fx 2 V : f ðx; x1; . . . ; xd�1Þ ¼ 0 for all x1; . . . ; xd�1 2 Vg;i.e., x2 ker f if and only if f (x)¼ 0, and the rank of f is rk f¼dimV� dim ker f.

Following Cohen and Helminck (1988, p. 2), we define, for any non-negative integer i, the ith domain of f, Ri( f )¼fx2V : rk f (x)¼ ig.

Proposition 4. sRi( f ) Ri( f ) for any isometry s of f.

Proof. First we show that rk(s � f )(x)¼ rk f (s�1x) for any automorphism

s2GL(V ).

1712 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

u 2 kerðs � f ÞðxÞ

, ðs � f ÞðxÞðu; v3; . . . ; vdÞ ¼ 0 for all v3; . . . ; vd 2 V

, s � f ðx; u; v3; . . . ; vdÞ ¼ 0 for all v3; . . . ; vd 2 V

, f ðs�1x; s�1u; s�1v3; . . . ; s�1vdÞ ¼ 0 for all v3; . . . ; vd 2 V

, f ðs�1x; s�1u;w3; . . . ;wdÞ ¼ 0 for all w3; . . . ;wd 2 V

, f ðs�1xÞðs�1u;w3; . . . ;wdÞ ¼ 0 for all w3; . . . ;wd 2 V

, s�1u 2 ker f ðs�1xÞ

, u 2 sker f ðs�1xÞ

So dim ker(s � f )(x)¼ dim[sker f (s�1x)]¼ dim[ker f (s

�1x)], hence rk(s � f )(x)¼rk f (s

�1x). If s is moreover an isometry, then s � f¼ f. Let x2Ri( f ), i.e.,rk f (x)¼ i. Then rk f (sx)¼ rk(s�1 � f )(x)¼ rk f (x)¼ i. Hence sx2Ri( f ), sox2 s�1Ri( f ). Therefore Ri( f ) s�1Ri( f ), i.e., sRi( f ) Ri( f ). &

Remark. If we put R0[R1[ � � � [Ri¼Ri, then Ri is still invariantunder isometry. Together with the next Proposition, this will show thatM(AltdV ) AltdV.

Proposition 5. If H¼V�AltdV is the alternating hyperbolic space withform f, then Rn( f )¼AltdV.

Proof. We choose a basis for H¼V�AltdV as follows: Let e1, . . . , enbe a basis for V. Let A¼fa : d! n j 1 a1 < � � � < ad ng. Thenf(ea1 . . . ead)�ja2Ag is a basis for AltdV. (For brevity we shall often writeea for (ea1 . . . ead)

�.) We see that dimH¼ nþ�nd

�. Suppose y2 ker f (x),

where x¼Pni¼1aieiþ

Pa2Aaaea and y¼P

ni¼1bieiþ

Pa2Abaea. This means

that, for all z1, . . . , zd�12H, f(x, y, z1, . . . , zd�1)¼ 0, i.e., for all tj2 n[A,j¼ 1, . . . , d� 1,

fX

aiei þX

aaea;X

biei þX

baea; et1 ; . . . ; etd�1

� �¼ 0 ð1Þ

Nontrivial equations arise only when at most one tj2A, so we distinguishtwo types of equations which can occur:

Type I: tj2 n for all j¼ 1, . . . , d� 1.

It suffices to consider those tj satisfying 1 t1 < � � � < td�1 n (i.e.,t2A0 ¼ ft : d � 1! n j 1 t1 < � � � < td�1 ng), because other choices ofthe tj will give, by alternation, trivial equations or the same equations

Alternating Forms of Higher Degree 1713

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

as for t2A0. Expanding (1) gives,

for all t 2 A0 :Xi

Xj

aibjf ðei; ej; et1 ; . . . ; etd�1Þ

þXi

Xa

aibaf ðei; ea; et1 ; . . . ; etd�1Þ

þXa

Xj

aabjf ðea; ej ; et1 ; . . . ; etd�1Þ

þXa

Xb

aabbf ðea; eb; et1;...etd�1Þ ¼ 0:

