mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 ide ...

is an invariant of a

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

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

on a

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 can ...

of positive characteristic (usually dividing the order 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 invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

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 that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

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, 17 ...

''q'' acting on the

ring Ring 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 :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

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 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 and ...

of the

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

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

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

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