App IV. Supplements to the Theory of Symmetric Groups

1. App IV. Supplements to the Theory of Symmetric Groups Lemma IV.1: xp = p x p–1 Let { h } & { v } be horizontal & vertical permutations of Young tableaux  . Let s, a & e be the associated symmetrizer, anti-symmetrizer, & irreducible symmetrizer, respectively. Then the corresponding quantities for the Young tableau p are given by Proof: Let x be h, v, s, a, or e.   QED Hence, only  need be considered explicitly.

2. Lemma IV.2: • For a given tableau , { h } & { v } are each a subgroup of Sn. • The following identities hold: Proof: • { h } = Sm where m n. Ditto { v }. 2. (Rearrangement theorem)

3. Lemma IV.3: Given  and p  Sn.  at least 2 numbers in one row of  which appear in the same column of  p.  Proof of  : If then Hence   p can be obtained from  in 2 steps: 1. which permutes numbers in same row of  2. which permutes numbers in same column of  p  p can't move 2 numbers in one row of  to the same column of  p. Negation of this completes the proof.

4. Proof of : Assume no 2 numbers are shared by a row of  and a column of  p  p can be obtained from  as followings: Starting from the 1st column of  p. Since these numbers are in different rows of  , they can be brought to the 1st column by a horizontal permutation. Repeating the procedure for the other columns of  p, we get , which differs from  p by a vertical permutation.  i.e., Negation of this completes the proof.

5. Lemma IV.4: Given  and p  Sn.  ( ~ denotes transpositions ) Proof: By Lemma IV.3,   (a,b) in the same row of  & the same column of  p Let t be the transposition of (a,b).  and Let   QED

6. Lemma IV.5: Given  and r  G.  Proof:  where  Lemma IV.4:    If then   QED

7. 1. Lemma IV.6: Given 2 distinct diagrams  > , 2. Proof: Let r be the permutation that brings the numbers of the 2 tableaux to the same sequential order ( 1st row left to right, then 2nd row … ) Since the diagrams are distinct, By lemma IV.3,  at 1 pair of numbers that appears simultaneously in one row of p and one column of q. Let t be the transposition of these 2 numbers. By lemma IV.2, 

8. Lemma IV.7: The linear group transformations on Vmn , spans the space K of all symmetry-preserving linear transformations. Proof: A  K  Obviously, A necessary & sufficient condition for { g  Gm } to span K is that  L = 0 where L is a linear functional by The symmetry-preserving version of L is Since A  K  We can assume

9. Let and consider where Since   is arbitrary    Repeating the argument gives QED