Some Representations of Special Linear Groups
نویسنده
چکیده
Mat r x ( F q ) o f a l l r b y r m a t r i c e s o v e r F q , b y C t h e g r o u p G L t i ( F q ) o f t h e u n i t s of M, and by S the special linear subgroup SL , (F q ) o f C . F o r a n a r b i t r a r y field F containing F q , l e t U s t a n d f o r t h e ( c o m m u t a t i v e ) p o l y n o m i a l a l g e b r a F[xl, , x r ] , a n d c o n s i d e r U g r a d e d a s u s u a l : U = E l ) U d w h e r e U d i s t h e ' s p a c e of homogeneous polynomials of degree d. In particular, U l i s t h e F s p a c e w i t h basi {x i, , x r } , s o M h a s a n a t u r a l a c t i o n o n U l ; t h i s e x t e n d s ( u n i q u e l y ) t o n action of M on all of U by graded algebra endomorphisms. Consider U an M-module (C-module, S-module) accordingly. This paper is concerned with the submodule structure of U. Obviously, each omogeneous component Ud is an M-submodule, and so is the sum of any set of the Ud. A s is well known, in the characteristic O analogue of this setup, these would be the only S-submodules (let alone M-submodules). I n prime characteristic there are other kinds of submodules as well, at least if one excludes (as I shall from now on) the rather trivial case r = 1. Namely, for each submodule Y, denote by YP the F-span of all the yP with y E Y: this is also a submodule (because on any commutative ring of characteristic p, the map u u P is a ring endomorphism which commutes with all other ring endomorphisms), and for instance O < U < Ud p u n l e s s d = O . O f c o u r s e , s u m s a n
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