mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a modular invariant of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

is an invariant of a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

acting Acting is an activity in which a story is told by means of its enactment by an actor who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad range of sk ...

on a

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 ...

of positive characteristic (usually dividing the

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

of the group). The study of modular invariants was originated in about 1914 by .

Dickson invariant

When ''G'' is the finite

general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

GL_''n''(F_''q'') over the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

F_''q'' of order a

prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...

''q'' acting on the

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

F_''q'' 'X''₁, ...,''X''_''n''in the natural way, found a complete set of invariants as follows. Write 'e''₁, ..., ''e''_''n''for 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 ...

of the

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

whose entries are ''X'', where ''e''₁, ..., ''e''_''n'' are non-negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s. For example, the Moore determinant ,1,2of order 3 is :

\begin x_1 & x_1^q & x_1^\\x_2 & x_2^q & x_2^\\x_3 & x_3^q & x_3^ \end

Then under the action of an element ''g'' of GL_''n''(F_''q'') these determinants are all multiplied by det(''g''), so they are all invariants of SL_''n''(F_''q'') and the ratios 'e''₁, ...,''e''_''n''thinsp;/ , 1, ..., ''n'' − 1are invariants of GL_''n''(F_''q''), called Dickson invariants. Dickson proved that the full ring of invariants F_''q'' 'X''₁, ...,''X''_''n''sup>GL_''n''(F_''q'') is a polynomial algebra over the ''n'' Dickson invariants , 1, ..., ''i'' − 1, ''i'' + 1, ..., ''n''thinsp;/ , 1, ..., ''n'' − 1for ''i'' = 0, 1, ..., ''n'' − 1. gave a shorter proof of Dickson's theorem. The matrices 'e''₁, ..., ''e''_''n''are divisible by all non-zero linear forms in the variables ''X''_''i'' with coefficients in the finite field F_''q''. In particular the Moore determinant , 1, ..., ''n'' − 1is a product of such linear forms, taken over 1 + ''q'' + ''q''² + ... + ''q''^{''n'' – 1} representatives of (''n'' – 1)-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

over the field. This factorization is similar to the factorization of the

Vandermonde determinant In algebra, the Vandermonde polynomial of an ordered set of ''n'' variables X_1,\dots, X_n, named after Alexandre-Théophile Vandermonde, is the polynomial: :V_n = \prod_ (X_j-X_i). (Some sources use the opposite order (X_i-X_j), which changes the ...

into linear factors.

References

* * * * * {{DEFAULTSORT:Modular Invariant Of A Group Invariant theory

Dickson invariant

See also

References