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

2018/12/14 by Alexander Heaton, Heaton, Alexander, Songpon Sriwongsa +3
Computer Science · Mathematics · #05E10 #20C30 #22E46 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1812.06211

openalex publication_date 2018/12/14 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

Let S be a principally embedded sl₂ subalgebra in sl_n for n > 2. A special\ncase of results of the third author and Gregg Zuckerman implies that there\nexists a positive integer b(n) such that for any finite-dimensional irreducible\nsl_n representation, V, there exists an irreducible S-representation embedding\nin V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou),\nthey prove that b(n)=n is the sharpest possible bound, and also address\nembeddings other than the principal one.\n These results concerning embeddings may by interpreted as statements about\nplethysm. Then, a well known result about these plethysms can be interpreted as\na "branching rule". Specifically, a (finite dimensional) representation of\nGL(n,C) will decompose into irreducible representations of the symmetric group\nwhen it is restricted to the subgroup consisting of permutation matrices. The\nquestion of which irreducible representations of the symmetric group occur with\npositive multiplicity is the topic of this paper, applying the previous work of\nLhou, Zuckerman, and the third author.\n

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