This page is a glossary of terms in invariant theory. For descriptions of particular invariant rings, see invariants of a binary form, symmetric polynomials. For geometric terms used in invariant theory see the glossary of classical algebraic geometry. Definitions of many terms used in invariant theory can be found in ( Sylvester 1853 ), ( Cayley 1860 ), ( Burnside & Panton 1881 ), ( Salmon 1885 ), ( Elliott 1895 ), ( Grace & Young 1903 ), ( Glenn 1915 ), ( Dolgachev 2012 ), and the index to the fourth volume of Sylvester's collected works includes many of the terms invented by him.
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped.
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 description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a tensor product of copies of it.
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.
In mathematics, a Bézout matrix is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester and Arthur Cayley and named after Étienne Bézout. Bézoutian may also refer to the determinant of this matrix, which is equal to the resultant of the two polynomials. Bézout matrices are sometimes used to test the stability of a given polynomial.
In algebra, the hyperdeterminant is a generalization of the determinant. Whereas a determinant is a scalar valued function defined on an n × n square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes.
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
In mathematics, Cayley's Ω process, introduced by Arthur Cayley, is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action.
The Classical Groups: Their Invariants and Representations is a mathematics book by Hermann Weyl, which describes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.
In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic. The study of modular invariants was originated in about 1914 by Dickson (2004).
In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y.
In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients that vanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced by Sylvester (1852); see Miller (2010). The word catalectic refers to an incomplete line of verse, lacking a syllable at the end or ending with an incomplete foot.
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions.
In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Arthur Cayley and discussed by Igor Dolgachev. In the same paper Cayley also introduced another similar invariant that he called the pippian, now called the Cayleyan.
In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or infinite degree. More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary form stabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space are called perpetuants. Perpetuants were introduced and named by Sylvester. MacMahon and Stroh classified the perpetuants. Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography.
In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.
In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical form for them.