By definition of f the first and last sums are zero (since they do nothave exactly one argument et, t2A), and using the alternation of f inthe middle sums we obtain:X

i

Xa

ðaiba � aabiÞf ðei; ea; et1 ; . . . ; etd�1Þ ¼ 0 for all t 2 A0:

Now fix t (i.e., choose one of these equations). Those terms on the LHSfor which i2 Im t (i.e., the image of t) are zero, so we consider thosefor which i 62 Im t. Suppose 1 � � � < tki < i < tki þ 1 < � � � n, where0 ki d� 1 and we define t0¼ 0, td¼1 to allow for i < t1 ori > td�1.Then

f ðei; ea; et1 ; . . . ; etd�1Þ ¼ �f ðea; ei; et1 ; . . . ; etd�1

Þ¼ �ð�1Þki f ðea; et1 ; . . . ; etki ; ei; etkiþ1

; . . . ; etd�1Þ

¼ ð�1Þki�1f ðea; et1 ; . . . ; etki ; ei; etkiþ1; . . . ; etd�1

Þ:So for each i 62 Im t, we haveX

a

ðaiba � aabiÞf ðei; ea; et1 ; . . . ; etd�1Þ

¼Xa

ð�1Þki�1ðaiba � aabiÞf ðea; et1 ; . . . ; ei; . . . ; etd�1Þ

¼ ð�1Þki�1ðaibt1...tki itkiþ1...td�1� at1...tki itkiþ1...td�1

biÞ;

since there is only one nonzero term in the middle expression, viz. the termfor which a1, . . . , ad corresponds exactly to t1, . . . , tki, i, tkiþ1, . . . , td�1.So the equation is

Pi(�1)ki�1(aibt1 . . . i . . . td�1

� at1 . . . i . . . td�1bi)¼ 0.

For an arbitrary injection a : d! n we define aa¼ E(s)aa0, where a0 2Aand sa¼ a0. Then we have

Pi(� 1)ki�1(� 1)ki(aibit1 . . . td�1

� ait1 . . . td�1bi)¼ 0.

HencePn

i¼1(aibit1 . . . td�1� ait1 . . . td�1

bi)¼ 0, for each t2A0.

1714 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

Type II: tj02A for exactly one j0, 1 j0 d� 1, and tj2 n for j 6¼ j0.

Suppose tj0¼ aj01 . . . aj0d¼ aj0 (by abuse of notation). Expanding (1)givesX

i

Xj

aibjf ðei; ej; et1 ; . . . ; eaj0 ; . . . ; etd�1Þ

þXi

Xa

aiba f ðei; ea; et1 ; . . . ; eaj0 ; . . . ; etd�1Þ

þXa

Xi

aabi f ðea; ei; et1 ; . . . ; eaj0 ; . . . ; etd�1Þ

þXa

Xb

aabb f ðea; eb; et1 ; . . . ; eaj0 ; . . . ; etd�1Þ ¼ 0:

By definition of f the last three sums are zero (since they do not haveexactly one argument et, t2A), so we obtain

Pi

Pkaibkf(ei, ek, et1, . . . ,

eaj0 , . . . , etd�1)¼ 0 for all aj02A, j0¼ 1, . . . , d� 1, and all tj2 n, j 6¼ j0.

As before, because of alternation, it is sufficient to consider t satisfy-ing 1 t1 < � � � < tj0 � 1 < tj0 þ 1 < � � � < td�1 n. Moreover, we needconsider only those t for which Im t Im aj0, since other t will give eithertrivial equations or the same equations as these t give.

