Hecke algebras at roots of unity.
Let mi be the number of nodes of with colour charge i. The energy of is de ned by P? E ( )= m0, and its weight by wt ( )= 0? in=01 mi i . (We use freely the standard b notations for roots, weights, etc. of sln, see e.g. 5].) Then, the formal character of the basic bln is representation V ( 0 ) of s ch V ( 0 )=
X
2 n
ewt ( ):
A realisation of V ( 0
) that has a basis naturally indexed by the set of n-regular partitions n was given in 5]. In 15], it was also shown how to describe combinatorially the branching functions of b tensor products of irreducible sln -modules. The branching function b;00 0 (q) is de ned such that, for each i 0, the multiplicity of the module V ( 00? i ) in the tensor product V ( ) V ( 0 ) is given by the coe cient of qi in b;00 0 (q). Therefore, in terms of characters, we have X 00? ch (V ( ))ch (V ( 0 ))= b; 0 (e )ch (V ( 00 )): The result of 15] relies on the notion of a path which we describe in the case where 0= 0 . b A path p is a sequence p= (p0; p1;:::), where pk 2 P, the weight lattice of sln . Given a dominant integral weight of level l for some l 0, and 2 n, a path p= p( ) is determined as follows. For k 1, de ne pk=+ k . Then for 0< k 1, recursively obtain pk?1 from pk by pk?1= pk? k?1? 0k; where i= i+1? i and as usual the indices are understood modulo n. We write 2 Y (; 0 ) if and only if all the coordinates pk of the path p( ) are dominant weights. Theorem 2.1 15] X qE( ): b;0 0 (q)=2Y (; 0 ) wt ( )= 0? ( mod )00 2P+
In the case when is of level one, the partitions appearing in this sum were characterised in 9]. We de ne the set FOW (n; j ) n as follows. Let= ( a1; a2;:::; ar ) 2 n . Then r 1 2 2 FOW (n; j ) if and only if either r= 1 or
ai+ i? i+1+ ai+1= 0 (mod n); for i= 1; 2;:::; r? 1 and j= ( 1? a1 ) mod n.
Theorem 2.2 9]
Y ( j; 0)= FOW (n; j ):
If we now de ne FOW (n; j; k) to be the subset of FOW (n; j ) comprising those partitions for which wt ( )= k+ j?k? j ( mod ), then Theorem 2.1 immediately yields the following:
Corollary 2.3 9] Let 0 j; k< n. Then the branching function b X k+ j?k E( )b j;0
k+ j?k j; 0 (q )
is given by:
(q)=
2FOW (n;j;k)
q
:
By means of this expression, the following was obtained: