Hecke algebras at roots of unity.
b The standard basis (v ) of F is labelled by all 2 and one has an sln -homomorphism d: F?! V ( 0 ) ' F=Vlow v 7?! v mod Vlow If F is identi ed with the Grothendieck ring of all Hm (v) for a generic v by writing S]= v, then d coincides with the decomposition map of modular representation theory. In order to introduce the canonical basis, one needs to q-deform the picture and to consider the q-Fock b space representation of Uq (sln ).
As an integrable highest weight module of an a ne algebra, G (n) has a canonical basis in the sense of Lusztig and Kashiwara. It turns out that this canonical basis coincides with p the natural basis given by the classes D( )] of the irreducible Hm ( n 1)-modules. To prove this, and to compute explicitly the canonical basis, one considers the Fock space b representation F of sln, wh
ich contains V ( 0 ) as its highest irreducible component: F= V ( 0 ) Vlow:
4 The Fock space representation of Uq (sbln)
b The q-Fock space F of Uq (sln ) has been described in 11] using a q-analogue of the Cli ord algebra. Another realization was given in 24, 19] in terms of semi-in nite q-wedge products. Let (v ) denote the standard basis of weight vectors of F . The Fock space F a ords an b integrable representation of Uq (sln ) whose decomposition into irreducible highest weight modules is given by M F= V ( 0? k ) p(k): (1)k 0
In 23], the lower crystal basis of F and its crystal graph structure were described. Let A Q (q) denote the ring of rational functions without a pole at q= 0. The lower crystal basis at q= 0 of F is the pair (L; B ) where L is the lower crystal lattice given by
L=
M2
Av;
and B is the basis of the Q -vector space L=qL given by B= fv mod qL j 2 g:~ The action of Kashiwara's operators fi on the element v 2 B corresponds to adding to a certain node of colour i which is called a good addable i-node. Likewise the action of ei corresponds to the removal from of a certain node of colour i which is called a good~ removable i-node. As observed in 22], these nodes are precisely those used by Kleshchev in his modular branching rule for symmetric groups 21], whence the terminology.~ For each 2, the largest integer k such that ei k ( ) 6= 0 (resp fi k ( ) 6= 0) is denoted by~"i ( ) (resp. 'i ( )). The crystal graph?n of F is disconnected and re ects the decomposition (1). The connected component of the empty partition is the crystal graph of V ( 0 ) and its vertices are labelled by n . We denote by fG( ); 2 n g the lower global basis of V ( 0 ). It is the Q q; q?1]-basis of the integral form VQ ( 0 ) of V ( 0 ) characterized by
G( ) v ( mod qL); G( )= G( ): Here, for v= xv; 2 V ( 0 ), v is de ned by v= xv;, where x 7! x is the Q -linear ring b automorphism of Uq (sln ) given by q= q?1; qh= q?h;5
ei= ei;
fi= fi;