Could you please explain example 2.1.9 with the table. w order and a56 CHAPTER 2. REPRESENTATIONS OF THE SYMMETRIC GROUPDefinition 2.1.5 Suppose An. LetM = C{{₁},... {tk}}.where (t₁)....() is a complete list of A-tabloids. Then MA is called thepermutation module corresponding to A. ■Since we are considering only row equivalence classes, we could list the rows ofMA in any order and produce an isomorphic module. Thus M" is defined forany composition (ordered partition) . As will be seen in the next examples.we have already met three of these modules.Example 2.1.6 If A = (n), thenwith the trivial action. ■M() = C{12}Example 2.1.7 Now consider A = (122). Each equivalence class {t} consistsof a single tableau, and this tableau can be identified with a permutation inone-line notation (by taking the transpose, if you wish). Since the action ofSn is preserved,M(1) CSand the regular representation presents itself. ■Example 2.1.8 Finally, if λ = (n-1, 1), then each A-tabloid is uniquelydetermined by the element in its second row, which is a number from 1 to n.As in Example 1.6.3, this sets up a module isomorphismM(-1,1)C{1,2,...,n},so we have recovered the defining representation. ■Example 2.1.9 Let us compute the full set of characters of these moduleswhen n = 3. In this case the only partitions are A = (3), (2, 1), and (1³). Fromthe previous examples, the modules M correspond to the trivial, defining,and regular representations of S3, respectively. These character values haveall been computed in Chapter 1. Denote the character of M by andrepresent the conjugacy class of S, corresponding to u by K. Then we havethe following table of characters:(3)(2,1)K(13)136K(2,1)110K(3)100

For the remainder of the section, we will define a product operation that ensures that such tables…

Example 2.1.9 Let us compute the full set of characters of these modules when n 3. In this case the only partitions are A = (3), (2, 1), and (1³). From the previous examples, the modules M correspond to the trivial, defining, and regular representations of S3, respectively. These character values have all been computed in Chapter 1. Denote the character of M by and represent the conjugacy class of S, corresponding to by K₁. Then we have the following table of characters: (3) (2,1) Ø(1³) K(13) K(21) K(3) 1 0 0- 1 3 6 1 1 0 Tro

Example 2.1.9 Let us compute the full set of characters of these modules when n 3. In this case the only partitions are A = (3), (2, 1), and (1³). From the previous examples, the modules M correspond to the trivial, defining, and regular representations of S3, respectively. These character values have all been computed in Chapter 1. Denote the character of M by and represent the conjugacy class of S, corresponding to by K₁. Then we have the following table of characters: (3) (2,1) Ø(1³) K(13) K(21) K(3) 1 0 0- 1 3 6 1 1 0 Tro

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.36P

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