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
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