Characterizing cofree representations of SLn x SLm

Abstract

The study, and in particular classification, of cofree representations has been an interest of research for over 70 years. The Chevalley-Shepard-Todd Theorem provides a beautiful intrinsic characterization for cofree representations of finite groups. Specifically, this theorem says that a representation V of a finite group G is cofree if and only if G is generated by pseudoreflections. Up until the late 1900s, with the exception of finite groups, all of the existing classifications of cofree representations of a particular group consist of an explicit list, as opposed to an intrinsic group-theoretic characterization. However, in 2019, Edidin, Satriano, and Whitehead formulated a conjecture which intrinsically characterizes stable irreducible cofree representations of connected reductive groups and verified their conjecture for simple Lie groups. The conjecture states that for a stable irreducible representation V of a connected reductive group G, V is cofree if and only if V is pure. In comparison to the classifications comprised of a list of cofree representations, this conjecture can be viewed as an analogue of the Chevalley–Shepard–Todd Theorem for actions of connected reductive groups. The aim of this thesis is to further expand upon the techniques formulated by Edidin, Satriano, and Whitehead as a means to work towards the verification of the conjecture for all connected semisimple Lie groups. The main result of this thesis is the verification of the conjecture for stable irreducible representations V\otimes W of SLn x SLm satisfying dim V >= n^2 and dim W >= m^2.

As the main group under study in this thesis is SLn x SLm, in Chapter 2 we provide a thorough analysis of the structure of irreducible representations of SLn from the view point of them being in one-to-one correspondence with irreducible representations of the Lie algebra Lie(SLn). The last section of Chapter 2 describes the general theory of irreducible representations of complex semisimple Lie algebras, with SLn as a toy example. In Chapter 3, we provide a brief introduction to Geometric Invariant Theory (GIT) and present the main results of the theory. We then discuss the history of GIT and the known characterization results for properties of representations that arise from GIT. In particular, we introduce cofree representations and the current classification results for cofree representations of certain classes of groups. We finish Chapter 3 by introducing pure representations and the conjecture formulated by Edidin, Satriano, and Whitehead. In Chapter 4, we verify that for all stable irreducible representations V\otimes W of SLn x SLm satisfying dim V >= n^2 and dim W >= m^2, V\otimes W is cofree if and only if V\otimes W if pure. This involves proving an upper bound on the dimension of pure representations of G_1 x G_2, with G_i connected reductive Lie groups. We also introduce two methods that can be used to show that a given representation is not pure. The last section in Chapter 4 discusses the difficulties and obstacles when trying to verify the conjecture for the remaining cases, namely when dim V < n^2 or dim W < m^2.