Glossary of invariant theory

Conventions

-an

Nouns ending in -an are often invariants named after people, as in Cayleyan, Hessian, Jacobian, Steinerian.

-ant

Nouns ending in -ant are often invariants, as in determinant, covariant, and so on.

-ary

Adjectives ending in -ary often refer to the number of variables of a form, as in unary, binary, ternary, quaternary, quinary, senary, septenary, octonary, nonary, denary.

-ic

Adjectives or nouns ending in -ic often refer to the degree of a form, as in linear or monic, quadric or quadratic, cubic, quartic or biquadratic, quintic, sextic, septic or septimic, octic or octavic, nonic, decic or decimic, undecic or undecimic, duodecic or duodecimic, and so on.

!$@

(a₀, a₁, ..., a_n)(x,y)ⁿ

Short for the form (n
0)a₀xⁿ + (n
1)a₁x^n–1y+ ... + (n
n)a_nyⁿ. When the first ) has a circumflex or arrow on top of it, this means that the binomial coefficients are omitted. The parentheses are sometimes overlapped: ( a 0 , … , a n ) ( x , y ) n

[]

See Sylvester (1853, Glossary p. 543–548)

(αβγ...)

The determinant of the matrix with entries α_i, β_i, γ_i,... For example, (αβ) means α₁β₂ – α₂β₁.

A

absolute

1.  The absolute invariant is essentially the j-invariant of an elliptic curve.2.  An absolute invariant is something fixed by a group action, in other words a (relative) invariant (something that transforms according to a character) where the character is trivial.

allotrious

See Sylvester (1853, Glossary p. 543–548), Archaic.

alternant

1.  An archaic term for the commutator AB–BA of two operators A and B. (Elliott 1895, p.144)2.  An alternant matrix is a matrix such that the entries of each column are given by some fixed function of a variable.

annihilator

An annihilator is a differential operator representing an element of a Lie algebra, so that invariants of a group are killed by the annihilators. (Elliott 1895, p.108)

anti-invariant

A relative invariant transforming according to a character of order 2 of a group such as the symmetric group.

anti-seminvariant

(Elliott 1895, p.126)

apocopated

See Sylvester (1853, Glossary p. 543–548). Archaic.

Arf invariant

An invariant of quadratic forms over a field of order 2.

Aronhold invariant

One of the two generators of degrees 4 and 6 of the ring of invariants of ternary cubic forms. (Dolgachev 2012, 3.1.1)

asyzygetic

Linearly independent.

B

Bezoutiant

A symmetric square matrix associated to two binary forms.

Bezoutic

See Sylvester (1853, Glossary p. 543–548). Archaic.

Bezoutiod

See Sylvester (1853, Glossary p. 543–548). Archaic.

bidegree

An ordered pair of integers, giving the degrees of a form relative to two sets of variables.

biform

A polynomial homogeneous in each of two sets of variables. In other words an element of S^mV×SⁿW, usually considered as a representation of GL_V×GL_W.

binary

Depending on 2 variables. Same as bivariate.

biquadratic

Same as quartic, meaning degree 4.

biternary

A biternary form is one in 6 variables, 3 transforming according to the fundamental representation of SL₃ and 3 transforming according to its dual.

bivariate

Depending on 2 variables. Same as binary.

Boolian invariant

An invariant for the orthogonal group. (Elliott 1895, p.344)

bordered Hessian

An alternative name for the reciprocant

bracket

An invariant given by either the pairing of a vector and a vector in the dual space, or the determinant of a matrix form by n vectors of an n-dimensional space (in other words their exterior product in the top exterior power).

