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fortran について書きます。

ケモナーがトランプ大統領を支持?www

American Furry

www.newsweek.com

モナーのいわゆるネトウヨ化もトランプのせい。
モナー界にまで影響を及ぼすトランプ大統領さすが!w

http://s.newsweek.com/sites/www.newsweek.com/files/styles/large/public/2017/11/20/fefurries01rtx1xlaz.jpg

ケモノのくせに人権とかいうのも笑える。

モナーは日米ともオタク界の最下層に位置するもので、自己の同性愛や幼児性愛傾向に直面できずに、動物との同一視を通して心理的な抜け道を通る嘆かわしい趣味だと思う。

American Fury (English Edition)

American Fury (English Edition)

American Fury

American Fury


思い出してもニャンダー仮面のタマちゃんと、ジュエルペット・サンシャインのルビーくらいにしか・・・

次期ポケモン映画のサトシが萌え化したと評判!

http://www.all-nationz.com/archives/1068831331.html

http://livedoor.blogimg.jp/all_nations/imgs/7/7/77eac153.jpg
http://livedoor.blogimg.jp/all_nations/imgs/c/4/c4dc3a5f.jpg
http://livedoor.blogimg.jp/all_nations/imgs/9/5/95780655.jpg

Schur マニュアルの問題を解く その5

6.2 Tutorial 1

問1

1. Study the output produced successively from the following commands “sk21,p” , “sk last,q”
and ”sk sk21,p,q” and explain the final result.

  • skew Young tablau

定義により P\equiv1/\prod(1+\alpha_i)=\sum_{m=0}(-1)^m\{m\}, Q\equiv\prod(1+\alpha_i)=\sum_{m=0}\{1^m\} であるから PQ=1、 よって逆変換になっているから結果は自明。

REP> sfn
Schur Function Mode    
SFN> sk 21,p
{21} - {2} - {1^2} + {1}
SFN> sk last, q
{21}
SFN> sk sk 21,p,q
{21}
SFN> 

\mathrm{sk}\, \lambda, P は、P と M に weight の偶奇性による位相因子が付いた形になっているので、U(n)→U(n-1) のブランチングに weight の偶奇性で符号が反転した位相が付いたものになる。

問2

2. Study the output produced successively from the following commands “i32,41”, “rd_i2,1” and
“makewt5,rd_i2,1”. Why is the output from the first and last command the same?<<

  • 内積の reduced expression

i32,41 は 5次対称群 S(5) における {32} と {41} の内積(指標の内積外積は指標の直積)。
rd_i 2,1 は規約表現の末尾がそれぞれ 2 と 1 になるものの内積の reduced expression 。したがって、weight が5になるように補う命令 makewt により、5次の対称群の規約表現が 2,1 に対して作られるので、i32,41 と同じ結果が得られるのは当然。

SFN> i32,41
{41} + {32} + {31^2} + {2^2 1}
SFN> rd_i 2,1
<3> + <21> + <2> + <1^2> + <1>
SFN> makewt 5, rd_i 2,1
<41> + <32> + <31^2> + <2^2 1>
SFN> 

問3

3. Consider the command sequence “wt4,o ser4,p,ser4,q”. Before entering the command what
output do you think will result? Now try it.

  • 有限長で打ち切った関数と逆関数のべき展開の積

当然1={0}になる。有限打ち切りの効果でカスが残ってしまうが、weight 4 命令で weight 4
より大きな項は捨てるので無問題。

SFN> wt4,o ser4,p,ser4,q
{0}
SFN> 

問4

4. You want a list of all the partitions of all the integers up to n where n is not too large, say
n < 20. Show that the command “ser n,f” yields the desired result and that “countc ser20,f”
shows that the total number of partitions for n = 0,1,...,0 is 2715.

  • 整数の分割

p(0)~p(20) までの和は 2714 であり、Schur の結果も 2714 と出る。問題文が間違っている。
F は全分割の母関数になっているので、これを展開すればあらゆる分割が得られる。

SFN> ser 4,f
{4} + {31} + {3} + {2^2} + {21^2} + {21} + {2} + {1^4} + {1^3} + {1^2} + {1} + {0}
SFN> countc ser 20,f                                                                                                               
CoeffSum = 2714
SFN> 

問5

5. Show that by issuing the command “countc wt-20ser20,f” there are 627 partitions of the integer
20. NB. On installations that are short on memory you might want to consider smaller
examples.

  • p(20)

ここでの計算は 20 の分割数 p(20) の計算と同じ。weight 20 の Young図を書き連ねて数えあげたようなもの。(但し Schur の内部的にはそういうドン臭いことはしていないらしいのだがw)

SFN> countc wt-20ser20,f
CoeffSum = 627
SFN> 

問6

6. Show that the command sequence “wt-105,ser105,t” generates just one partition. Why?

  • T 母関数

T 母関数は、階段状の young 図に対応する分割を生成するので、n(n+1)/2 に合致する数の weight をもつものだけが許される。

SFN> wt-105,ser105,t
{14 13 12 11 10 987654321}

SFN> ser 105,t
{14 13 12 11 10 987654321} + {13 12 11 10 987654321} + {12 11 10 987654321} + {11 10 987654321} + {10 987654321} + {987654321}
 + {87654321} + {7654321} + {654321} + {54321} + {4321} + {321} + {21} + {1} + {0}
SFN> yo last
                                                                           
