by Bessenrodt, Christine; Olsson, Jørn B.

The elementary divisors of the reduced spin 2–decomposition matrix for the double covers of the finite symmetric groups are described. In addition, the maximal power of 2 dividing all spin character values on a fixed conjugacy class corresponding to a cycle type with odd parts only is determined.

DOI: 10.1023/A:1009936318838
Print publication date: 9/1/2000
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by Leclerc, Bernard

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an nth root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space representation of
$$U_q \left( {\widehat{{\text{sl}}}_n } \right)$$
. It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible subrepresentation generated by the vacuum vector. As a consequence, using the works of Ariki and Varagnolo and Vasserot, it is possible to give a new proof of Lusztig”s character formula for the simple U

_v
(sl
_r
)-modules at roots of unity, which does not involve representations of
$$\widehat{{\text{sl}}}_r $$
of negative level.

DOI: 10.1023/A:1009984201999
Print publication date: 9/1/2000
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by Marubayashi, Hidetoshi; Yi, Zhong

Let R be a commutative ring, let G be a finite group acting on R as automorphisms of R and let R * G be the skew group ring. By using the decomposition subgroups of G, the inertial subgroups of G, the properties of the coefficient ring R and the properties of the fixed subring R

^G
, some necessary and sufficient conditions for R * G to be a prime Goldie ring, a semi-hereditary order in a simple Artinian ring, or a Prüfer order in a simple Artinian ring are given.

DOI: 10.1023/A:1009932217929
Print publication date: 9/1/2000
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by Parshall, Brian; Scott, Leonard; Wang, Jian-pan

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J.
(1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent” Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.

DOI: 10.1023/A:1009980101091
Print publication date: 9/1/2000
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by Schauenburg, Peter

We analyze the structure of a Hopf algebra H that has a Hopf subalgebra H ” and a left H “-module coalgebra projection onto H “. In this situation H ≅ H ” ⊗ Q for Q = H / H “⁺
H, and the Hopf algebra structure on H can be recovered from suitable structures on Q, among others an in general nonassociative multiplication. The construction of H from H ” and Q generalizes Radford biproduct, double crossproducts, and certain bicrossproducts. Further examples are Hopf algebras with a triangular decomposition, like all quantized enveloping algebras. In an appendix, we improve a standard criterion for a bicrossproduct A
^α ⋈_τ
B of two Hopf algebras to be a Hopf algebra, and we show that in this case the antipode of the bicrossproduct is bijective if the antipodes of the factors are.

DOI: 10.1023/A:1009993517021
Print publication date: 9/1/2000
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