Brioschi covariant

This is a degree 12 order 9 covariant of ternary cubic forms, introduced by Brioschi (1863). (Dolgachev 2012, 3.4.3)

C

canonical form

A particularly simple representation of a form, such as a sum of powers of linear forms, or with many zero coefficients. For example, the canonical form of a binary form of degree 2m+1 is a sum of m+1 powers of linear forms.

canonisant

canonizant

A covariant of a form, given by the catalecticant of the penultimate emanant. It is related to the canonical form of a form. For example, the canonizant of a binary form of degree 2n–1 has degree n and order n. (Elliott 1895, p.21)

catalecticant

An invariant vanishing on forms that are the sum of an unusually small number of powers of linear forms.

Cayley Ω process

A certain differential operator used for constructing invariants.

Cayleyan

A contravariant.

characteristic

See Sylvester (1853, Glossary p. 543–548)

class

The class of a contravariant or concomitant is its degree in the covariant variables. See also degree and order.

Clebsch invariant

(Dolgachev 2012, p.283)

co-Bezoutiant

See Sylvester (1853, Glossary p. 543–548). Archaic.

cogredient

Transforming according to the natural representation of a linear group. (Elliott 1895, p.55)

combinant

A joint relative invariant of several forms of the same degree, that is unchanged if a multiple of one of the forms is added to another. Essentially a relative invariant of a product of two general linear groups. (Elliott 1895, p.340) Sylvester (1853, Glossary p. 543–548) (Salmon 1885, p.161)

combinative

Related to invariants of a product of groups. For example a combinative covariant is a covariant of a product of two groups.

commutant

A generalization of the determinant to arrays of dimension greater than 2. (Cayley 1860)

complete

A complete system of invariants is a set of generators for the ring of invariants.

concomitant

A relative invariant of GL(V) acting on the polynomials over Sⁿ(V)⊕V⊕V*.

conjunctive

See Sylvester (1853, Glossary p. 543–548)

connex

A form in two sets of variables, one set corresponding to a vector space and the other to its dual, or in other words an element of the symmetric algebra of V⊕V* for a vector space V. Introduced by Clebsch.

continuant

A determinant of a tridiagonal matrix.(Salmon 1885, p.18)

contragredient

Transforming according to the dual of the natural representation of a linear group. (Elliott 1895, p.74)

contravariant

A relative invariant of GL(V) acting on the polynomials over Sⁿ(V)⊕V.

convolution

A method of constructing invariants from two other invariants. (Glenn 1915, p.87)

covariancy

(Elliott 1895, p.83)

covariant

1.  (Noun) A relative invariant of GL(V) acting on the polynomials over Sⁿ(V)⊕V*.2.  (Adjective) Invariant under the action of a group, especially for functions between two spaces acted on by the group.

cross ratio

The cross ratio is an invariant of 4 points of a projective line.

cubic

(Adjective) Degree 3(Noun) A form of degree 3

cubicovariant

A covariant of degree 3, in particular an order 3 degree 3 covariant of a binary cubic given by the Jacobian of the cubic and its Hessian.. (Elliott 1895, p.50)

cubinvariant

An invariant of degree 3.

cubo-

Used to form compound adjectives such as cubo-linear, cubo-quadric, and so on, indicating the bidegree of something. For example, cubo-linear means having degree 3 in the first of two sets of variables and degree 1 in the second.

cumulant

The numerator or denominator of a continued fraction, often expressed as a determinant. Sylvester (1853, Glossary p. 543–548).

D

decic

decimic

(Adjective) Degree 10(Noun) A form of degree 10

degree

1.  The degree of a form is the total power of the variables in it.2.  The degree of an invariant or covariant or contravariant means its degree in terms of the coefficients of the form. The degree of a form considered as a form is usually not its degree when considered as a covariant.3.  Some authors exchange the meanings of "degree" and "order" of a covariant or concomitant.

denary

Depending on 10 variables

determinant

The determinant is a joint invariant of n vectors of an n-dimensional space.

dialytic

Sylvester's dialytic method is a method for calculating resultants, essentially by expressing them as the determinant of a Sylvester matrix. See Sylvester (1853, Glossary p. 543–548). Archaic.

differentiant

Another name for an invariant of a binary form. Archaic.

