2007/01/17 by Andrea Jedwab, Susan Montgomery, Jedwab, Andrea +1 · 2 citations
Mathematics · #16W30 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.math/0701495
openalex publication_date 2007/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the representations and their Frobenius-Schur indicators of two semisimple Hopf algebras related to the symmetric group Sn, namely the bismash products Hn = kCn# kSn-1 and its dual Jn = k^Sn-1# kCn = (Hn)^*, where k is an algebraically closed field of characteristic 0. Both algebras are constructed using the standard representation of Sn as a factorizable group, that is Sn = Sn-1Cn = CnSn-1. We prove that for Hn, the indicators of all simple modules are +1. For the dual Hopf algebra Jn = k^Sn1# kCn, the indicator can have values either 0 or 1. When n = p, a prime, we obtain a precise result as to which representations have indicator +1 and which ones have 0; in fact as p → ∞, the proportion of simple modules with indicator 1 becomes arbitrarily small. We also prove a result about Frobenius-Schur indicators for more general bismash products H =kG# kF, coming from any factorizable group of the form L = FG such that F≅ Cp. We use the definition of Frobenius-Schur indicators for Hopf algebras, as described in work of Linchenko and the second author, which extends the classical theorem of Frobenius and Schur, for a finite group G.