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Hermite reciprocity

In mathematics, Hermite's law of reciprocity, introduced by Hermite, states that the degree m covariants of a binary form of degree n correspond to the degree n covariants of a binary form of degree m

Quippian

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 i

The Classical Groups

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 responsibl

Geometric invariant theory

In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in

Ternary quartic

In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.

Transvectant

In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process.

Hall algebra

In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by but forgotten until it was rediscover

Reynolds operator

In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid

Gram's theorem

In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group can be defined by absolute invariants. (Dieudonné & Carrell , p. 31).

Capelli's identity

In mathematics, Capelli's identity, named after Alfredo Capelli, is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation th

Symbolic method

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 invar

Perpetuant

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

Invariant estimator

In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea

Covariant (invariant theory)

In invariant theory, a branch of algebra, given a group G, a covariant is a G-equivariant polynomial map between linear representations V, W of G. It is a generalization of a classical convariant, whi

Hilbert–Mumford criterion

In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert and David Mumford, characterizes the semistable and stable points of a group action on a vector space in terms of eigenvalues

Littlewood–Richardson rule

In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur fu

Newton's identities

In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials

Chevalley–Shephard–Todd theorem

In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if a

Radical polynomial

In mathematics, in the realm of abstract algebra, a radial polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, i

Riemann invariant

Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the

Catalecticant

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 f

Haboush's theorem

In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-ve

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 the

Molien's formula

In mathematics, Molien's formula computes the generating function attached to a linear representation of a group G on a finite-dimensional vector space, that counts the homogeneous polynomials of a gi

Evectant

In mathematical invariant theory, an evectant is a contravariant constructed from an invariant by acting on it with a differential operator called an evector. Evectants and evectors were introduced by

Glossary of invariant theory

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 theo

Kostant polynomial

In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a

Zariski's finiteness theorem

In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case. Precisely, it states: Given a normal domain A, f

Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indetermin

Canonizant

In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical form for them.

Chevalley–Iwahori–Nagata theorem

In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the r

Invariants of tensors

In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the characteristic polynomial , where is the

Ring of symmetric functions

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring

Modular invariant theory

In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular inv

Hilbert's syzygy theorem

In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important

Hilbert's basis theorem

In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.

Bracket ring

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 rega

Hilbert's fourteenth problem

In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that k

Hodge bundle

In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has a

Quantum invariant

In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.

Ternary cubic

In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.

Vermeil's theorem

In differential geometry, Vermeil's theorem essentially states that the scalar curvature is the only (non-trivial) absolute invariant among those of prescribed type suitable for Albert Einstein’s theo

Quaternary cubic

In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space.

Nullform

In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by Hilbert. (Dieudonné & Carrell , , p.57).

Osculant

In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurface that vanishes if the hypersurface touches itself, or an invariant of several hypersur

Invariant of a binary form

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 acti

Standard monomial theory

In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis o

Differential invariant

In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundame

Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or i

Semi-invariant of a quiver

In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space Vi to each vertex and a linear map V(α): V(s(α)) → V(t(α)) to each arrow α,

Hesse's principle of transfer

In geometry, Hesse's principle of transfer (German: Übertragungsprinzip) states that if the points of the projective line P1 are depicted by a rational normal curve in Pn, then the group of the projec

Cayley's Ω process

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. As

Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal i

Kempf–Ness theorem

In algebraic geometry, the Kempf–Ness theorem, introduced by George Kempf and Linda Ness, gives a criterion for the stability of a vector in a representation of a complex reductive group. If the compl

Bracket algebra

In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that L is a proper signed al

Invariant polynomial

In mathematics, an invariant polynomial is a polynomial that is invariant under a group acting on a vector space . Therefore, is a -invariant polynomial if for all and . Cases of particular importance

Trace identity

In mathematics, a trace identity is any equation involving the trace of a matrix.

Schur polynomial

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the comple

Young–Deruyts development

In mathematics, the Young–Deruyts development is a method of writing invariants of an action of a group on an n-dimensional vector space Vin terms of invariants depending on at most n–1 vectors (Dieud

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