discriminant

The discriminant of a form in n variables is the multivariate resultant of the n differentials with respect to each of the variables. For binary forms the discriminant vanishes if the form has multiple roots and is essentially the same as the discriminant of a polynomial of 1 variable. The discriminant of a form vanishes when the corresponding hypersurface has singularities (as a scheme).

disjunctive

See Sylvester (1853, Glossary p. 543–548)

divariant

An alternative name for a concomitant suggested by Salmon (1885, p.121)

duodecic

duodecimic

(Adjective) Degree 12(Noun) A form of degree 12

E

effective

See Sylvester (1853, Glossary p. 543–548)

effluent

See Sylvester (1853, Glossary p. 543–548). Archaic.

eliminant

De Morgan's name for the (multivariate) resultant, an invariant of n forms in n variables that vanishes if they have a common nonzero solution. (Elliott 1895, p.16)

emanant

The rth emanant of a binary form in variables x_i is a covariant given by the action of the rth power of the differential operator Σy_i∂/∂x_i. This is essentially the same as polarization. (Elliott 1895, p.56) Sylvester (1853, Glossary p. 543–548)

endoscopic

See Sylvester (1853, Glossary p. 543–548). Archaic.

equianharmonic contravariant

A weight 4 contravariant of binary quartics (Dolgachev 2012, 6.4)

evectant

A contravariant given by the action of an evector.

evector

A differential operator constructed from a binary form.

excess

The excess of a polynomial in the coefficients a₀,...a_p of a form of degree p is ip–2w, where p is the degree of the polynomial and w is its weight. (Elliott 1895, p.141)

exoscopic

See Sylvester (1853, Glossary p. 543–548). Archaic.

extensor

An element of the kth exterior power of a vector space that can be written as the exterior product of k vectors.

extent

The extent of a polynomial in a₀, a₁,... is the largest value of p such that the polynomial involves a_p. (Elliott 1895, p.138)

F

facient

One of the variables of a form (Cayley 1860)

facultative

A facultative point is one where a given function is positive. (Salmon 1885, p.243)

form

A homogeneous polynomial in several variables, also called a quantic.

functional determinant

An archaic name for Jacobians

fundamental

1.  The first fundamental theorem describes generators (called brackets) for the ring of invariant polynomials on a sum of copies of a vector space V and its dual (for the special linear group of V). The second fundamental theorem describes the syzygies between the generators.2.  For fundamental scale see Sylvester (1853, Glossary p. 543–548). Archaic.3.  A fundamental invariant is an element of a set of generators for a ring of invariants.4.  A fundamental system is a set of generators (for a ring of invariants, covariants, and so on).

G

Gordan

Named for Paul Gordan.1.  Gordan's theorem states that the ring of invariants of a binary form (or several binary forms) is finitely generated.

grade

The highest power of a bracket factor in the symbolic expression for an invariant. (Glenn 1915, 4.8)

gradient

A homogeneous polynomial in a₀, ..., a_p all of whose terms have the same weight, where a_n has weight n. (Elliott 1895, p.138) Archaic.

Gröbner basis

A basis for an ideal of a ring of polynomials chosen according to some rule to make computations easier.

ground form

An element of a minimal set of homogeneous generators for the invariants of a form. Archaic.

H

hectic

A joke term for a form of degree 100.

harmonic contravariant

A weight 6 contravariant of binary quartics (Dolgachev 2012, 6.4)

harmonizant

A bilinear invariant of two forms whose vanishing means they are polar. (Dolgachev 2012, p.75)

Hermite

Named after Charles Hermite1.  The Hermite contravariant is a degree 12 class 9 contravariant of ternary cubics. (Dolgachev 2012, 3.4.3)2.  Hermite's law of reciprocity 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.3.  The Hermite invariant is the degree 18 skew invariant of a binary quintic.

Hessian

A covariant of a form u, given by the determinant of the matrix with entries ∂²u/∂x_i∂x_j.

