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| In [[mathematics]], the '''Dedekind [[zeta function]]''' of an [[algebraic number field]] ''K'', generally denoted ζ<sub>''K''</sub>(''s''), is a generalization of the [[Riemann zeta function]]—which is obtained by specializing to the case where ''K'' is the [[rational number]]s '''Q'''. In particular, it can be defined as a [[Dirichlet series]], it has an [[Euler product]] expansion, it satisfies a [[functional equation (L-function)|functional equation]], it has an [[analytic continuation]] to a [[meromorphic function]] on the [[complex plane]] '''C''' with only a [[simple pole]] at ''s'' = 1, and its values encode arithmetic data of ''K''. The [[extended Riemann hypothesis]] states that if ''ζ''<sub>''K''</sub>(''s'') = 0 and 0 < Re(''s'') < 1, then Re(''s'') = 1/2. | | In mathematics, the '''Bruhat decomposition''' (named after [[François Bruhat]]) G = BWB into cells can be regarded as a general expression of the principle of [[Gauss–Jordan elimination]], which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the [[Schubert cell]] decomposition of Grassmannians: see [[Weyl group]] for this. |
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| The Dedekind zeta function is named for [[Richard Dedekind]] who introduced them in his supplement to [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]].<ref>{{harvnb|Narkiewicz|2004|loc=§7.4.1}}</ref>
| | More generally, any group with a [[(B,N) pair]] has a Bruhat decomposition. |
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| ==Definition and basic properties== | | ==Definitions== |
| Let ''K'' be an [[algebraic number field]]. Its Dedekind zeta function is first defined for complex numbers ''s'' with [[real part]] Re(''s'') > 1 by the Dirichlet series
| | *''G'' is a [[connected space|connected]], [[reductive group|reductive]] [[algebraic group]] over an [[algebraically closed field]]. |
| | *''B'' is a [[Borel subgroup]] of ''G'' |
| | *''W'' is a [[Weyl group]] of ''G'' corresponding to a maximal torus of ''B''. |
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| :<math>\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}</math> | | The '''Bruhat decomposition''' of ''G'' is the decomposition |
| | :<math>G=BWB =\coprod_{w\in W}BwB</math> |
| | of ''G'' as a disjoint union of [[double coset]]s of ''B'' parameterized by the elements of the Weyl group ''W''. (Note that although ''W'' is not in general a subgroup of ''G'', the coset ''wB'' is still well defined.) |
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| where ''I'' ranges through the non-zero [[ideal (ring theory)|ideals]] of the [[ring of integers]] ''O''<sub>''K''</sub> of ''K'' and ''N''<sub>''K''/'''Q'''</sub>(''I'') denotes the [[absolute norm]] of ''I'' (which is equal to both the [[Index of a subgroup|index]] [''O''<sub>''K''</sub> : ''I''] of ''I'' in ''O''<sub>''K''</sub> or equivalently the [[cardinality]] of [[quotient ring]] ''O''<sub>''K''</sub> / ''I''). This sum converges absolutely for all complex numbers ''s'' with [[real part]] Re(''s'') > 1. In the case ''K'' = '''Q''', this definition reduces to that of the Riemann zeta function.
