Введение
Свойство в теории групп
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
В математике, если X – конечное множество, содержащее не менее двух элементов, то перестановки X (то есть биективные функции из X в X) делятся на два класса равного размера: четные перестановки и нечетные перестановки. Если какое-либо полное упорядочение X фиксировано, четность (нечетность или четность) перестановки σ может быть определена как четность числа инверсий для σ, то есть пар элементов x, y из X, таких что x < y и σ(x) > σ(y). Знак, сигнатура или сигнум перестановки σ обозначается sgn(σ) и определяется как +1, если σ четная, и −1, если σ нечетная. Сигнатура определяет знакопеременный характер симметрической группы Sn. Другое обозначение для знака перестановки дается более общим символом Леви-Чивиты (εσ), который определяется для всех отображений из X в X и принимает значение ноль для небиективных отображений. Знак перестановки может быть явно выражен как:
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
sgn(σ) = (−1)^(N(σ))
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
где N(σ) – число инверсий в σ.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
Альтернативно, знак перестановки σ можно определить из ее разложения в произведение транспозиций как:
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
sgn(σ) = (−1)^m
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
где m – число транспозиций в разложении. Хотя такое разложение не является единственным, четность числа транспозиций во всех разложениях одинакова, что означает, что знак перестановки определен однозначно.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
Свойства
Пермутация идентичности — это четная перестановка. Кроме того, мы видим, что четные перестановки образуют подгруппу Sn. Это ядро гомоморфизма sgn. Нечетные перестановки не могут образовывать подгруппу, поскольку композиция двух нечетных перестановок является четной, но они образуют смежный класс An (в Sn). Если n > 1, то в Sn количество четных перестановок равно количеству нечетных; следовательно, An содержит n!/2 перестановок. (Причина в том, что если σ четная, то (1 2)σ нечетная, а если σ нечетная, то (1 2)σ четная, и эти два преобразования обратны друг другу.) Цикл является четным тогда и только тогда, когда его длина нечетна. Это следует из формул, таких как… На практике, чтобы определить, является ли данная перестановка четной или нечетной, перестановку записывают в виде произведения непересекающихся циклов. Перестановка нечетная, если и только если эта факторизация содержит нечетное число циклов четной длины. Другой способ определения четности или нечетности перестановки — построить соответствующую матрицу перестановки и вычислить ее определитель. Значение определителя совпадает с четностью перестановки. Каждая перестановка нечетного порядка должна быть четной. Перестановка (1 2)(3 4) в A4 показывает, что обратное утверждение неверно в общем случае.
In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation is odd if and only if this factorization contains an odd number of even length cycles. Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value of the determinant is the same as the parity of the permutation. Every permutation of odd order must be even. The permutation (1 2)(3 4) in A4 shows that the converse is not true in general.
Обобщения
Парность можно обобщить на группы Коксетера: определяется функция длины ℓ(v), зависящая от выбора образующих (для симметрической группы – смежных транспозиций), и тогда отображение v ↦ (−1)ℓ(v) дает обобщённое знаково-ориентированное отображение.