Question

Suppose that you have a square with the vertices labeled A,B,C,D. The following operations are defined using tableau notation: r

1 =( AB BC CD DA )s 1 =( AC BB CA DD ) a. Find the composition r 1​ ∘r 1
​ and write it using tableau notation.

Answer 1

The composition of the operation r1 with itself results in the vertices of the square being permuted as A to C, B to D, C to A, and D to B, which in tableau notation is (AC BD CA DB).

Step-by-step explanation:

The question is about finding the composition of an operation r1 with itself using tableau notation. Tableau notation is a way to represent a permutation of elements. In this case, it presents the permutation of the vertices of a square. The operation r1 is defined as (AB BC CD DA), which means A goes to B, B goes to C, C goes to D, and D goes to A.

When we compose r1 with itself (r1 ∘ r1), we look at where each vertex goes after the first operation and then apply r1 again. So after the first r1, A goes to B, and applying r1 again, B goes to C, therefore A will go to C. This process is repeated for each vertex, resulting in:

• A goes to C
• B goes to D
• C goes to A
• D goes to B

Writing this in tableau notation gives us the composition (AC BD CA DB).