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Wikipedia:Today's featured article/requests/Affine symmetric group

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Affine symmetric group

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This is the archived discussion of the TFAR nomination for the article below. Subsequent comments should be made on the appropriate discussion page (such as Wikipedia talk:Today's featured article/requests). Please do not modify this page.

The result was: scheduled for Wikipedia:Today's featured article/October 8, 2023 by - Dank (push to talk) 23:23, 5 September 2023 (UTC)[reply]

The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group S̃3
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group 3

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. They may also be defined as collections of permutations (rearrangements) of the integers 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. Each affine symmetric group is an infinite extension of a finite symmetric group, and many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. 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. (Full article...)