Abstract: We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent upper triangular matrices with coefficients in a finite field, and NCSym, the symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are Hopf isomorphic. This allows developments in each to be transfered. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

The resulting theory is very nicely behaved — there is a rich combinatorics describing induction and restriction along with an elegant formula for the values of superclasses on superclasses. The combinatorics is described in terms of set partitions (the symmetric groups theory involves integer partitions) and the combinatorics seems akin to tableau combinatorics. At the same time, supercharacter theory is rich enough to serve as a substitute for ordinary character theory in some problems.