Mathematics – Representation Theory
Scientific paper
2009-07-18
Mathematics
Representation Theory
32 pages, revised abstract and introduction, and reorganized the structure of the paper
Scientific paper
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 $B$. The algebra $B$ 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 $n \times (n+1)$ submatrix induces an isomorphism of $\B$ to a different branching algebra $\B'$. The algebra $\B'$ 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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