Abstract
Branching of symplectic groups is not multiplicity free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra \({\mathcal{B}}\). The algebra \({\mathcal{B}}\) is a graded algebra whose components encode the multiplicities of irreducible representations of Sp 2n–2 in irreducible representations of Sp 2n . Our first theorem states that the map taking an element of Sp 2n to its principal n × (n + 1) submatrix induces an isomorphism of \({\mathcal{B}}\) to a different branching algebra \({\mathcal{B}^{\prime}}\). The algebra \({\mathcal{B}^{\prime}}\) encodes multiplicities of irreducible representations of GL n–1 in certain irreducible representations of GL n+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp 2n to Sp 2n–2 is canonically an irreducible module for the n-fold product of SL 2. In particular, this induces a canonical decomposition of the multiplicity spaces into one-dimensional spaces, thereby resolving the multiplicities.
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This work was supported in part by an ARCS fellowship.
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Yacobi, O. An analysis of the multiplicity spaces in branching of symplectic groups. Sel. Math. New Ser. 16, 819–855 (2010). https://doi.org/10.1007/s00029-010-0033-z
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DOI: https://doi.org/10.1007/s00029-010-0033-z