Witt vectors, semirings, and total positivity

@article{Borger1970WittVS,
  title={Witt vectors, semirings, and total positivity},
  author={James M. Borger},
  journal={arXiv: Combinatorics},
  year={1970},
  pages={273-329}
}
  • J. Borger
  • Published 11 October 2013
  • Mathematics
  • arXiv: Combinatorics
We extend the big and $p$-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the $p$-typical case, it uses positivity with respect to an apparently new basis of the $p$-typical symmetric functions. We also give explicit descriptions of the big Witt vectors of the natural numbers and of the nonnegative… 
Boolean Witt vectors and an integral Edrei–Thoma theorem
A subtraction-free definition of the big Witt vector construction was recently given by the first author. This allows one to define the big Witt vectors of any semiring. Here we give an explicit
The $\gamma$-filtration on the Witt ring of a scheme
The K-ring of symmetric vector bundles over a scheme X, the so-called Grothendieck-Witt ring of X, can be endowed with the structure of a (special) $\lambda$-ring. The associated $\gamma$-filtration
Cyclotomic factors of necklace polynomials
Necklace polynomials $M_d(x)$ play an important role in number theory, combinatorics, dynamics, and representation theory. In this paper we introduce and analyze the \emph{cyclotomic factor
Symmetric Representation Rings are $\lambda$-Rings
The representation ring of an affine algebraic group scheme can be endowed with the structure of a (special) $\lambda$-ring. We show that the same is true for the ring of symmetric representations,
Symmetric representation rings are λ-rings
The representation ring of an affine algebraic group scheme can be endowed with the structure of a (special) λ-ring. We show that the same is true for the ring of symmetric representations, i.e., for

References

SHOWING 1-10 OF 10 REFERENCES
Representable Functors and Operations on Rings
Introduction The main aim of this article is to describe the mechanics of certain types of operations on rings (e.g. A-operations on special A-rings or differentiation operators on rings with
On the generating functions of totally positive sequences I
A real matrix, finite or infinite, is called totally positive if and only if all its minors, of all orders = 1, 2,..., are non-negative. An infinite sequence $$ {a_0},{a_1},{a_2}, \ldots ,\quad
Lectures on curves on an algebraic surface
These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over
Collected Papers
THIS volume is the first to be produced of the projected nine volumes of the collected papers of the late Prof. H. A. Lorentz. It contains a number of papersnineteen in all, mainly printed
Zyklische Körper und Algebren der Charakteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p.
----------------------------------------------------Nutzungsbedingungen DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung
Stochastic processes
Eléments de Mathématique
Die unzerlegbaren
  • positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Z. 85
  • 1964
Total positivity
  • Vol. I. Stanford University Press, Stanford, Calif
  • 1968