Irreducible Character Restrictions to Maximal Subgroups of Low-Rank Classical Groups of Type B and C
IRREDUCIBLE CHARACTER RESTRICTIONS TO MAXIMAL SUBGROUPS OF LOW-RANK CLASSICAL GROUPS OF TYPE B AND C KEMPTON ALBEE, MIKE BARNES, AARON PARKER, ERIC ROON, AND A.A. SCHAEFFER FRY Abstract Representations are special functions on groups that give us a way to study abstract groups using matrices, which are often easier to understand. In particular, we are often interested in irreducible representations, which can be thought of as the building blocks of all representations. Much of the information about these representations can then be understood by instead looking at the trace of the matrices, which we call the character of the representation. This paper will address restricting characters to subgroups by shrinking the domain of the original representation to just the subgroup. In particular, we will discuss the problem of determining when such restricted characters remain irreducible for certain low-rank classical groups. 1. Introduction Given a finite group G, a (complex) representation of G is a homomorphism Ψ: G ! GLn(C). By summing the diagonal entries of the images Ψ(g) for g 2 G (that is, taking the trace of the matrices), we obtain the corresponding character, χ = Tr◦Ψ of G. The degree of the representation Ψ or character χ is n = χ(1). It is well-known that any character of G can be written as a sum of so- called irreducible characters of G. In this sense, irreducible characters are of particular importance in representation theory, and we write Irr(G) to denote the set of irreducible characters of G. Given a subgroup H of G, we may view Ψ as a representation of H as well, simply by restricting the domain.
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