By alternation, we can write these equations asP

i

Pk> i (aibk� akbi)

f (ei, ek, et1, . . . , eaj0, . . . , etd�1)¼ 0, for all aj02A, j0¼ 1, . . . , d� 1, and all

t such that 1 t1 < � � � < tj0�1 < tj0þ1 < � � � < td�1 n, Im t Im aj0.Now f (ei, ek, et1, . . . , eaj0, . . . , etd�1

)¼± f (eaj0, et1, . . . , ei, . . . , ek, . . . , etd�1), soX

i

Xk>i

ðaibk � akbiÞf ðei; ek; et1 ; . . . ; eaj0 ; . . . ; etd�1Þ

¼ �Xi

Xk>i

ðaibk � akbiÞf ðeaj0 ; et1 ; . . . ; ei; . . . ; ek; . . . ; etd�1Þ:

Fix 1 j0 d� 1, aj02A and t such that 1 t1 < � � � < tj0�1 < tj0þ1 <� � � < td�1 n and Im t Im aj0. (2)

Then exactly one pair (i, k) with i < k will give a nonzero term in thedouble sum, viz. the two elements in Im aj0nIm t. So we get aibk� akbi¼ 0(for this pair). If we range through all choices of j0, aj0 and t satisfying theconditions (2) above, we obtain equations aibk� akbi¼ 0 for all i, k2 n.

We now show AltdV Rn( f ). If x2AltdV, i.e., a1¼ � � � ¼ an¼ 0,then all Type II equations are redundant and Type I equations become:Pn

k¼1akt1 . . . td�1bk¼ 0. Hence the ba are arbitrary, so dim ker f (x)�

�nd

�,

i.e., rk f (x) nþ�nd

��

�nd

�¼ n, hence x2Rn( f ), as required.

Finally we show that Rn( f ) AltdV. Suppose x 62AltdV, i.e., someai 6¼ 0. Re-numbering, if necessary, we suppose a1 6¼ 0. From Type IIequations we have: arbs� asbr¼ 0 for all r, s2 n. For all s� 2

Alternating Forms of Higher Degree 1715

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

(taking r¼ 1) we have bs¼ (as=a1)b1¼ csb1, and for s¼ 1 we put c1¼ 1. Itis easy to see that all Type II equations are satisfied by thebs : arbs� asbr¼ ar(as=a1)b1� as(ar=a1)b1¼ 0. Now substitute these bs intothe Type I equations:Xn

k¼1

ðakt1...td�1b1 � akbkt1...td�1

Þ ¼ 0 for all t 2 A0:

If t2A0 and 1 62 Im t we can write the equation as b1t1 . . . td�1¼

(1=a1)[a1t1 . . . td�1c1b1þ

Pnk¼2(akt1 . . . td�1

b1� akbkt1 . . . td�1)]. We check that all

Type I equations are satisfied by the bi and the ba. It is obviously sufficientto check only the remaining Type I equations, i.e., those for which12 Im t. Suppose that tj¼ 1 for some j. Type I equations are thenPn

i¼1ða1it1...btjtj ...td�1bi � aib1it1...btjtj ...td�1

Þ ¼ 0, for all t2A0. The first term is

zero, so we havePn

i¼2ða1it1...btjtj ...td�1bi � aib1it1...btjtj ...td�1

Þ ¼ 0. Substitute for

bi and b1it1...btjtj ...td�1

ði � 2Þ:Xni¼2

a1it1...btjtj ...td�1

cib1 � ðai=a1Þ(a1it1...btjtj ...td�1

c1b1

"þXnk¼2

akit1...btjtj ...td�1

ckb1 � akbkit1...btjtj ...td�1

� �)#¼ �

Xni¼2

Xnk¼2

cickakit1...btjtj ...td�1� ciakbkit1...btjtj ...td�1

� �¼ 0;

because of alternation of the aa and bb. So all the bi and bb can be writtenin terms of b1 and the bt1 . . . td�1k

