An Analysis of the Multiplicity Spaces in Branching of Symplectic Groups

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 $B$. The algebra $B$ is a graded algebra whose components encode the multiplicities of irreducible representations of $S{p}_{2n-2}$ in irreducible representations of $S{p}_{2n}$. Our first theorem states that the map taking an element of $S{p}_{2n}$ 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 $G{L}_{n-1}$ in certain irreducible representations of $G{L}_{n+1}$. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of $S{p}_{2n}$ to $S{p}_{2n-2}$ is canonically an irreducible module for the $n$-fold product of $S{L}_2$. In particular, this induces a canonical decomposition of the multiplicity spaces into one dimensional spaces, thereby resolving the multiplicities.

32 pages, revised abstract and introduction, and reorganized the structure of the paper

References in corpus (2)

• A Basis for Representations of Symplectic Lie Algebras
• A Basis for the Symplectic Group Branching Algebra