Lowest 𝔰𝔩(2)-types in 𝔰𝔩(𝔫)-representations

Abstract

Abstract. Fix n ≥ 3. Let s be a principally embedded sl 2 -subalgebra in sl n . A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite dimensional irreducible sl n -representation, V , there exists an irreducible s-representation embedding in V with dimension at most b(n). We prove that b(n) = n is the sharpest possible bound. We also address embeddings other than the principal one.The exposition involves an application of the Cartan-Helgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the "branching algebra" introduced by Roger Howe, Eng-Chye Tan, and the second author.

Branching from the General Linear Group to the Symmetric Group and the Principal Embedding

“…In Section 4 we saw that finding the multiplicity of F λ k inside F µ (Sym C k ) (plethysm) was equivalent to finding the multiplicity of Y µ n inside F λ n (branching). In this section we will apply the main theorem from [4] on the plethysm side to guarantee non-zero multiplicity of certain irreps when k = l(λ) = 2. Thus we will also have guaranteed non-zero multiplicity of certain branching multiplicities as well.…”

“…First we will show that a certain branching multiplicity will be equal to a certain plethysm multiplicity. Later, we will use this fact to re-interpret the main theorem of [4] in terms of the branching from GL n to S n . We regard F λ as a functorial operator on the category of vector spaces -often called the Schur functor.…”

“…As an example, consider decomposing irreps of GL 4 into a direct sum of irreps of S 4 . But as described above, only decompose the irreps of GL 4 given by F (d) 4 , which are equivalent to the dth symmetric power of C 4 . We have the symmetric group on four letters S 4 and its irreducible representations Y µ 4 where µ is from the set µ ∈ {(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)} Consider the first few.…”

“…Restricting to SL m+1 it is again irreducible. By the main theorem of [4] we are guaranteed the existence of some subrepresentation isomorphic to Sym d C 2 , an irrep of SL 2 , for some d ∈ {0, 1, 2, . .…”

Branching from the General Linear Group to the Symmetric Group and the Principal Embedding

“…In this paper we show a relationship between the principal embedding and branching from GL n to the symmetric group. Our main tool is the following theorem proved in [2] which was anticipated in [3]: Theorem 1. Fix n ≥ 3 and a principal sl 2 -subalgebra, s, of sl n .…”