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On Infinitesimal Schur Algebras
Author(s) -
Doty Stephen R.,
Nakano Daniel K.,
Peters Karl M.
Publication year - 1996
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-72.3.588
Subject(s) - mathematics , homomorphism , representation theory , monoid , infinitesimal , pure mathematics , schur algebra , multiplicative function , polynomial , equivalence (formal languages) , discrete mathematics , combinatorics , algebra over a field , mathematical analysis , classical orthogonal polynomials , gegenbauer polynomials , orthogonal polynomials
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.