, where k� 2 and 1 62 Im t. Now#(fbi, i� 2g[ fb1t1 . . . td�1

g)¼ n� 1þ�n� 1d � 1

�, hence rk f (x)� n� 1þ�

n� 1d � 1

�. Since we assume n > d, n� 1 > d� 1, so

�n� 1d � 1

�> 1, and hence

rk f (x) > n� 1þ 1¼ n. Thus x 62Rn( f ), as required. &

We now return to the determination of G(H, f ). Since M isinvertible and B¼ 0, A is also invertible, i.e., A2GL(V ). Now theisometry condition gives, for all v1, . . . , vdþ1 and y1, . . . , ydþ1,

fh

A 0C D

� �v1y1

� �; . . . ; A 0

C D

� �vdþ1ydþ1

� �i¼ f v1

y1

� �; . . . ; vdþ1

ydþ1

� �h i, i.e.,

ðCv1 þDy1ÞðAv2; . . . ;Avdþ1Þ � ðCv2 þDy2ÞðAv1;Av3; . . . ;AvdÞþ � � � þ ð�1ÞdðCvdþ1 þDydþ1ÞðAv1; . . . ;AvdÞ

¼ y1ðv2; . . . ; vdþ1Þ � y2ðv1; v3; . . . ; vdþ1Þþ � � � þ ð�1Þdydþ1ðv1; . . . vdÞ: ð3Þ

1716 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

Choose y2¼ � � � ¼ ydþ1¼ 0, v1¼ 0. Then Dy1(Av2, . . . ,Avdþ1)¼y1(v2, . . . , vdþ1), i.e., A�1 � (Dy1)(v2, . . . , vdþ1)¼ y1(v2, . . . , vdþ1) for allv2, . . . , vdþ1 and all y1. Hence A�1 � (Dy1)¼ y1, i.e., Dy1¼A � y1 for ally1. This means that D is determined by A.

In order to describe C, put y1¼ � � � ¼ ydþ1¼ 0 in (3):

Xdþ1

i¼1

ð�1Þi�1CviðAv1; . . . ; cAviAvi; . . . ;Avdþ1Þ ¼ 0 for all v1; . . . ; vdþ1:

Define f :Vdþ1!F by f(v1, . . . , vdþ1)¼Cvdþ1(Av1, . . . ,Avd)¼A�1 �(Cvdþ1)(v1, . . . , vd). f is linear in v1, . . . , vdþ1 because C is linear in vdþ1

and Cvdþ1 is linear in v1, . . . , vd. f is alternating in v1, . . . , vd becauseCvdþ1 is alternating. f also satisfies:

Xdþ1

i¼1

ð�1Þi�1fðv1; . . . ; bvivi; . . . ; vdþ1; viÞ

¼Xdþ1

i¼1

ð�1Þi�1CviðAv1; . . . ; cAviAvi; . . . ;Avdþ1Þ ¼ 0:

So f is a multilinear form satisfying:

(i) f is alternating in v1, . . . , vd ;(ii)

Pdþ1i¼1 (�1)i�1f(v1, . . . , bvivi, . . . , vdþ1, vi)¼ 0.

Hence C is determined by a (dþ 1)-multilinear form f satisfying theabove conditions.

As a set we may thus identify G(H, f ) with GL(V )�F, where Fdenotes the space of (dþ 1)-multilinear forms f satisfying conditions (i)and (ii). We can recover M=

�A 0C D

�2G(H, f) from (A, f)2GL(V )�

using Dy¼A � y and Cvdþ1(v1, . . . , vd)¼f(A�1v1, . . . ,A�1vdþ1).

We now establish that G(H, f )ffiGL(V )fflF, the semidirect productof GL(V ) and F. We obtain the operation on GL(V )�F by using the

compatibility requirement: Suppose Mi¼�Ai 0Ci Di

�corresponds to

(Ai, fi) for i¼ 1, 2. Now suppose M1M2¼M¼�A 0C D

�and (A1, f1)

(A2, f2)¼ (A, f). Then M¼�

A1A2 0C1A2 þD1C2 D1D2

�, so obviously A¼A1A2,

D¼D1D2, C¼C1A2þD1C2, and f must satisfy

fðv1; . . . ; vdþ1Þ ¼ ðC1A2 þD1C2Þvdþ1ðAv1; . . . ;AvdÞ¼ C1ðA2vdþ1ÞðAv1; . . . ;AvdÞ þD1ðC2vdþ1ÞðAv1; . . . ;AvdÞ¼ f1ðA�1

