Episode 2347
Affine symmetric group
Sun, 2023-Oct-08 00:23 UTC
Length - 2:06
Direct Link
Welcome to featured Wiki of the Day where we read the summary of the featured Wikipedia article every day.
The featured article for Sunday, 8 October 2023 is Affine symmetric group.
The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections of permutations (rearrangements) of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied as part of the fields of combinatorics and representation theory.
A finite symmetric group consists of all permutations of a finite set. Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation.
The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups.
This recording reflects the Wikipedia text as of 00:23 UTC on Sunday, 8 October 2023.
For the full current version of the article, see Affine symmetric group on Wikipedia.
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This has been Gregory Neural. Thank you for listening to featured Wiki of the Day.
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