Hecke algebras at roots of unity.
for all h 2 P _ and 0 i< n. The upper global crystal basis fGup ( ); 2 n g of V ( 0 ) 17] is the basis adjoint to fG( )g with respect to the inner product h; i de ned by
From now on we x n 2 and write Hm instead of Hm ( n 1). It has been conjectured b in 22] and proved in 2] that at q= 1, the upper global crystal basis of the Uq (sln )-module V ( 0 ) coincides in the identi cation V ( 0 ) ' G (n) with the basis ( D]) of G (n). This was also proved by Grojnowski, using the results of 10]. This implies that
hv;; v; i= 1 hqh v; v0 i= hv; qh v0 i; for all v; v0 2 V ( 0 ), h 2 P _ and 0 i< n.
hei v; v0 i= hv; fi v0 i p
Theorem 4.1 If 2 n(m) and the coe cients c (q) are de ned by: n?1 ! X X up upi=0
ei G ( )=
2 n(m?1)
c (q)G ( );
then,
D#Hm?1]= Hm
M2 n (m?1)
c (1) D]:
5 Jantzen-Seitz modules of Hecke algebrasWe are now in position to prove the following result.ar ) 2 n (m). Then D#Hm is irreducible, i.e. 2 r Hm?1 JS (n), if and only if either r= 1 or ai+
i? i+1+ ai+1 0( mod n) for i= 1; 2;:::; r? 1.
Theorem 5.1 Let= ( a1;:::; 1
Proof: By Theorem 4.1, D# is irreducible if and only if ei Gup ( )= Gup ( ) for some . It is known (see 18], eqs. 5.3.8.{ 5.3.10.) that
ei Gup ( )="i ( )]q Gup (~i )+ e E i (q) 2 q2?"i (
X6=
E i (q) Gup ( )
(2)
where E i (q) is a Laurent polynomial invariant under q 7! q?1 such that) Z q]:
If"i ( )= 1, then"i ( )]q= 1, and E i (q)= 0 since for all k the coe cients of qk and q?k in E i are equal. Therefore ei Gup ( )= Gup ( ) where= ei . On the other hand, if~ ei Gup ( )= Gup ( ) for some, then (2) implies that"i ( )= 1 so that= ei .~ Hence we have obtained that D# is irreducible if and only if"i ( )= 1 for some i 2 f0; 1;:::; n? 1g and"j ( )= 0 for j 6= i. Using crystal graph theory (see e.g. Lemma 5.1 and Theorem 5.3 of 15]), we see that ( ) h 0; hj i for j= 0; 1;:::; n? 1 if and only if 2 Y ( 0; 0). Therefore, D# is j irreducible if and only if 2 Y ( i; 0) for some i. Then, by Theorem 2.1, the Theorem is proved.
Example 5.2 The argument considered in the above proof may be illustrated by reference b to the;-connected component of the crystal graph?3 of sl3 given (up to weight 8) in Fig. 1. Here, those partitions for which 2 JS (n) have been highlighted with an asterisk. As in the above proof, these partitions correspond to the nodes b in this crystal graph for which, for some j,"j (b) 1 and"i (b)= 0 for all i 6= j .6