1 Av1; . . . ;A�11 Avd ;A2vdþ1Þ þ A1 � ðC2vdþ1ÞðAv1; . . . ;AvdÞ

Alternating Forms of Higher Degree 1717

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

¼ f1ðA2v1; . . . ;A2vdþ1Þ þ C2vdþ1ðA2v1; . . . ;A2vdÞ¼ A�1

2 � f1ðv1; . . . ; vdþ1Þ þ f2ðv1; . . . ; vdþ1Þ:

This shows that the operation on GL(V )�F is

ðA1;f1ÞðA2;f2Þ ¼ ðA1A2;A�12 � f1 þ f2Þ;

so the product is semidirect as claimed. We have thus proved:

Proposition 6. G(H(V ), f )¼GL(V )fflF, where F is the space of formsof degree dþ 1 on V satisfying two symmetry conditions:

(i) Alternation in the first d variables; and(ii) A (signed) Jacobi identity.

V. A CO-SCHUR FUNCTOR

The space F turns out to be isomorphic to a certain co-Schur functor(Akin et al., 1982, Definition II.1.3 p. 220). First we observe that it is easyto show that condition (ii) can be written in other ways:

(ii)0Pdþ1

i¼1 E(si)si �f¼ 0, where si¼ (i, dþ 1) for 1 i dþ 1.(ii)00 f(v1, . . . , vdþ1)þ (� 1)df(v2, . . . , vdþ1,þf(v3, . . . , vdþ1, v1, v2)

þ � � � þf(vd, vdþ1, v1, . . . , vd�1)þ (�1)df(vdþ1, v1, . . . , vd)¼ 0.(ii)000

Pdþ1i¼1 E(g

i�1)gi�1 �f¼ 0, where g¼ (1, 2, . . . , dþ 1).

The last two of these give a Jacobi identity which alternates in signwhen deg(f)¼ dþ 1 is even.

The next result is proved by a direct method, which contrasts withKeet’s use of the letter-place algebra to describe a space of forms definedby similar conditions (Keet, 1994, 4.8 Proposition p. 1589).

Proposition 7. The spaceF is isomorphic to the co-Schur functor Km(V�),

where m is the partition (2, 1d�1), i.e., the Young diagram

(with d blocks in the first column).

1718 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

The Proposition is proved as follows:

(a) We describe a standard basis for Km(V�), and use it to show that

F Km(V�).

(b) We show that every basis element of Km(V�) satisfies (i) and (ii),

which implies that Km(V�) F.

The co-Schur functor Km(V�) may be described as follows:

Let fv1, . . . , vng be a basis for V, with fx1, . . . , xng the dual basis forV �. The co-Schur functor is the image of dm

0 (V �) :Sm(V�)!^m�(V �), i.e.,

dm0 (V �) :S2(V

�)�V � �� � ��V � !^d(V �)�V �, where m~¼ (d, 1)¼

:

Now dm0(V �)[(xp1

Jxpdþ1)� xp2�� � �� xpd] can be visualized as follows

(Akin et al., 1982, top of p. 222):

(The rows on the LHS of ! are multilinearized.)Now take the exterior product down the columns on the RHS, and

tensor the results to obtain:

d 0mðV �Þ½ðxp1K xpdþ1

Þ � xp2 � � � � � xpd Þ�¼ ðxp1 ^ � � � ^ xpd Þ � xpdþ1

þ ðxpdþ1^ xp2 ^ � � � ^ xpd Þ � xp1 :

By the Standard Basis Theorem (Akin et al., 1982, Theorem II.3.16p. 235), a basis for Km(V

�) is given by those elements for which the dia-gram is co-standard, i.e., for which p1 < � � �< pd and p1 pdþ1.

Proof of (a). An arbitrary (dþ 1)-multilinear form f can be written(uniquely) asf¼P

i2I(a j i1 . . . idþ1)xi1� � � � � xidþ1, where I¼fi : d þ 1! ng.

