Abstract
We derive a closed formula for the tensor product of a family of mixed tensors using Deligne’s interpolating category underline {Rep}(GL_{0}). We use this formula to compute the tensor product of a family of irreducible GL(n|n)-representations. This includes the tensor product of any two maximal atypical irreducible representations of GL(2|2).
Highlights
For the classical group GL(n) the tensor product decomposition L(λ) ⊗ L(μ) = cλνμL(ν)ν between two irreducible representations is given by the Littlewood-Richardson rule for the Littlewood-Richardson coefficients cλνμ
A classical result from Berele and Regev [1] and Sergeev [21] shows that the fusion rule between direct summands of tensor powers V ⊗r of the standard representation V km|n is again given by the Littlewood-Richardson rule
Since the decomposition of the tensor product of two indecomposable elements is known for Rep(GLm−n) by results of Comes and Wilson [7], we obtain an analogous decomposition law once we describe the image Fm|n(X) of an arbitrary indecomposable object X in Rep(GLm−n). This was achieved in [13] based on results by Brundan and Stroppel [6] on the interplay between Khovanov algebras and Walled Brauer algebras
Summary
Since the decomposition of the tensor product of two indecomposable elements is known for Rep(GLm−n) by results of Comes and Wilson [7], we obtain an analogous decomposition law once we describe the image Fm|n(X) of an arbitrary indecomposable object X in Rep(GLm−n). This was achieved in [13] based on results by Brundan and Stroppel [6] on the interplay between Khovanov algebras and Walled Brauer algebras. [13], these results give a decomposition law for their tensor products, covering in particular the decomposition between any two irreducible GL(m|1)-representations
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