31 research outputs found
A basis theorem for the degenerate affine oriented Brauer-Clifford supercategory
We introduce the oriented Brauer-Clifford and degenerate affine oriented
Brauer-Clifford supercategories. These are diagrammatically defined monoidal
supercategories which provide combinatorial models for certain natural monoidal
supercategories of supermodules and endosuperfunctors, respectively, for the
Lie superalgebras of type Q. Our main results are basis theorems for these
diagram supercategories. We also discuss connections and applications to the
representation theory of the Lie superalgebra of type Q.Comment: 37 pages, many figures. Version 3 replaces the partial results from
the previous versions with a proof by the first author of a basis theorem for
cyclotomic quotients at all levels. Various other minor corrections and
revisions were mad
Presenting Schur superalgebras
We provide a presentation of the Schur superalgebra and its quantum analogue
which generalizes the work of Doty and Giaquinto for Schur algebras. Our
results include a basis for these algebras and a presentation using weight
idempotents in the spirit of Lusztig's modified quantum groups.Comment: 28 pages, to appear in the Pacific Journal of Mathematic
Cohomology and Support Varieties for Lie Superalgebras II
In \cite{BKN} the authors initiated a study of the representation theory of
classical Lie superalgebras via a cohomological approach. Detecting subalgebras
were constructed and a theory of support varieties was developed. The dimension
of a detecting subalgebra coincides with the defect of the Lie superalgebra and
the dimension of the support variety for a simple supermodule was conjectured
to equal the atypicality of the supermodule. In this paper the authors compute
the support varieties for Kac supermodules for Type I Lie superalgebras and the
simple supermodules for . The latter result verifies our
earlier conjecture for . In our investigation we also
delineate several of the major differences between Type I versus Type II
classical Lie superalgebras. Finally, the connection between atypicality,
defect and superdimension is made more precise by using the theory of support
varieties and representations of Clifford superalgebras.Comment: 28 pages, the proof of Proposition 4.5.1 was corrected, several other
small errors were fixe