Hilbert

Named after David HilbertA Hilbert series is a formal power series whose coefficients are dimensions of spaces of invariants of various degrees.Hilbert's theorem states that the ring of invariants of a finite-dimensional representation of a reductive group is finitely generated.

homographic

1.  A homographic transformation is a transformation taking x to (ax+b)/(cx+d).2.  A homographic relation between x and y is a relation of the form axy + bx + cy + d=0 .

hyperdeterminant

An invariant of a multidimensional array of coefficients, generalizing the determinant of a 2-dimensional array.

I

identity covariant

A form considered as a covariant of degree 1.

immanant

A generalization of the determinant and permanent of a matrix

inertia

The signature of a real quadratic form. See Sylvester (1853, Glossary p. 543–548)

integral rational function

A polynomial.

intercalations

See Sylvester (1853, Glossary p. 543–548). Archaic.

intermediate invariant

An invariant of two forms constructed from two invariants of each of the forms. (Elliott 1895, p.23)

intermutant

A special form of permutant. (Cayley 1860)

invariant

1.  (Adjective) Fixed by the action of a group2.  (Noun) An absolute invariant, meaning something fixed by a group action.3.  (Noun) A relative invariant, meaning something transforming according to a character of a group. In classical invariant theory it often refers to relatively invariant polynomials in the coefficients of a quantic, considered as a representation of a general linear group.

involutant

See Sylvester's collected papers, volume IV, page 135

irreducible

Not expressible as a polynomial in things of smaller degree.

isobaric

All terms having the same weight. (Elliott 1895, p.32)

J

Jacobian

A covariant of n forms f_i in n variables x_j, given by the determinant of the matrix with entries ∂f_i/∂x_j.

joint invariant

A relative invariant for polynomials over reducible representation of a group, in particular a relative invariant for a several binary forms.

K

kenotheme

Sylvester (1853, Glossary p. 543–548) defines this as "A finite system of discrete points defined by one or more homogeneous equations in number one less than the number of variables contained therein." This may mean an intersection of n hypersurfaces in n-dimensional projective space. Archaic.

L

linear

Degree 1

lineo-

Used to form compound adjectives such as lineo-linear, lineo-quadric, and so on, indicating the bidegree of something. For example, lineo-linear means having degree 1 in each of two sets of variables. In particular the lineo-linear invariant of two binary forms has degree 1 in the coefficients of each form. (Elliott 1895, p.54)

Lüroth invariant

A degree 54 invariant vanishing on Lüroth quartics (nonsingular quartic plane curves containing the 10 vertices of a complete pentalateral). (Dolgachev 2012, p.295)

M

meicatalecticizant

Sylvester's original term for what he later renamed the catalecticant. Archaic.

mixed concomitant

A concomitant that involves both covariant and contravariant variables, in other words one that is not a covariant or contravariant. (Elliott 1895, p.77)

modular

Defined over a finite field.

modulus

An alternative name for the determinant of a linear transformation. (Elliott 1895, p.3)

monic

1.  Adjective. Having leading coefficient 1.2.  Adjective. Having degree 1.3.  Noun. A form of degree 1.

monotheme

See Sylvester (1853, Glossary p. 543–548). Archaic.

N

nonary

Depending on 9 variables

nonic

(Adjective) Degree 9(Noun) A form of degree 9

nullcone

The cone of nullforms

nullform

A form on which all invariants with zero constant term vanish.

