6-6節 Exercises 7-10
問7
7. Use the command to construct the character table of the symmetric group .
- S4 の指標表つくり
共役類ρは power sum 対称関数、規約表現は Schur 関数で表わされるとして、その変換行列を求めればよい。結果は縦に求まる。
DP> sfn Schur Function Mode SFN> p_to_s 1^4 {4} + 3{31} + 2{2^2} + 3{21^2} + {1^4} SFN> p_to_s 1^22 {4} + {31} - {21^2} - {1^4} SFN> p_to_s 13 {4} - {2^2} + {1^4} SFN> p_to_s 4 {4} - {31} + {21^2} - {1^4} SFN> p_to_s 2^2 {4} - {31} + 2{2^2} - {21^2} + {1^4} SFN>
問8
8. Show that the command sequence
brings to the screen the value of the characteristic of .
- 上記 S4 対称群の指標を例として
上記問7から特定の項を抜き出しているだけ。
SFN> ? compare COMPare Format:-Comp EXPR1,EXPR2 Modes:-REP, SFN Description:-Compares EXPR1 and EXPR2, and creates a new EXPR in which the multiplicities are the products of the corresponding multiplicities in EXPR1 and EXPR2. Example:-SFN> ->comp 42 +31, 5 +4.42 +5.31 +2^2 4{42} + 5{31} SFN> ->comp 2.42 -3.31, 5 +4.42 +5.31 +2.2^2 8{42} - 15{31} SFN> SFN> p_to_s 1^4 {4} + 3{31} + 2{2^2} + 3{21^2} + {1^4} SFN> compare 4, p_to_s 1^4 {4} SFN> compare 31, p_to_s 1^4 3{31} SFN> compare 2^2, p_to_s 1^4 2{2^2} SFN> compare 21^2, p_to_s 1^4 3{21^2} SFN> compare 1^4, p_to_s 1^4 {1^4} SFN>
以下各共役類に対して繰り返し。
問9
9. A person wishes to be able to form the product of two functions and using the Littlewood-Richardson rule retaining only terms whose first part is . Show that the command sequence
will achieve the desired result.
- Young 図で考えれば自明
len n は分割 λ の長さ n 以下のものを返す。長さ n ピッタリが欲しい時は -n。
λ={2,1}, μ={4}, n=5 の場合の例を以下に示す。
SFN> o 21,4 {61} + {52} + {51^2} + {421} SFN> yo last OOOOOO OOOOO OOOOO OOOO O OO O OO O O SFN> yo conj last OOO OOO OO OO OO O OO O O O O O O O O O O O O O SFN> yo len 5 last OOO OOO OO OO O OO O O O O O O O O SFN> yo conj last OOOOO OOOOO OOOO OO O OO O O SFN> last {52} + {51^2} + {421} SFN> conj len 5 conj o 21,4 {52} + {51^2} + {421} SFN>
問10
10. Repeat previous item for the combination of functions.
- Q-関数よく分からないアル
取りあえず、こういう事かなと。
SFN> ? S_TO_QsymmFn S_TO_QsymmFn Format:-S_To_Q EXPR Modes:-SFN Description:-Treats EXPR as a list of S-functions of type S(x,-1) and transforms them into a list of Q-functions. Example:- SFN> ->s_to_q 31 Q_{4} + Q_{31} SFN> SFN> conj len 5 s_to_q o q_to_sd 21, q_to_sd 4 Q_{321^2} + 2Q_{2^2 1^3} + 2Q_{21^5} SFN>