Benjamin Steinberg

The representation theory of the monoid of affine transformations of a vector space over a finite field

The monoid $\mathrm{Aff}(n,q)$ of all affine transformations of the vector space $\mathbb F_q^n$ is a naturally occuring monoid that can be viewed from several perspectives: it is the semidirect product of the monoid $M(n,q)$ of $n\times n$ matrices over $\mathbb F_q$ with the additive group of $\mathbf F_q^n$; it is also an the endomorphism monoid of an independence algebra (the vector space $\mathbb F_q^n$ with affine combinations as a collection of binary operations, one for each scalar).

In this talk we consider the representation theory of this monoid over the field of complex numbers. We compute the global dimension of the complex algebra of this monoid using a combination of algebraic and topological techniques. The dramatis personae of this story are Putcha–Okniński's semisimplicity of $\mathbb CM(n,q)$, Nico's results on the global dimension of regular monoids (aka the theory of quasihereditary algebras) and Solomon's irreducible representation of the affine general linear group on the top homology of the order complex of the proper part of the lattice of flats of the affine matroid.