| | == Examples == |
| | Let ''G'' be the [[general linear group]] '''GL'''<sub>n</sub> of invertible <math>n \times n</math> matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group ''W'' is isomorphic to the [[symmetric group]] ''S<sub>n</sub>'' on ''n'' letters, with [[permutation matrices]] as representatives. In this case, we can take ''B'' to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix ''A'' as a product ''U<sub>1</sub>PU<sub>2</sub>'' where ''U<sub>1</sub>'' and ''U<sub>2</sub>'' are upper triangular, and ''P'' is a permutation matrix. Writing this as ''P = U<sub>1</sub><sup>-1</sup>AU<sub>2</sub><sup>-1</sup>'', this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row ''i'' (resp. column ''i'') to row ''j'' (resp. column ''j'') if ''i>j'' (resp. ''i<j''). The row operations correspond to ''U<sub>1</sub><sup>-1</sup>'', and the column operations correspond to ''U<sub>2</sub><sup>-1</sup>''. |
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| ===Euler product===
| | The [[special linear group]] '''SL'''<sub>n</sub> of invertible <math>n \times n</math> matrices with [[determinant]] 1 is a [[semisimple algebraic group|semisimple group]], and hence reductive. In this case, ''W'' is still isomorphic to the symmetric group ''S<sub>n</sub>''. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in '''SL'''<sub>n</sub>, we can take one of the nonzero elements to be -1 instead of 1. Here ''B'' is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of '''GL'''<sub>n</sub>. |
| The Dedekind zeta function of ''K'' has an Euler product which is a product over all the [[prime ideal]]s ''P'' of ''O''<sub>''K''</sub> | |
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| :<math>\zeta_K (s) = \prod_{P \subseteq \mathcal{O}_K} \frac{1}{1 - (N_{K/\mathbf{Q}}(P))^{-s}},\text{ for Re}(s)>1.</math>
| | == Geometry == |
| | The cells in the Bruhat decomposition correspond to the [[Schubert cell]] decomposition of Grassmannians. The dimension of the cells corresponds to the [[length function|length]] of the word ''w'' in the Weyl group. [[Poincaré duality]] constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the [[fundamental class]]), and corresponds to the [[longest element of a Coxeter group]]. |
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| This is the expression in analytic terms of the [[Dedekind domain|uniqueness of prime factorization of the ideals]] ''I'' in ''O''<sub>''K''</sub>. The fact that, for Re(''s'') > 1, ζ<sub>''K''</sub>(''s'') is given by a product of non-zero numbers implies that it is non-zero in this region.
| | ==Computations== |
| | The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the ''q''-polynomial<ref>[http://math.ucr.edu/home/baez/week186.html This Week's Finds in Mathematical Physics, Week 186]</ref> of the associated [[Dynkin diagram]]. |
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| ===Analytic continuation and functional equation=== | | ==See also== |
| [[Erich Hecke]] first proved that ''ζ''<sub>''K''</sub>(''s'') has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at ''s'' = 1. The [[Residue (complex analysis)|residue]] at that pole is given by the [[analytic class number formula]] and is made up of important arithmetic data involving invariants of the [[unit group]] and [[class group]] of ''K''. | | * [[Lie group decompositions]] |
| | | * [[Birkhoff factorization]], a special case of the Bruhat decomposition for affine groups. |
| The Dedekind zeta function satisfies a functional equation relating its values at ''s'' and 1 − ''s''. Specifically, let Δ<sub>''K''</sub> denote [[Discriminant of an algebraic number field|discriminant]] of ''K'', let ''r''<sub>1</sub> (resp. ''r''<sub>2</sub>) denote the number of [[real place]]s (resp. [[complex place]]s) of ''K'', and let
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| :<math>\Gamma_\mathbf{R}(s)=\pi^{-s/2}\Gamma(s/2)</math>
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| :<math>\Gamma_\mathbf{C}(s)=2(2\pi)^{-s}\Gamma(s)</math>
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| where Γ(''s'') is the [[Gamma function]]. Then, the function
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| :<math>\Lambda_K(s)=\left|\Delta_K\right|^{s/2}\Gamma_\mathbf{R}(s)^{r_1}\Gamma_\mathbf{C}(s)^{r_2}\zeta_K(s)</math>
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| satisfies the functional equation
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| :<math>\Lambda_K(s)=\Lambda_K(1-s).\;</math>
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| ==Special values==
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| Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field ''K''. For example, the analytic class number formula relates the residue at ''s'' = 1 to the [[class number (number theory)|class number]] ''h''(''K'') of ''K'', the [[regulator of an algebraic number field|regulator]] ''R''(''K'') of ''K'', the number ''w''(''K'') of roots of unity in ''K'', the absolute discriminant of ''K'', and the number of real and complex places of ''K''. Another example is at ''s'' = 0 where it has a zero whose order ''r'' is equal to the [[rank of an abelian group|rank]] of the unit group of ''O''<sub>''K''</sub> and the leading term is given by
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| :<math>\lim_{s\rightarrow0}s^{-r}\zeta_K(s)=-\frac{h(K)R(K)}{w(K)}.</math>
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| Combining the functional equation and the fact that Γ(''s'') is infinite at all integers less than or equal to zero yields that ''ζ''<sub>''K''</sub>(''s'') vanishes at all negative even integers. It even vanishes at all negative odd integers unless ''K'' is [[totally real number field|totally real]] (i.e. ''r''<sub>2</sub> = 0; e.g. '''Q''' or a [[real quadratic field]]). In the totally real case, [[Carl Ludwig Siegel]] showed that ''ζ''<sub>''K''</sub>(''s'') is a non-zero rational number at negative odd integers. [[Stephen Lichtenbaum]] conjectured specific values for these rational numbers in terms of the [[algebraic K-theory]] of ''K''.
