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Consider the symmetric group 33 consisting of the permutations of three objects and ViEW the three basis vectors of R3 as the three objects that etc permutei This leads to the following three-dimensional rep I)" of 33: on] n10 DL{e}=(1l} a), D"(b)=(1 i] a). DID [101 Here the rep is oniy Specied for the two generators :- and h of S: whit]: in cycle nota- tion are given by c = [123} and b: (12}. [a] Show that DL indeed forms a rep of 5'3. (1}) Show using the characters of D" that D" is not an irrep of S; and decompose D" into irreps of 33 Using the character table of 33+ (c) Decompose BL 13' DL into irreps of 5'3 using the character table of 33. (d) Identify the direction in R:1 that is invariant under the action of the rep BL of 3;. Please answer all 4 sub-parts with working. Thank you