We now consider what happens when f2F. By (i), if is¼ it forsome 1 s < t d, then f( . . . vis� � � � � vit . . . )¼ (a j . . . is . . . it . . . )¼ 0.

Alternating Forms of Higher Degree 1719

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

So f¼Pi2I1(a j i1 . . . idþ1)xi1� � � � � xidþ1

, where I1¼fi2 I j i1, . . . ,id aredistinctg.

Choose j2 I1. Then for p2Sd, we have (by (i)): p�f(vj1� � � � �vjd� vjdþ1

)¼ E(p)f(vj1� � � � � vjd� vjdþ1), i.e.,X

i2I1ðaji1 . . . idþ1Þðxi1 � � � � � xidþ1

Þðvjp1 � � � � � vjpd � vjdþ1Þ

¼ EðpÞXi2I1

ðaji1 . . . idþ1Þðxi1 � � � � � xidþ1Þðvj1 � � � � � vjdþ1

Þ;

i.e., (a j jp1 . . . jpdjdþ1)¼ E(p)(a j j1 . . . jdþ1).Let I2¼fi2 I1 j i1 < � � �< idg. Clearly, the set of all ip1 . . . ipdidþ1 as i

ranges over I2 and p ranges over Sd is just I1. Hence

f ¼Xi2I2

Xp2Sd

ðajip1 . . . ipd idþ1Þxip1 � � � � � xipd � xidþ1

¼Xi2I2

Xp2Sd

EðpÞðaji1 . . . idþ1Þxip1 � � � � � xipd � xidþ1

¼Xi2I2

ðaji1 . . . idþ1ÞXp2Sd

EðpÞxip1 � � � � � xipd � xidþ1:

Using the embedding ^d(V �) ,!T d(V �), we can identify f with

d!Xi2I2

ðaji1 . . . idþ1Þðxi1 ^ � � � ^ xid Þ � xidþ1: ð4Þ

Now suppose that for some j2 I2 we have j1 > jdþ1. By (ii), we have:(�1)df[(vj1^ � � �^vjd)� vjdþ1

]¼�Pdr¼1(�1)r�1f[(vj1^ � � �^ bvjrvjr . . .^vjdþ1

)� vjr ],

i.e., (�1)d(a j j1 . . . jdþ1)¼�Pdr¼1(�1)r�1(a j j1 . . . jr . . . jdþ1 jr)¼�Pd

r¼1

(�1)r�1(� 1)d�1(a j jdþ1j1 . . . jr . . . jdjr).Hence

ðajj1 . . . jdþ1Þ ¼Xdr¼1

ð�1Þr�1ðajjdþ1j1 . . . bjrjr . . . jd jrÞ: ð5Þ

This shows that any coefficient (a j j) for which jdþ1 < j1 can be written asa sum of coefficients (a j j s ) with all j s distinct, and js1 < � � �< jsd , j

s1 < jsdþ1

for all s. Also, for any other coefficient (a j l) for which ldþ1 < l1, the lt

(which occur on the RHS of (5)) are distinct from the js.

1720 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

It follows that the sum (4) may be taken over I3¼fi2 I2 j i1 idþ1g;also, since the coefficients on the RHS of (5) are all distinct, any coeffici-ent (a j i) with i2 I3 occurs at most twice in the sum. It remains to showthat every term in the sum can be written as (a j i1 . . . idþ1)[(xi1^ � � �^ xid)�xidþ1

þ (xidþ1^ xi2^� � �^ xid)� xi1], where i2 I3. We consider three cases:

(i) If idþ1¼ i1, then (xi1^� � �^ xid)� xidþ1¼ 1

2[(xi1^� � �^ xid)� xidþ1þ

(xidþ1^ xi2^� � �^ xid)� xi1)].

(ii) If idþ1¼ ir, for some r¼ 2, . . . , d, then (xi1^� � �^ xid)� xidþ1¼

(xi1^� � �^ xid)� xidþ1þ (xidþ1

^ xi2^� � �^ xid)� xi1, since the lastterm is zero.

