Bracket ring (original) (raw)
In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding. 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).
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| dbo:abstract | In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding. For given d ≤ n we define as formal variables the brackets [λ1 λ2 ... λd] with the λ taken from {1,...,n}, subject to [λ1 λ2 ... λd] = − [λ2 λ1 ... λd] and similarly for other transpositions. The set Λ(n,d) of size 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[xi,j] in nd indeterminates given by mapping [λ1 λ2 ... λd] to the determinant of the d by d matrix consisting of the columns of the xi,j 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. 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). (en) |
| dbo:wikiPageExternalLink | http://www.math.ufl.edu/~white/stanley1.ps https://web.archive.org/web/19971115024847/http:/www.math.ufl.edu/~white/stanley1.ps |
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| dbo:wikiPageLength | 4067 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1079027433 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Minor_(linear_algebra) dbr:Determinant dbr:Mathematics dbr:Matrix_(mathematics) dbr:Subring dbr:Grassmannian dbr:Bracket_algebra dbc:Invariant_theory dbr:Ideal_(ring_theory) dbr:Kernel_(algebra) dbr:Linear_subspace dbr:Field_(mathematics) dbr:Ring_(mathematics) dbr:Invariant_theory dbr:Academic_Press dbc:Algebraic_geometry dbr:Advances_in_Mathematics dbr:Transposition_(mathematics) dbr:Plücker_embedding dbr:Polynomial dbr:Set_(mathematics) dbr:Image_(mathematics) dbr:Ring_homomorphism dbr:Springer-Verlag dbr:Generic_matrix |
| dbp:wikiPageUsesTemplate | dbt:Algebra-stub dbt:Citation dbt:Reflist |
| dcterms:subject | dbc:Invariant_theory dbc:Algebraic_geometry |
| rdfs:comment | In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding. 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). (en) |
| rdfs:label | Bracket ring (en) |
| owl:sameAs | freebase:Bracket ring wikidata:Bracket ring https://global.dbpedia.org/id/4bAAT |
| prov:wasDerivedFrom | wikipedia-en:Bracket_ring?oldid=1079027433&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Bracket_ring |
| is dbo:wikiPageRedirects of | dbr:Covariant_bracket dbr:Contravariant_bracket |
| is dbo:wikiPageWikiLink of | dbr:Bracket_algebra dbr:Plücker_embedding dbr:Covariant_bracket dbr:Contravariant_bracket |
| is foaf:primaryTopic of | wikipedia-en:Bracket_ring |