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From Groupprops

Contents 1 Bruhat ordering 2 Young diagrams and tableaux under the Robinson-Schensted correspondence

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S3.
View element structure of particular groups | View other specific information about symmetric group:S3

This article discusses symmetric group:S3, the symmetric group of degree three. We denote its elements as acting on the set {1,2,3}, written using cycle decompositions, with composition by function composition where functions act on the left. The multiplication table is:

Element	()	(1,2)	(2,3)	(3,1)	(1,2,3)	(1,3,2)
()	()	(1,2)	(2,3)	(3,1)	(1,2,3)	(1,3,2)
(1,2)	(1,2)	()	(1,2,3)	(1,3,2)	(2,3)	(1,3)
(2,3)	(2,3)	(1,3,2)	()	(1,2,3)	(1,3)	(1,2)
(3,1)	(3,1)	(1,2,3)	(1,3,2)	()	(1,2)	(2,3)
(1,2,3)	(1,2,3)	(1,3)	(1,2)	(2,3)	(1,3,2)	()
(1,3,2)	(1,3,2)	(2,3)	(1,3)	(1,2)	()	(1,2,3)

Since this group is a complete group (i.e., every automorphism is inner and the center is trivial), the classification of elements up to conjugacy is the same as the classification up to automorphisms. Further, since cycle type determines conjugacy class for symmetric groups, the conjugacy classes are parametrized by cycle types, which in turn are parametrized by unordered integer partitions of 3.

There are three conjugacy classes:

1. 3 = 1 + 1 + 1: The identity element, which has three cycles of size one each. (1)
2. 3 = 2 + 1: The conjugacy class of transpositions, comprising (1,2),(2,3),(1,3). (3)
3. 3 = 3: The conjugacy class of 3-cycles, comprising (1,2,3),(1,3,2). (2)

Bruhat ordering

The symmetric group of degree three can be viewed as a Coxeter group, with generators s₁ = (1,2) and s₂ = (2,3). The presentation is:

.

We can thus consider a Bruhat ordering on the elements of the symmetric group of degree three. Note that the Bruhat ordering depends on the specific choice of transpositions we use to generate the group, which in turn depends on an implicit order of the elements {1,2,3} that the group acts on (up to reversal). Thus, the Bruhat ordering is not invariant under conjugation.

The Bruhat ordering on the symmetric group of degree three has the special feature (no longer true for higher degree) that any two elements with distinct Bruhat lengths are comparable in the order. In the Bruhat ordering, there are four levels based on Bruhat length:

1. length 0: The identity element is the only element of length zero. (1)
2. length 1: s₁ = (1,2) and s₂ = (2,3) are the only elements of length one. Note that these are in the same conjugacy class, but they are not a complete conjugacy class. (2)
3. length 2: s₁s₂ = (1,2,3) and s₂s₁ = (1,3,2) are the only elements of length two. Note that these are precisely one conjugacy class. (2)
4. length 3: s₁s₂s₁ = (1,3), which is the antidiagonal permutation, i.e., the permutation whose matrix is the antidiagonal. It sends every x to n − x. This is in the same conjugacy class as the length one permutations. (1)

Young diagrams and tableaux under the Robinson-Schensted correspondence

There are three possible Young diagrams with three blocks: the single column, the single row, and the upside-down L-shape, with a column of length two and a column of length one. The corresponding number of ways of filling these diagrams is 1,1,2. Under the Robinson-Schensted correspondence:

1. The identity map corresponds to the single column. (1)
2. The antidiagonal permutation, i.e., the transposition (1,3), corresponds to the single row. (1)
3. The other four permutations, i.e., the two transpositions (1,2) and (2,3) and the two 3-cycles (1,2,3) and (1,3,2), correspond to the inverted L shape. (2² = 4, because there are 2 ways of getting a tableau with that shape).