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| ==Relations to other ''L''-functions==
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| For the case in which ''K'' is an [[abelian extension]] of '''Q''', its Dedekind zeta function can be written as a product of [[Dirichlet L-function]]s. For example, when ''K'' is a [[quadratic field]] this shows that the ratio
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| :<math>\frac{\zeta_K(s)}{\zeta_{\mathbf{Q}}(s)}</math>
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| is the ''L''-function ''L''(''s'', χ), where χ is a [[Jacobi symbol]] used as [[Dirichlet character]]. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet ''L''-function is an analytic formulation of the [[quadratic reciprocity]] law of Gauss.
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| In general, if ''K'' is a [[Galois extension]] of '''Q''' with [[Galois group]] ''G'', its Dedekind zeta function is the [[Artin L-function|Artin ''L''-function]] of the [[regular representation]] of ''G'' and hence has a factorization in terms of Artin ''L''-functions of [[irreducible representation|irreducible]] [[Artin representation]]s of ''G''.
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| The relation with Artin L-functions shows that if ''L''/''K'' is a Galois extension then <math>\frac{\zeta_L(s)}{\zeta_K(s)}</math> is holomorphic (<math>\zeta_K(s)</math> "divides" <math>\zeta_L(s)</math>): for general extensions the result would follow from the [[Artin conjecture (L-functions)|Artin conjecture for L-functions]].<ref name=Mar19>Martinet (1977) p.19</ref>
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| Additionally, ''ζ''<sub>''K''</sub>(''s'') is the [[Hasse–Weil zeta function]] of [[Spectrum of a ring|Spec]] ''O''<sub>''K''</sub><ref>{{harvnb|Deninger|1994|loc=§1}}</ref> and the [[motivic L-function|motivic ''L''-function]] of the [[motive (algebraic geometry)|motive]] coming from the [[cohomology]] of Spec ''K''.<ref>{{harvnb|Flach|2004|loc=§1.1}}</ref>
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| ==Arithmetically equivalent fields==
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| Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. {{harvs|txt | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | year=2002 | volume=2369 }} used [[Gassmann triple]]s to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.
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| ==Notes== | | ==Notes== |
| {{reflist}}
| | <references/> |
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| ==References== | | ==References== |
| | *[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2. |
| | *[[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics)'', ISBN 3-540-42650-7 |
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| *{{Citation | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | editor1-last=Kohel | editor1-first=David R. | editor2-last=Fieker | editor2-first=Claus | title=Algorithmic number theory (Sydney, 2002) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Comput. Sci. | isbn=978-3-540-43863-2 | doi=10.1007/3-540-45455-1_6 | mr=2041074 | year=2002 | volume=2369 | chapter=On arithmetically equivalent number fields of small degree | pages=67–79}}
| | [[Category:Lie groups]] |
| *Section 10.5.1 of {{Citation
| | [[Category:algebraic groups]] |
| | last=Cohen
| |
| | first=Henri
| |
| | author-link=Henri Cohen (number theorist)
| |
| | title=Number theory, Volume II: Analytic and modern tools
| |
| | publisher=Springer
| |
| | location=New York
| |
| | series=[[Graduate Texts in Mathematics]]
| |
| | volume=240
| |
| | year=2007
| |
| | isbn=978-0-387-49893-5
| |
| | mr=2312338
| |
| | doi=10.1007/978-0-387-49894-2
| |
| }}
| |
| *{{Citation
| |
| | last=Deninger
| |
| | first=Christopher
| |
| | contribution=''L''-functions of mixed motives
| |
| | title=Motives, Part 1
| |
| | series=Proceedings of Symposia in Pure Mathematics
| |
| | publisher=[[American Mathematical Society]]
| |
| | volume=55.1
| |
| | year=1994
| |
| | pages=517–525
| |
