Abstract
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 combinatorial description of the big Witt vectors of the Boolean semiring. We do the same for two variants of the big Witt vector construction: the Schur Witt vectors and the p-typical Witt vectors. We use this to give an explicit description of the Schur Witt vectors of the natural numbers, which can be viewed as the classification of totally positive power series with integral coefficients, first obtained by Davydov. We also determine the cardinalities of some Witt vector algebras with entries in various concrete arithmetic semirings.
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References
Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.: On the generating functions of totally positive sequences. Proc. Natl. Acad. Sci. USA 37, 303–307 (1951)
Aissen, M., Schoenberg, I.J., Whitney, A.M.: On the generating functions of totally positive sequences I. J. Anal. Math. 2, 93–103 (1952)
Bloch, S.: Algebraic \(K\)-theory and crystalline cohomology. Inst. Hautes Études Sci. Publ. Math. 47, 187–268 (1977)
Borger, J.: \(\Lambda \)-Rings and the field with one element. arXiv:0906.3146v1
Borger, J.: Witt vectors, semirings, and total positivity. In: Thas, K. (ed.) Absolute Arithmetic and \({\mathbb{F}}_1\)-Geometry. European Mathematical Society Publishing House, Zurich (2015). arXiv:1310.3013v1
Chatzistamatiou, A.: Big de Rham–Witt cohomology: basic results. Doc. Math. 19, 567–599 (2014)
Connes, A.: The Witt construction in characteristic one and quantization. In: Connes, A., Gorokhovsky, A., Lesch, M., Pflaum, M., Rangipour, D. (eds.) Noncommutative Geometry and Global Analysis, volume 546 of Contemporary Mathematics, pp. 83–113. American Mathematical Society, Providence (2011)
Connes, A., Consani, C,: The universal thickening of the field of real numbers. arXiv:1202.4377v2
Davydov, A.A.: Totally positive sequences and \(R\)-matrix quadratic algebras. J. Math. Sci. (N.Y.) 100(1), 1871–1876 (2000). (Algebra, 12)
Edrei, A.: On the generating functions of totally positive sequences II. J. Anal. Math. 2, 104–109 (1952)
Golan, J.S.: The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, volume 54 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow (1992)
Hesselholt, L.: The big de Rham–Witt complex. Acta Math. 214(1), 135–207 (2015)
Hesselholt, L., Madsen, I.: On the \(K\)-theory of nilpotent endomorphisms. In: Greenlees, J.P.C. (ed.) Homotopy Methods in Algebraic Topology (Boulder, CO, 1999), volume 271 of Contemporary Mathematics, pp. 127–140. American Mathematical Society, Providence (2001)
Illusie, L.: Finiteness, duality, and Künneth theorems in the cohomology of the de Rham–Witt complex. In: Algebraic Geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Mathematics, pp. 20–72. Springer, Berlin (1983)
Kerov, S.V.: Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, volume 219 of Translations of Mathematical Monographs. American Mathematical Society, Providence (2003). (Translated from the Russian manuscript by N. V. Tsilevich, with a foreword by A. Vershik and comments by G. Olshanski)
Kerov, S., Okounkov, A., Olshanski, G.: The boundary of the Young graph with Jack edge multiplicities. Int. Math. Res. Not. 4, 173–199 (1998)
Langer, A., Zink, T.: De Rham–Witt cohomology for a proper and smooth morphism. J. Inst. Math. Jussieu 3(2), 231–314 (2004)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, Oxford University Press, New York (1995). (With contributions by A. Zelevinsky, Oxford Science Publications)
Okounkov, A.: On representations of the infinite symmetric group. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240(Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2), 166–228, 294 (1997)
Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, volume 203 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2001)
Salem, R.: Power series with integral coefficients. Duke Math. J. 12, 153–172 (1945)
Schoenberg, I.J.: Some analytical aspects of the problem of smoothing. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 351–370. Interscience Publishers, New York (1948)
Stanley, R.P.: Enumerative Combinatorics. Vol. 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997). (With a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 original)
Stanley, R.P.: Enumerative Combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin)
Thoma, E.: Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Z. 85, 40–61 (1964)
van Leeuwen, M.A.A.: The Littlewood–Richardson rule, and related combinatorics. arXiv:math/9908099v1
Vershik, A.M., Kerov, S.V.: Asymptotic theory of the characters of a symmetric group. Funktsional. Anal. i Prilozhen. 15(4), 15–27 (1981)
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James Borger: Supported by the Australian Research Council, DP120103541 and FT110100728.
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Borger, J., Grinberg, D. Boolean Witt vectors and an integral Edrei–Thoma theorem. Sel. Math. New Ser. 22, 595–629 (2016). https://doi.org/10.1007/s00029-015-0198-6
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DOI: https://doi.org/10.1007/s00029-015-0198-6