Schur-finiteness in Λ-rings

We introduce the notion of a Schur-finite element in a λ-ring. Since the beginning of algebraic K-theory in [G57], the splitting principle has proven invaluable for working with λ-operations. Unfortunately, this principle does not seem to hold in some recent applications, such as the K-theory of motives. The main goal of this paper is to introduce the subring of Schur-finite elements of any λ-ring, and study its main properties, especially in connection with the virtual splitting principle. A rich source of examples comes from Heinloth’s theorem [Hl], that the Grothendieck group K0(A) of an idempotent-complete Q-linear tensor category A is a λring. For the category M of effective Chow motives, we show that K0(V ar) → K0(M ) is not an injection, answering a question of Grothendieck. When A is the derived category of motives DMgm over a field of characteristic 0, the notion of Schur-finiteness in K0(DMgm) is compatible with the notion of a Schur-finite object in DMgm, introduced in [Mz]. We begin by briefly recalling the classical splitting principle in Section 1, and answering Grothendieck’s question in Section 2. In section 3 we recall the Schur polynomials, the Jacobi-Trudi identities and the Pieri rule from the theory of symmetric functions. Finally, in Section 4, we define Schur-finite elements and show that they form a subring of any λ-ring. We also state the conjecture that every Schur-finite element is a virtual sum of line elements. Notation. We will use the term λ-ring in the sense of [Ber, 2.4]; we warn the reader that our λ-rings are called special λ-rings by Grothendieck, Atiyah and others; see [G57] [AT] [A]. A Q-linear category A is a category in which each hom-set is uniquely divisible (i.e., a Q-module). By a Q-linear tensor category (or QTC) we mean a Q-linear category which is also symmetric monoidal and such that the tensor product is Q-linear. We will be interested in QTC’s which are idempotent-complete. Date: October 26, 2012. Weibel’s research was supported by NSA and NSF grants.