(iii) If the is, 1 s dþ 1, are all distinct, we suppose1 i1 < � � �< ir�1 < idþ1 < irþ1 < � � �< id n, i.e., idþ1 is inthe rth position in the sequence. Then (re-labelling).

ðaji2 . . . idþ1 . . . id i1Þ½ðxi2 ^ � � � ^ xidþ1^ � � � ^ xid Þ � xi1 �

¼ ðajk1 . . . kr�1 . . . kdkdþ1Þ½ðxi2 ^ � � � ^ xidþ1^ � � � ^ xid Þ � xi1 �

¼Xdl¼1

ð�1Þl�1ðajkdþ1k1 . . . bklkl . . . kdklÞ� ½ðxi2 ^ � � � ^ xidþ1

^ � � � ^ xid Þ � xi1 �

(since k1 < � � �< kd and kdþ1 < k1, we can use (5)).The (r� 1)th term in the sum is

ð�1Þr�2ðajkdþ1k1 ...dkr�1kr�1 ...kdkr�1Þ½ðxi2^���^xidþ1^���^xid Þ�xi1 �

¼ð�1Þr�2ðaji1i2 ...id idþ1Þ½ðxi2^���^xidþ1^���^xid Þ�xi1 �

¼ð�1Þr�2ð�1Þr�2ðaji1 ...idþ1Þ½ðxidþ1^xi2^���^xid Þ�xi1 �

¼ðaji1 ...idþ1Þ½ðxidþ1^xi2^���^xid Þ�xi1 �

So the coefficient of (a j i1...idþ1) in this case is also(xi1^ ��� ^ xid)� xidþ1

þ (xidþ1^xi2^ ��� ^ xid)� xi1, as required.

Proof of (b). We show that a basis element F¼ (xp1^ � � � ^ xpd)�xpdþ1

þ (xpdþ1^ xp2^� � �^ xpd)� xp1 of Km(V

�) satisfies (i) and (ii) of Propo-sition 6. Then by linearity of the permutation action, all elements ofKm(V

�) satisfy (i) and (ii).Since F2^d(V �)�V � ffi (^dV )� �V � ffi (^dV�V )�, it is obvious

that F satisfies (i). To show that F satisfies (ii), we observe first thattwo (apparently) different actions of the symmetric group on aform are in fact the same. We consider only a basis element

Alternating Forms of Higher Degree 1721

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

f¼ xi1� � � � � xik2Tk(V �), as both actions can be extended to an arbi-trary form by linearity. The two actions are:

1. p�f (vj1� � � � � vjk)¼ f(vjp1� � � � � vjpk) (action of p on a form – seeSec. 2).

2. p � f¼ p � (xi1� � � � � xik)¼ xip�11� � � � � xip�1k

(action of p on atensor).

Now

p � f(vj1� � � � � vjk)¼ f(vjp1� � � � � vjpk)

¼ ðxi1 � � � � � xikÞðvjp1 � � � � � vjpk Þ¼ xi1ðvjp1Þ . . . xikðvjpkÞ;

whilep � f(vj1� � � � � vjk)¼ (xip�11

� � � � � xip�1k)(vj1� � � � � vjk)

¼ xip�11ðvj1Þ . . . xip�1k

ðvjkÞ:

It is obvious that the two actions are identical. Hence to show thatPdþ1i¼1 E(si)si �F¼ 0, we show instead that

Pdþ1i¼1 E(si)si �F¼ 0.

