On Infinitesimal Schur Algebras

One approach to the representation theory of the general linear group G = GL n (over an infinite field, in the defining characteristic) depends on the equivalence between rational representations of the multiplicative monoid M = M n of n × n matrices and polynomial representations of GL n . This equivalence reduces polynomial representation theory of G to representation theory of certain finite‐dimensional algebras S ( n , d ), the (ordinary) Schur algebras. Another standard approach in prime characteristic is to study the representations of GL n through representations of the group schemes G r T associated to the r th Frobenius kernel G r of G and a maximal torus T of G . These two methods have led us to construct a module category for a monoid scheme, M r D , which combines facets of both theories. The monoid M r D bears the same relation to G r T as does M to G. We are led to a polynomial representation theory of G r T and to certain associated subalgebras S ( n , d ) r of S ( n , d ). These subalgebras are the infinitesimal Schur algebras.