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In mathematics, the '''bracket ring''' is the [[subring]] of the ring of [[polynomial]]s ''k''[''x''<sub>11</sub>,...,''x''<sub>''dn''</sub>] generated by the ''d'' by ''d'' [[Minor (linear algebra)|minors]] of a generic ''d'' by ''n'' [[matrix (mathematics)|matrix]] (''x''<sub>''ij''</sub>).<br />
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The bracket ring may be regarded as the ring of polynomials on the image of a [[Grassmannian]] under the [[Plücker embedding]].<ref>{{citation | last1=Björner | first1=Anders | last2=Las Vergnas | author2-link=Michel Las Vergnas | first2=Michel | last3=Sturmfels | first3=Bernd | author3-link=Bernd Sturmfels | last4=White | first4=Neil | last5=Ziegler | first5=Günter | title=Oriented matroids | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=46 | publisher=[[Cambridge University Press]] | year=1999 | isbn=0-521-77750-X | zbl=0944.52006 | page=79 }}</ref><br />
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For given ''d'' ≤ ''n'' we define as formal variables the ''brackets'' [λ<sub>1</sub> λ<sub>2</sub> ... λ<sub>''d''</sub>] with the λ taken from {1,...,''n''}, subject to [λ<sub>1</sub> λ<sub>2</sub> ... λ<sub>''d''</sub>] = − [λ<sub>2</sub> λ<sub>1</sub> ... λ<sub>''d''</sub>] and similarly for other transpositions. The set Λ(''n'',''d'') of size <math>\binom{n}{d}</math> generates a polynomial ring ''K''[Λ(''n'',''d'')] over a field ''K''. There is a homomorphism Φ(''n'',''d'') from ''K''[Λ(''n'',''d'')] to the polynomial ring ''K''[''x''<sub>''i'',''j''</sub>] in ''nd'' indeterminates given by mapping <br />
[λ<sub>1</sub> λ<sub>2</sub> ... λ<sub>''d''</sub>] to the determinant of the ''d'' by ''d'' matrix consisting of the columns of the ''x''<sub>''i'',''j''</sub> indexed by the λ. The ''bracket ring'' ''B''(''n'',''d'') is the image of Φ. The kernel ''I''(''n'',''d'') of Φ encodes the relations or ''syzygies'' that exist between the minors of a generic ''n'' by ''d'' matrix. The projective variety defined by the ideal ''I'' is the (''n''−''d'')''d'' dimensional Grassmann variety whose points correspond to ''d''-dimensional subspaces of an ''n''-dimensional space.<ref>Sturmfels (2008) pp.78–79</ref><br />
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To compute with brackets it is necessary to determine when an expression lies in the ideal ''I''(''n'',''d''). This is achieved by a ''straightening law'' due to Young (1928).<ref>Sturmfels (2008) p.80</ref><br />
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==See also==<br />
* [[Bracket algebra]]<br />
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==References==<br />
{{reflist}}<br />
*{{Citation | last1=Dieudonné | first1=Jean A. | last2=Carrell | first2=James B. | title=Invariant theory, old and new | doi=10.1016/0001-8708(70)90015-0 | mr=0255525 | year=1970 | journal=Advances in Mathematics | issn=0001-8708 | volume=4 | pages=1–80 | zbl=0196.05802 }}<br />
*{{Citation | last1=Dieudonné | first1=Jean A. | last2=Carrell | first2=James B. | title=Invariant theory, old and new | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-215540-6 | doi=10.1016/0001-8708(70)90015-0 | mr=0279102 | year=1971 | zbl=0258.14011 }}<br />
*{{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | title=Algorithms in Invariant Theory | series=Texts and Monographs in Symbolic Computation | others= | edition=2nd | publisher=[[Springer-Verlag]] | year=2008 | isbn=3211774165 | zbl=1154.13003 }}<br />
*{{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | last2=White | first2=Neil | title=Stanley decompositions of the bracket ring | mr=1096453 | year=1990 | journal=Mathematica Scandinavica | issn=0025-5521 | volume=67 | issue=2 | pages=183–189 | url=http://www.math.ufl.edu/~white/stanley1.ps | zbl=0727.13005 }}<br />
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[[Category:Invariant theory]]<br />
[[Category:Algebraic geometry]]<br />
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{{algebra-stub}}</div>93.187.84.100