Algebraic groups over the field with one element

Lorscheid, Oliver

doi:10.1007/s00209-011-0855-1

Cited by 16 publications

(21 citation statements)

References 15 publications

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“…More generally one may consider finite incidence geometries over finite fields F q (compare [5]) where there again may be interesting degenerate geometric objects associated to "the field with one element." There has been much recent work on developing notions of generalized algebraic geometry "over F 1 ", of which we may mention [6], [7], [8], [9], [14], [16], [17], [18], [28], [32], [34]. For connections with motives, see [22,Appendix].…”

Section: 2mentioning

confidence: 99%

“…Only a restricted class of algebraic varieties yield statistics having the rational function interpolation property. The rational function interpolation property is known to hold for counting points over F q on nonsingular toric varieties defined over Q, compare [17], [18]. One can more generally evaluate statistics associated with vector bundles and cohomology of local systems for such varieties, viewed over finite fields F q , and admit them as supplying data "over F 1 " when they have the rational function interpolation property.…”

Section: 2mentioning

confidence: 99%

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A family of measures on symmetric groups and the field with one element

Lagarias

2016

Journal of Number Theory

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Abstract. For each n ≥ 1 this paper considers a one-parameter family of complex-valued measures on the symmetric group Sn, depending on a complex parameter z. For parameter values z = q = p f this measure describes splitting probabilities of monic degree n polynomials over Fq [X], conditioned on being square-free. It studies these measures in the case z = 1, and shows that they have an interesting internal structure having a representation theoretic interpretation. These measures may encode data relevant to the hypothetical "field with one element F1". It additionally studies the case z = −1, which also has a representation theoretic interpretation.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

A family of measures on symmetric groups and the field with one element

Lagarias

2016

Journal of Number Theory

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“…[2], [6], [7] and [9]). By now, there are different versions in existence, so C. Soulé's paper [8] and a number of others [3][4][5]10], all of them interrelated with each other.…”

mentioning

confidence: 99%

Vector bundles over projective spaces. The case $${\mathbb F_1}$$

2011

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Abstract. Over the field of one element, vector bundles over n-dimensional projective spaces are considered. It is shown that all line bundles are tensor powers of the Hopf bundle and all vector bundles are direct sums of line bundles. This is in complete analogy to the case of the projective line over an arbitrary classical field, but drastically simpler in comparison with projective spaces of higher dimensions.Mathematics Subject Classification (2010). Primary 14D20; 14J60, 14T99. Keywords. Fields with one element, Vector bundles, Projective spaces.In recent years there has been quite a bit of activity concerning the field of one element, F 1 (see e.g. [2], [6], [7] and [9]). By now, there are different versions in existence, so C. Soulé's paper [8] and a number of others [3][4][5]10], all of them interrelated with each other. We follow here the approach of Anton Deitmar in [4], which is closest to standard algebraic geometry.The purpose of this paper is to study vector bundles over the projective line and, more generally, on n-dimensional projective space. As the classification of vector bundles on P 1 is rather immediately related with the Euclidean algorithm, it was our hope that something combinatorially interesting might show up in this context, as the underlying group here is the symmetric group.However it seems that the situation at least for this question is too simple. Over P 1 our result is identical with Grothendiecks result that any vector bundle on P 1 is a direct sum of line bundles. The latter are classified up to isomorphisms by their degree. The same holds true for vector bundles on P n over F 1 , so here the situation is drastically simpler than in the classical case.Formally, this project came up as the Diplom-thesis of the second author in (2008), written under guidance of the last and with crucial observations of the first author.

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“…Simple examples include projective space, for which #P (n−1) (F 1 ) = n, and Grassmanians, for which #Gr(k, n)(F 1 ) = n k , see [36]. Several different forms of geometry over F 1 were developed in recent years (see [35] for a general overview).…”

Section: #V (γ)mentioning

confidence: 99%

Quantum field theory overF1

Bejleri

Marcolli

2013

Journal of Geometry and Physics

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Abstract. In this paper we discuss some questions about geometry over the field with one element, motivated by the properties of algebraic varieties that arise in perturbative quantum field theory. We follow the approach to F 1 -geometry based on torified schemes. We first discuss some simple necessary conditions in terms of Euler characteristic and classes in the Grothendieck ring, then we give a blowup formula for torified varieties and we show that the wonderful compactifications of the graph configuration spaces, that arise in the computation of Feynman integrals in position space, admit an F 1 -structure. By a similar argument we show that the moduli spaces of curvesM 0,n admit an F 1 -structure. We also discuss conditions on hyperplane arrangements, a possible notion of embedded F 1 -structure and its relation to Chern classes, and questions on Chern classes of varieties with regular torifications.

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Algebraic groups over the field with one element

Cited by 16 publications

References 15 publications

A family of measures on symmetric groups and the field with one element

A family of measures on symmetric groups and the field with one element

Vector bundles over projective spaces. The case $${\mathbb F_1}$$

Quantum field theory overF1

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