algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

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

for

classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or s ...

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 descri ...

. 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

The theorem states that the ring of

GL(V)

-invariant polynomial functions on

^p \times 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

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

s over the complex numbers. Then the only

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

-invariant prime ideals 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 value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...

s of all the

k \times k

- minors.

First fundamental theorem

Second fundamental theorem for general linear group

Notes

References

Further reading