O

octavic

octic

(Adjective) Degree 8(Noun) A form of degree 8

octonary

Depending on 8 variables

Omega process

order

1.  The degree of a covariant or concomitant in the variables of a form.2.  Some authors interchange the meaning of "degree" and "order" of a covaraint.3.  See Sylvester (1853, Glossary p. 543–548)

ordinary

An ordinary invariant means a relative invariant, in other words something transforming according to a character of a group, as opposed to an absolute invariant.

osculant

An invariant of several forms of the same degree generalizing the tact-invariant of two forms, equal to the discriminant if the number of forms is 1, and to the multivariate resultant if the number of forms is the number of variables. Salmon (1885, p.171)

P

partial transvectant

partition

An expression of a number as a sum of positive integers.(Elliott 1895, p.119)

peninvariant

Same as seminvariant. (Cayley 1860)

permanent

A variation of the determinant of a matrix

permutant

(Cayley 1860)

perpetuant

Roughly an irreducible covariant of a form of infinite order.

persymmetrical

A persymmetrical matrix is a Hankel matrix. See Sylvester (1853, Glossary p. 543–548). Archaic.

Pfaffian

A square root of the determinant of a skew-symmetric matrix.

pippian

An old name for the Cayleyan.

plagiogonal

Related to or fixed by the orthogonal group of some quadratic form. See Sylvester's collected papers, volume I, page 357

plexus

A set of generators of an ideal, especially if the number of generators needed is larger than the codimension of the corresponding variety.

polarization

A method of reducing the degree of something by introducing extra variables.

principiant

A reciprocant that is invariant under homographic substitutions, up to a constant facts. See Sylvester's collected papers, vol IV, page 382

projective invariant

1.  An invariant of the projective general linear group.2.  An invariant of a central extension of a group.

protomorph

A set of protomorphs is a set of seminvariants, such that any seminvariant is a polynomial in the protomorphs and the inverse of the first protomorph. (Elliott 1895, p.206)

Q

quadratic

quadric

(Adjective) Degree 2(Noun) A form of degree 2

quadricovariant

A covariant of degree 2. (Salmon 1885, p.261)

quadrinvariant

An invariant of degree 2. Sylvester (1853, Glossary p. 543–548).

quadro-

Degree 2. Used to form compound adjectives such as quadro-linear, quadro-quadric, and so on, indicating the bidegree of something. For example, quadro-linear means having degree 2 in the first of two sets of variables and degree 1 in the other.

quantic

An archaic name for a homogeneous polynomial in several variables, now usually called a form.

quartic

(Adjective) Degree 4(Noun) A form of degree 4

quarticovariant

A covariant of degree 4.

quartinvariant

An invariant of degree 4

quarto-

Used to form compound adjectives such as quarto-linear, quarto-quadric, and so on, indicating the bidegree of something. For example, quarto-linear means having degree 4 in the first of two sets of variables and degree 1 in the other.

quaternary

Depending on 4 variables

quinary

Depending on 5 variables.

quintic

(Adjective) Degree 5(Noun) A form of degree 5

quintinvariant

An invariant of degree 5.

quippian

R

rational integral function

A polynomial.

reciprocal

The reciprocal of a matrix is the adjugate matrix.

reciprocant

1.  A contravariant of a ternary form, giving the equation of a dual curve. (Elliott 1895, p.400)

reciprocity

Exchanging the degree of a form with the degree of an invariant. For example, Hermite's law of reciprocity states that the degree p invariants of a form of degree n correspond to the degree n invariants of a form of degree p. (Elliott 1895, p.137)

reducible

Expressible as a polynomial in things of smaller degree.

relative invariant

Something transforming according to a 1-dimensional character of a group, often a power of the determinant. Same as ordinary invariant.

resultant

1.  A joint invariant of two binary forms that vanishes when they have a common root. More generally a (multivariate) resultant is a joint invariant of n forms in n variables that vanishes if they have a common nontrivial zero. Sometimes called an eliminant in older books.2.  An archaic term for the determinant

revenant

Suggested by Sylvester (collected works vol 3, page 593) as an alternative name for a perpetuant.

Reynolds operator

Projection onto the fixed vectors

rhizoristic

See Sylvester (1853, Glossary p. 543–548). Archaic.

