Speaker: Sagar Shrivastava (ISU)
Title: Branching Multiplicities of Symplectic Groups as $SL_2$ Representations
Abstract: Branching rules provide a systematic framework for decomposing irreducible representations of a Lie group when restricted to a subgroup. While the branching rules for the General Linear and Orthogonal groups are notably multiplicity-free, the Symplectic group $Sp(2n)$ presents a more complex structure where multiplicities can exceed one. The combinatorial landscape of these multiplicities was first explicitly described by Lepowsky. More recently, in 2009, Wallach and Yacobi demonstrated that these multiplicities are not merely integers, but correspond to the dimensions of specific representation spaces of $SL_2 \cong Sp(2)$.
In this two-part seminar, we present an alternative, more transparent proof of this correspondence. Unlike traditional approaches, our method bypasses the use of complex partition function machinery. We will show how these multiplicity spaces emerge naturally, assuming only a working knowledge of the Weyl Character Formula.
In the first part, we would introduce the machinery and the way to use that machinery to deduce the branching laws in the multiplicity free cases.
In the second part, we would show how that machinery allows us to compute the multiplicity in the case of the symplectic group.