RSOS models and Jantzen-Seitz representations of Hecke algeb(2)

3 Modular representationsThe set S partitions FOW (n; j; k) has a de nite meaning in modular representation theory. of Indeed, j;k FOW (n; j; k) labels the modular representations of Sm in characteristic n that remain irreducible under reduction to Sm?1 14, 20]. However, in the case of FOW (n; j; k), n can be any positive integer, whereas in the case of Sm, it has to be a prime number. This di erence can be removed by working in the context of Hecke algebras at an nth root of unity, where the Jantzen-Seitz problem still makes sense. The Hecke algebra Hm (v) of type Am?1, is the C (v )-algebra generated by the elements T1;:::; Tm?1 subject to the relations

Ti Ti+1 Ti= Ti+1 Ti

Ti+1; Ti Tj= Tj Ti ji? j j> 1; Ti2= (v? 1)Ti+ v:is semisimple, and its irreducible representations are parametrised by (m). A convenient realisation of the representation labelled by is the Specht module S in which the entries of the representation matrices of the generators are elements of Z v]. In the non-generic case when v is a primitive nth root of unity, Hm (v) is no longer semisimple in general, and its representations are not necessarily completely reducible. The full set of irreducible representations is indexed by n (m) (see 4]). The irreducible module labelled by 2 n (m) is denoted by D . De ne JS (n) to be the set of all partitions that label the irreducible representations of p p Hm ( n 1) which remain irreducible under reduction to Hm?1 ( n 1). One of the aims of this work is to show that the partitions in JS (n) are de ned by the same conditions as in the case of symmetric groups, and to explain why

Sm, through identifying Ti with the simple transposition (i; i+1) 2 Sm. In fact, for generic values of v, Hm (v) is isomorphic to the group algebra of the symmetric group Sm . Hence, it

In the case v= 1, Hm (v) may be identi ed with the group algebra of the symmetric group

JS (n)=

j;k

FOW (n; j; k):

To tackle this problem, it is convenient to consider, as was done in 22, 2], the Grothenp p dieck group G0 (Hmp n 1)) of the category of nitely generated Hm ( n 1)-modules. The el( ements of G0 (Hm ( n 1)) are classes M] of modules, where M1]= M2] if and only if the simple composition factors of M1 and M2 occur with identical multiplicities (the order of the composition factors in the series is disregarded). The sum is de ned by M]+ N]= M N]. It is known that this is a free abelian group with as basis the set f D L of classes of irre]g p p ducible Hm ( n 1)-modules. Then de ne G (n) as the direct sum G (n)= m G0 (Hm ( n 1)). b As observed in 22], G (n) can be identi ed with the basic representation V ( 0 ) of sln, the action of the Chevalley generators ei, fi being given by the i-restriction and i-induction operators, as de ned in the fties by G. de B. Robinson in the case of symmetric groups. 4

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;

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 produ …… 此处隐藏：6038字，全部文档内容请下载后查看。喜欢就下载吧 ……