Schur マニュアルの問題を解く　その４

6-6節 Exercises 7-10

問７

7. Use the command $\mathrm{p\_to\_s}$ to construct the character table of the symmetric group $S_4$ .

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。

f:id:fortran66:20171210135515p:plain

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
$\mathrm{compare} \,\lambda, \,\mathrm{p\_to\_s} \,\rho \,\,\,\,\,\lambda,\rho \vdash n$
brings to the screen the value of the characteristic $\chi^\lambda_\rho$ of $S_n$ .

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 $S-$ functions $\lambda$ and $\mu$ using the Littlewood-Richardson rule retaining only terms whose first part is $\leq n$ . Show that the command sequence
$\mathrm{conj \,len \,n \,conj \,o} \,\lambda,\mu$
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 $Q-$ 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>