INTRODUCTION

9

Pentad Identity

0

=

lx1x2x311::1-1xlx2x411::1

+

lx1x3x411::1-1x2x3x411:~1

Sextet Identity

O

=

IX1

Y1

Septet Identity

O

=

\Xl

Y1

x

4

llx2x3l

+

\x2 x3\lx1x4l

Y2 Y1 Y2

_,x2 X4\lxlx31

+

lx3 X4\lxlx21

Y1 Y2 Y1 Y2

It is also convenient to introduce the subalgebra H[7rn]

C

H[7rn] generated over k

by the elements lg;

I

and the subring

H(i

[1r n]

C

H+ [1r n] generated over k by the

elements

~~:\,

p $ q. In Chapter 5 we show that H[7rn] = if+[7rn] EB if-[7rn] is a

direct sum of four H(i[7rn]-modules as follows.

H(i[7rn]{l} EB H(i[7rn]{lgrgsgtl}, r s t

H(i[7rn]{lg;l} EB H(i[7rn]{lgjgk}, j k.

The brackets { } indicate module generators. We also find formulas in H[7rn] for

all products of these module generators over

H(i

[1r nl· For example, the Determinant

Identity is a formula for products of the generators

lgrgsgt

l-

In the Addendum to Chapter 5, we outline a proof that all the

H(i

[1r n]-module

relations among the

{lgrgsg11},

{lgjgkl}, and {lg;l} are implied, respectively, by

Pentad, Sextet, and Septet Identities, with group generators substituted for the

X;

and y1

.

We show that H+[7rn] and H[7rn] are free modules of rank 2n over if+[7rn]

and H[7rnJ, respectively, with basis all square-free monomials in the elements {g{}.

The squares are given by (gt)

2

=

~~:I

+

1, which is a simple formula from Chapter

2. We also show that all the relations between the algebra generators of H(i[7rn]

=

k

[\~:\]

are implied by the Rank Three Identity, that is, by the vanishing of all

4 x 4 minor determinants of the matrix

S

=

(\;;I).

These results combine to give

presentations of H[7rn] and H+ [7rn]· In fact, we give two presentations. If

n

=

3,

our results explicitly present

H+ [7r:3] and H[7ra] as free modules of rank 2 and 8,

respectively, over a polynomial ring on 6 generators. This presentation coincides

with a known result from invariant theory.

If n = 4, we explicitly present H+[1r4

]

and H[1r4

]

as free modules of rank 8 and 32, respectively, over a polynomial ring on

9 generators.

In Chapter 6, we give an alternate derivation of one of the presentations of

H+ [1r,] as a commutative ring. The idea here is to construct a certain represen-

tation

p, :

7f

11

--

SL(2, A[1r,]) and prove that

Pn

lu+[rr.,j= H+[7rn]

--

A[1r,]/

C