mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...

, the bracket ring is the

subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

of the

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s ''k'' 'x''₁₁,...,''x''_''dn''generated by the ''d''-by-''d'' minors of a generic ''d''-by-''n''

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

(''x''_''ij''). The bracket ring may be regarded as the ring of polynomials on the

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

of a

Grassmannian In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...

under the

Plücker embedding In mathematics, the Plücker map embeds the Grassmannian \mathrm(k,V), whose elements are ''k''-Dimension (vector space), dimensional Linear subspace, subspaces of an ''n''-dimensional vector space ''V'', either real or complex, in a projective sp ...

. For given ''d'' ≤ ''n'' we define as formal variables the ''brackets'' �₁ λ₂ ... λ_''d''with the λ taken from , subject to �₁ λ₂ ... λ_''d''= − �₂ λ₁ ... λ_''d''and similarly for other transpositions. The

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

Λ(''n'',''d'') of size

\binom

generates a polynomial ring ''K'' �(''n'',''d'')over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K''. There is a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

Φ(''n'',''d'') from ''K'' �(''n'',''d'')to the polynomial ring ''K'' 'x''_''i'',''j''in ''nd'' indeterminates given by mapping �₁ λ₂ ... λ_''d''to the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

of the ''d'' by ''d'' matrix consisting of the columns of the ''x''_''i'',''j'' indexed by the λ. The ''bracket ring'' ''B''(''n'',''d'') is the image of Φ. The

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

''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).Sturmfels (2008) p.80

References

* * * * Invariant theory Algebraic geometry {{algebraic-geometry-stub

See also

References