Math 2.0

Group theory and generalizations Definition

Sign of a permutation

Builds on Decomposition of a permutation into transpositions

Definition

Let n≥2n \geq 2 and let σ∈Sn\sigma \in S_n. Write σ=τ1τ2⋯τm\sigma = \tau_1 \tau_2 \cdots \tau_m as a product of transpositions. The sign of σ\sigma is sgn⁡(σ)=(−1)m\operatorname{sgn}(\sigma) = (-1)^m.

Remarks

The parity of mm does not depend on the decomposition, so the sign is well defined. Equivalently, sgn⁡(σ)=(−1)N(σ)\operatorname{sgn}(\sigma) = (-1)^{N(\sigma)}, where N(σ)N(\sigma) is the number of pairs i<ji < j with σ(i)>σ(j)\sigma(i) > \sigma(j). The sign of a permutation is the determinant of its permutation matrix.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptSymmetric group, definition, no evidence levelSymmetric groupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementSign of a permutation, definition, no evidence levelSign of a permutationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cycle notation concept
  2. Group concept
  3. Symmetric group definition
  4. Disjoint cycles commute statement
  5. Transposition definition
  6. Decomposition of a permutation into disjoint cycles statement
  7. Decomposition of a permutation into transpositions statement
  8. Sign of a permutation definition

Where this leads

13 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, lead section
About this page
Tag0SK90238BH
URIm2://0SK90238BH
Aliasessgn
Kinddefinition
WritingW0 stub
MSC20B30
DefinesSign of a permutation
UsesDecomposition of a permutation into transpositions

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by