algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the first and second fundamental theorems of invariant theory concern the generators and relations of the

ring of invariants In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the f ...

in the

ring of polynomial functions In mathematics, the ring of polynomial functions on a vector space ''V'' over a field ''k'' gives a coordinate-free analog of a polynomial ring. It is denoted by ''k'' 'V'' If ''V'' is finite dimensional and is viewed as an algebraic variety, t ...

for

classical group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...

s (roughly, the first concerns the generators and the second the relations). The theorems are among the most important results of

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 ...

. Classically the theorems are proved over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s. But characteristic-free invariant theory extends the theorems to 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 ...

of arbitrary characteristic.

First fundamental theorem for $\operatorname(V)$

The theorem states that the ring of

\operatorname(V)

-invariant polynomial functions on

^p \oplus V^q

is generated by the functions

\langle \alpha_i ,  v_j \rangle

, where

\alpha_i

are in

V^*

and

v_j \in V

Second fundamental theorem for general linear group

Let ''V'', ''W'' be

finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

s over the complex numbers. Then the only

\operatorname(V) \times \operatorname(W)

-invariant

prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...

s in

\mathbb operatorname(V, W) /math> are the determinant ideal I_k = \mathbb operatorname(V, W)_k generated by 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 ...

s of all the

k \times k

minor Minor may refer to: Common meanings * Minor (law), a person not under the age of certain legal activities. * Academic minor, a secondary field of study in undergraduate education Mathematics * Minor (graph theory), a relation of one graph to an ...

First fundamental theorem for $\operatorname(V)$

Second fundamental theorem for general linear group

Notes

References

Further reading

First fundamental theorem for \operatorname(V)

Second fundamental theorem for general linear group

Notes

References

Further reading

First fundamental theorem for $\operatorname(V)$