S

Salmon invariant

A degree 60 invariant vanishing on ternary quartics with an inflection bitangent. (Dolgachev 2012, 6.4)

Scorza covariant

A covariant of ternary quartics. (Dolgachev 2012, 6.3.4)

semicovariant

An analogue of seminvariants for covariants. See (Burnside & Panton 1881, p.329)

semi-invariant

seminvariant

1.  The leading term of a covariant, also called its source. (Grace & Young 1903, section 33)2.  An invariant of the group of upper triangular matrices.

senary

Depending on 6 variables. (Rare)

septenary

Depending on 7 variables

septic

septimic

(Adjective) Degree 7(Noun) A form of degree 7

sextic

(Adjective) Degree 6(Noun) A form of degree 6

sexticovariant

A covariant of degree 6

sextinvariant

An invariant of degree 6 (Salmon 1885, p.262)

signaletic

See Sylvester (1853, Glossary p. 543–548). Archaic.

singular

1.  See Sylvester (1853, Glossary p. 543–548)

skew

A skew invariant is a relative invariant of a group G that changes sign under an element of order 2 in its abelianization. In particular for the general linear group it changes sign under elements of determinant –1, and for the symmetric group it changes sign under odd permutations. For binary forms skew invariants are the invariants of odd weight. They do not exist for binary quadrics, cubics, or quartics, but do for binary quintics. (Elliott 1895, p.112)

source

The source of a covariant is its leading term, when the covariant is considered as a form. Also called a seminvariant. (Elliott 1895, p.126)

Steinerian

symbolic

The symbolic method is a way of representing invariants, that repeatedly uses the identification of the symmetric power of a vector space with the symmetric elements of a tensor power.

syrrhizoristic

Sylvester (1853, Glossary p. 543–548) defined this as "A syrrhizoristic series is a series of disconnected functions which serve to determine the effective intercalations of the real roots of two functions lying between any assigned limits." Archaic. This term does not seem to have been used (or understood) by anyone other than Sylvester.

syzygant

(Elliott 1895, p.198)

syzygetic

See Sylvester (1853, Glossary p. 543–548)

syzygy

A linear or algebraic relation, especially one between generators of a ring or module.

T

tacinvariant

tact invariant

An invariant of one or two ternary forms that vanishes if the corresponding curve touches itself, or if the two curves touch each other. It is generalized by the osculant.

tamisage

Sylvester's name for his method of guessing the degrees of a generating set of invariants or covariants by examining the generating function.(Elliott 1895, p.175). Archaic.

tantipartite

An archaic term for multilinear. (Cayley 1860)

Tschirnhaus transformation

ternary

Depending on 3 variables

Toeplitz invariant

An invariant of nets of quadrics in 3-dimensional projective space that vanishes on nets with a common polar pentahedron. (Dolgachev 2012, p.51)

transfer

A method of constructing contravariants of forms in n+1 variables from invariants of forms in n variables. (Dolgachev 2012, 3.4.2)

transvectant

An invariant formed from n invariants in n variables using Cayley's omega process. (Elliott 1895, p.71)

trinomial

A polynomial with at most three non-zero coefficients.

U

ueberschiebung

Transvectant. (Elliott 1895, p.171)

umbrae

umbral

See Sylvester (1853, Glossary p. 543–548)

unary

Depending on 1 variable. Same as univariate.

undecic

undecimic

(Adjective) Degree 11(Noun) A form of degree 11

unimodular

Having determinant 1

unitarian trick

Finite-dimensional representations of a semisimple Lie group are equivalent to finite-dimensional representations of a compact form, and are therefore completely reducible.

univariate

Depending on 1 variable. Same as unary.

universal concomitant

The pairing between a vector space and its dual, considered as a concomitant. (Elliott 1895, p.77)

W

weight

1.  The power of the determinant appearing in the formula for transformation of a relative invariant.2.  A character of a torus3.  See Sylvester (1853, Glossary p. 543–548)4.  The weight of a_i is i, and the weight of a product of monomials is the sum of their weights.

XYZ

zeta

ζ

A product of squared differences. See Sylvester (1853, Glossary p. 543–548)