| | editor-last=Jannsen
| |
| | editor-first=Uwe
| |
| | editor2-last=Kleiman
| |
| | editor2-first=Steven
| |
| | editor3-last=Serre
| |
| | editor3-first=Jean-Pierre
| |
| | editor3-link=Jean-Pierre Serre
| |
| | isbn=978-0-8218-1635-6
| |
| | url=http://wwwmath.uni-muenster.de/u/deninger/about/publikat/cd22.ps
| |
| }}
| |
| *{{Citation
| |
| | last=Flach
| |
| | first=Mathias
| |
| | contribution=The equivariant Tamagawa number conjecture: a survey
| |
| | url=http://www.math.caltech.edu/papers/baltimore-final.pdf
| |
| | title=Stark's conjectures: recent work and new directions
| |
| | publisher=[[American Mathematical Society]]
| |
| | series=Contemporary Mathematics
| |
| | volume=358
| |
| | pages=79–125
| |
| | isbn=978-0-8218-3480-0
| |
| | editor-last=Burns
| |
| | editor-first=David
| |
| | editor2-last=Popescu
| |
| | editor2-first=Christian
| |
| | editor3-last=Sands
| |
| | editor3-first=Jonathan
| |
| | editor4-last=Solomon
| |
| | editor4-first=David
| |
| }}
| |
| *{{citation | last=Martinet | first=J. | chapter=Character theory and Artin L-functions | pages=1-87 | title=Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975 | editor1-last=Fröhlich | editor1-first=A. | editor1-link=Albrecht Fröhlich | publisher=Academic Press | year=1977 | isbn=0-12-268960-7 | zbl=0359.12015 }}
| |
| *{{Citation
| |
| | last=Narkiewicz
| |
| | first=Władysław
| |
| | title=Elementary and analytic theory of algebraic numbers
| |
| | edition=3 | at=Chapter 7
| |
| | year=2004
| |
| | publisher=Springer-Verlag
| |
| | location=Berlin
| |
| | series=Springer Monographs in Mathematics
| |
| | isbn=978-3-540-21902-6
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| | mr=2078267
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| }}
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| [[Category:Zeta and L-functions]] | | [[ja:ブリュア分解]] |
| [[Category:Algebraic number theory]]
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In mathematics, the Bruhat decomposition (named after François Bruhat) G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of Grassmannians: see Weyl group for this.
More generally, any group with a (B,N) pair has a Bruhat decomposition.
Definitions
The Bruhat decomposition of G is the decomposition
of G as a disjoint union of double cosets of B parameterized by the elements of the Weyl group W. (Note that although W is not in general a subgroup of G, the coset wB is still well defined.)
Examples
Let G be the general linear group GLn of invertible matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group W is isomorphic to the symmetric group Sn on n letters, with permutation matrices as representatives. In this case, we can take B to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix A as a product U1PU2 where U1 and U2 are upper triangular, and P is a permutation matrix. Writing this as P = U1-1AU2-1, this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row i (resp. column i) to row j (resp. column j) if i>j (resp. i<j). The row operations correspond to U1-1, and the column operations correspond to U2-1.
The special linear group SLn of invertible matrices with determinant 1 is a semisimple group, and hence reductive. In this case, W is still isomorphic to the symmetric group Sn. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SLn, we can take one of the nonzero elements to be -1 instead of 1. Here B is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of GLn.
Geometry
The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of Grassmannians. The dimension of the cells corresponds to the length of the word w in the Weyl group. Poincaré duality constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the fundamental class), and corresponds to the longest element of a Coxeter group.
Computations
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the q-polynomial[1] of the associated Dynkin diagram.
See also
Notes
References
- Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
- Bourbaki, Nicolas, Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics), ISBN 3-540-42650-7
ja:ブリュア分解