The linear algebra of the Han–Monsky representation ring

Abstract

From the perspective of Classical Linear Algebra, we analyze a family of finite dimensional algebras related to to the ‘Representation Ring’ introduced by Han and Monsky [Some surprising Hilbert-Kunz functions. Math. Z., 214:119–135, 1992]. This ring plays a central role in the theory of Hilbert-Kunz multiplicity, a modern numerical invariant of certain positive characteristic commutative rings. The matrices representing these structures have very rich properties with applications to commutative algebra and tantalizing connections to number theory. We construct an eigenbasis that effectively diagonalizes the Han-Monsky product, opening the door for new results in Hilbert-Kunz theory. Using this basis, we study the classical properties of this family of matrices.