Representation of the Symmetric Group

Over the last 112 years since Frobenius and Burnside initiated the study of representation theory, matrix algebra has played a central role with the indispensable idea of a group character [16]. Recently, point groups and crystallographic groups have been studied in geometric algebra [37, 40]. In this chapter, representations of the symmetric group are studied in the geometric algebra $${\mathbb{G}}_{n,n}$$ of neutral signature. The representation depends heavily upon the definition of a new twisted symmetric product, which does not depend upon a matrix representation, although it is equivalent to it.