Representations of the Oriented Brauer Category
Datum
2015-08-18
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Verlag
University of Oregon
Zusammenfassung
We study the representations of a certain specialization $\mathcal{OB}(\delta)$ of the oriented Brauer category in arbitrary characteristic $p$. We exhibit a triangular decomposition of $\mathcal{OB}(\delta)$, which we use to show its irreducible representations are labelled by the set of all $p$-regular bipartitions. We then explain how its locally finite dimensional representations can be used to categorify the tensor product $V(-\varpi_{m'}) \otimes V(\varpi_{m})$ of an integrable lowest weight and highest weight representation of the Lie algebra $\mathfrak{sl}_{\Bbbk}$. This is an example of a slight generalization of the notion of tensor product categorification in the sense of Losev and Webster and is the main result of this paper. We combine this result with the work of Davidson to describe the crystal structure on the set of irreducible representations. We use the crystal to compute the decomposition numbers of standard modules as well as the characters of simple modules assuming $p = 0$. We give another proof of the classification of irreducible modules over the walled Brauer algebra. We use this classification to prove that the irreducible $\mathcal{OB}(\delta)$-modules are infinite dimensional unless $\delta = 0$, in which case they are all infinite dimensional except for the irreducible module labelled by the empty bipartition, which is one dimensional.