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Kazhdan–Lusztig Cells, q ‐Schur Algebras and James' Conjecture
Author(s) -
Geck Meinolf
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610700001873
Subject(s) - mathematics , schur algebra , representation theory , pure mathematics , root of unity , semisimple lie algebra , weyl group , conjecture , simple module , type (biology) , combinatorics , field (mathematics) , algebra over a field , simple (philosophy) , quantum , affine lie algebra , current algebra , physics , classical orthogonal polynomials , ecology , philosophy , gegenbauer polynomials , epistemology , quantum mechanics , orthogonal polynomials , biology
We consider the Dipper–James q ‐Schur algebra S q ( n , r ) k , defined over a field k and with parameter q ≠ 0. An understanding of the representation theory of this algebra is of considerable interest in the representation theory of finite groups of Lie type and quantum groups; see, for example, [ 6 ] and [ 11 ]. It is known that S q ( n , r ) k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when S q ( n , r ) k is not semisimple. Then we have a corresponding decomposition matrix which records the multiplicities of the simple modules in certain ‘standard modules’ (or ‘Weyl modules’). A major unsolved problem is the explicit determination of these decomposition matrices.