An Indicator Formula for the Hopf Algebra $$k^{S_{n-1}}\#kC_n$$

Abstract

AbstractThe semisimple bismash product Hopf algebra$$J_n=k^{S_{n-1}}\#kC_n$$Jn=kSn-1#kCnfor an algebraically closed fieldkis constructed using the matched pair actions of$$C_n$$Cnand$$S_{n-1}$$Sn-1on each other. In this work, we reinterpret these actions and use an understanding of the involutions of$$S_{n-1}$$Sn-1to derive a new Froebnius-Schur indicator formula for irreps of$$J_n$$Jnand show that fornodd, all indicators of$$J_n$$Jnare nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all 2-dimensional irreps of$$J_n$$Jnand Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of$$J_n$$Jnand use these formulas to show that nonzero indicators become rare for largen.