By antisymmetrization (Sec. 2), we have

F ¼ ðxp1 ^ � � � ^ xpd Þ � xpdþ1þ ðxpdþ1

^ xp2 ^ � � � ^ xpd Þ � xp1

¼ ðxp1 ^ � � � ^ xpd Þ � xpdþ1þ ðxps11 ^ xps12 ^ � � � ^ xps1d Þ � xps1ðdþ1Þ

¼X

a2Sdþ1ðdþ1ÞEðaÞxpa1 � � � � � xpad � xpaðdþ1Þ

þX

b2Sdþ1ðdþ1ÞEðbÞxps1b1 � � � � � xps1bd � xps1bðdþ1Þ ;

where s1¼ (1, dþ 1) and Sdþ1(dþ 1) denotes the subgroup of Sdþ1 whichfixes dþ 1, and which may be identified with Sd. Then we have

Xdþ1

i¼1

EðsiÞsi � F

¼Xdþ1

i¼1

Xa2Sdþ1ðdþ1Þ

EðaÞEðsiÞxpasi1 � � � � � xpasi ðdþ1Þ

þXdþ1

i¼1

Xb2Sdþ1ðdþ1Þ

EðbÞEðsiÞxps1bsi1 � � � � � xps1bsi ðdþ1Þ :

1722 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

It is easy to see that as i ranges from 1 to dþ 1 and a (resp. b) rangesover Sdþ 1(dþ 1), asi (resp. s1bsi) ranges over Sdþ1.

Put asi¼ g and s1bsi¼ d. Then E(g)¼ E(a)E(si) and E(d)¼ E(s1)�E(b)E(si)¼� E(b)E(si). Hence we have

Xdþ1

i¼1

EðsiÞsi � F ¼X

g2Sdþ1

EðgÞxpg1 � � � � � xpgðdþ1Þ

�X

d2Sdþ1

EðdÞxpd1 � � � � � xpdðdþ1Þ ¼ 0;

as required. &

Remarks. We have thus proved that the dual of the space T of tensorsv1� � � � � vdþ12T T dþ1(V ) satisfying the symmetry conditions

(i) s�(v1� � � � � vdþ1)¼ E(s)v1��� vdþ1 for s2Sd; and(ii) v1�� � �� vdþ1þ v2� v3� � � � � vdþ1� v1þ � � � þ vdþ1� v1� � � �

� � � � vd¼ 0

corresponds to the co-Schur functor Km(V�), where m¼ (2, 1d�1).

This is the analogue of the antisymmetrization isomorphism in thealternating case (Sec. 2), or polarization in the symmetric case (Keet,1993, p. 417). Since we have no analogue of the exterior (or symmetric)algebra for these forms, there is no possibility of describing the isomorph-ism in terms of Hopf algebra structure. We can only give the above expli-cit description, as we did in Sec. 2 for alternating tensors.

The structure of G( f ), where f is alternating hyperbolic, differs fromthe symmetric case only in that the co-Schur functor Km(V

�) replaces theSchur functor Ll(V

�) (cf. Keet, 1993, Sec. 4.4 p. 432).

REFERENCES

Akin, K., Buchsbaum, D. A., Weyman, J. (1982). Schur functors andSchur complexes. Adv. Math. 44:207–278.

Artin, E. (1957). Geometric Algebra, New York: Interscience.Berezin, F. A. (1987). Introduction to Superanalysis. Dordrecht: Reidel.Cohen, A. M., Helminck, A. G. (1988). Trilinear alternating forms on a

vector space of dimension 7. Comm. Alg. 6(1):1–25.

Alternating Forms of Higher Degree 1723

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014

Gurevich, G. B. (1964). Foundations of the Theory of Algebraic Invariants.Translated by Radok, J. R. M., Spencer, A. J. M. Groningen:Noordhoff.

Harrison, D. K. (1975). A Grothendieck ring of higher degree forms.J. Alg. 35:123–138.

Keet, A. (1993). Higher degree hyperbolic forms. Quaest. Math.16(4):413–442.

Keet, A. (1994). The Lie algebra of a higher degree form and a Schurfunctor. Comm. Alg. 22(5):1577–1601.

MacLane, S., Birkhoff, G. (1967). Algebra. New York: Macmillan.Scharlau, W. (1985). Quadratic and Hermitian Forms. Berlin: Springer.

Received October 31, 2001Revised August 28, 2002

1724 Omar

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

13:

04 0

3 D

ecem

ber

2014