0.
We use such a matrix A = (ai,j) 2 G(U) to denote an element
P0a0,1,1P1a1,2,2P0a0,2,2P2a2,3,3P1a1,3,3P0a0,3,3. . .
in U. Then, G(U) is a basis of U. The total degree `(A) of A is an infi-
P 1 P k-1
nite tuple (`0(A), `1(A), . .).defined by `k(A) = s=k+1ak,s- s=0 as,k. The
weight of A is w(A) = (i0, i0+ i1, . .).if `(A) = (i0, i1, . .).. We also have
that as a multi-degree, `(ab) = `(a)+ `(b) and w(ab) = w(a)+ w(b) for all
a, b 2 U.
It is easy to check that for a 2 U(n) and `(a) = (i0, . .,.in) and w(a) =
(j1, . .,.jn), i0+ . .+.in = 0 and jk > 0 for k = 1, . .,.n. For a 2 U and `(a)*
* =
(i0, i1, . .).and w(a) = (j0, j1, . .)., there is an n such that i0+ . .+.ik = *
*jk = 0
for k > n.
P
Definition 1.2 For n > 1, let (n) = {(i0, i1, . .,.in) | (i0, . .,.in) is*
* a
P
permutation of (0, . .,.n)}. For 0 < n < p_and ff, fi 2! (n), ff = (i0, . .,.i*
*n),
i0, . .,.in
fi = (j0, . .,.jn), we define D(fi, ff)= D 2 U(n) U(Gn) as
j0, . .,.jn
follows. D(ff, fi) 6= 0 only if there are 0 6 s < t 6 n and 0 6 M < N 6 n
such that is = jt = N and it = js = M and ik = jk otherwise.
_ !
. . .N, is+1, is+2, . .,. it-1, M, . . .
D
. . .M, is+1, is+2, . .,. it-1, N, . . .
3
0 1
0 1 as,s+1 as,s+2 . . .as,t
X t-1Y BB as+1,s+2 . . .as+1,tCC
= @ Z(M, N; iu, xu)AB C , (0 omitted)
u=s+1 @ . . . . . .A
at-1,t
P u-1
where xu = v=sav,uand the sum is taken throughout all matrices (ai,j) 2
G(U(n)) such that `t(ai,j) = M- N and `s(ai,j) = N- M and `k(ai,j) = 0
otherwise. The coefficient is defined by ( 1 if t = s+ 1 )
8 _ !
>>> N- k- x -1
>>> ifk < M
>< M - k
_ ! -1
Z(M, N; k, x) = > k- N k- M- 1
>>>(-1)N-M-x ________ ifk > N, x 6 N- M
>>> k- M x
: 0 otherwise
_ !
. .,.N, is+1, . .,.it-1, M, . . .
Notice that for N> M, D 6= 0 if and only if
. .,.M, is+1, . .,.it-1,_N, . . . !
. .,.N, M, . . . N-M
for all s < k < t, either ik < M or ik > N and D = Ps,s+1.
. .,.M, N, . . .
Definition 1.3 For 0 < n < p, we define M(n) to be the free right U(n)
P P
module generated by the set (n). For ff 2 (n), we still use ff to denote the
corresponding generator in M(n). The right U(n)-module homomorphism ffi
P