  OOOOOOOOOOOOOO   OOOOOOOOOOOOO   OOOOOOOOOOOO   OOOOOOOOOOO   OOOOOOOOOO 
  OOOOOOOOOOOOO    OOOOOOOOOOOO    OOOOOOOOOOO    OOOOOOOOOO    OOOOOOOOO  
  OOOOOOOOOOOO     OOOOOOOOOOO     OOOOOOOOOO     OOOOOOOOO     OOOOOOOO   
  OOOOOOOOOOO      OOOOOOOOOO      OOOOOOOOO      OOOOOOOO      OOOOOOO    
  OOOOOOOOOO       OOOOOOOOO       OOOOOOOO       OOOOOOO       OOOOOO     
  OOOOOOOOO        OOOOOOOO        OOOOOOO        OOOOOO        OOOOO      
  OOOOOOOO         OOOOOOO         OOOOOO         OOOOO         OOOO       
  OOOOOOO          OOOOOO          OOOOO          OOOO          OOO        
  OOOOOO           OOOOO           OOOO           OOO           OO         
  OOOOO            OOOO            OOO            OO            O          
  OOOO             OOO             OO             O                        
  OOO              OO              O                                       
  OO               O                                                       
  O                                                                        
                                                                           
                                                                           
  OOOOOOOOO   OOOOOOOO   OOOOOOO   OOOOOO   OOOOO   OOOO   OOO   OO   O   .
  OOOOOOOO    OOOOOOO    OOOOOO    OOOOO    OOOO    OOO    OO    O         
  OOOOOOO     OOOOOO     OOOOO     OOOO     OOO     OO     O               
  OOOOOO      OOOOO      OOOO      OOO      OO      O                      
  OOOOO       OOOO       OOO       OO       O                              
  OOOO        OOO        OO        O                                       
  OOO         OO         O                                                 
  OO          O                                                            
  O                                                                        
                                                                           

SFN> 

問7

7. Try analysing the following command sequence
“wt4,pl,ser4,m,4”
and then try running it first as a single command and then try to reproduce the result by issuing
a sequence of three separate commands. The above sequence is actually relevant to a problem
involving the symplectic nuclear model.

  • plethysm

plethysm の日本語訳語は『重合』らしいのだが、対称性を加味した冪のイメージと全く合わない。

SFN> wt4,pl,ser4,m,4
5{4} + 2{31} + 3{3} + 2{2^2} + {21} + 2{2} + {1} + {0}
SFN> ser 4, m
{4} + {3} + {2} + {1} + {0}
SFN> pl last, 4
{16 } + {15 } + {14 2} + {14 1} + 2{14 } + {13 3} + 2{13 2} + 2{13 1} + 3{13 } + 2{12 4} + 3{12 3} + {12 2^2} + {12 21} + 5{12 2}
 + 4{12 1} + 5{12 } + {11 41} + 3{11 4} + {11 32} + 2{11 31} + 5{11 3} + 2{11 2^2} + 3{11 21} + 7{11 2} + {11 1^2} + 5{11 1}
 + 5{11 } + 2{10 6} + {10 51} + 3{10 5} + 2{10 42} + 4{10 41} + 8{10 4} + 3{10 32} + 5{10 31} + 10{10 3} + {10 2^3} + {10 2^2 1}
 + 5{10 2^2} + 6{10 21} + 12{10 2} + 2{10 1^2} + 8{10 1} + 7{10 } + {961} + 3{96} + {952} + 3{951} + 5{95} + {943} + {9421}
 + 5{942} + {941^2} + 8{941} + 11{94} + {93^2} + {9321} + 6{932} + {931^2} + 9{931} + 13{93} + {92^3} + 2{92^2 1} + 7{92^2}
 + {921^2} + 9{921} + 14{92} + 3{91^2} + 9{91} + 7{9} + {8^2} + {87} + 2{862} + 3{861} + 6{86} + {8521} + 4{852} + {851^2} + 6{851}
 + 8{85} + 2{84^2} + 3{843} + {842^2} + 3{8421} + 11{842} + 2{841^2} + 13{841} + 16{84} + {83^2} + {832^2} + 3{8321} + 10{832}
 + 2{831^2} + 12{831} + 16{83} + 2{82^3} + 4{82^2 1} + 11{82^2} + 2{821^2} + 12{821} + 17{82} + 3{81^2} + 10{81} + 8{8} + {7^2 1^2}
 + {7^2 1} + {763} + 3{762} + {761^2} + 5{761} + 6{76} + {7531} + 2{753} + 2{7521} + 6{752} + 3{751^2} + 9{751} + 8{75} + {74^2 1}
 + 3{74^2} + 2{7431} + 6{743} + {742^2} + 5{7421} + 14{742} + 5{741^2} + 17{741} + 16{74} + {73^2 1} + 3{73^2} + {732^2} + 4{7321}
 + 12{732} + 4{731^2} + 15{731} + 16{73} + 2{72^3} + 4{72^2 1} + 11{72^2} + 3{721^2} + 13{721} + 16{72} + {71^3} + 4{71^2} + 9{71}
 + 7{7} + {6^2 4} + 2{6^2 3} + {6^2 2^2} + {6^2 21} + 5{6^2 2} + 5{6^2 1} + 7{6^2} + {654} + {6531} + 3{653} + {652^2} + 3{6521}
 + 8{652} + 2{651^2} + 9{651} + 9{65} + {64^2 2} + 2{64^2 1} + 5{64^2} + {6432} + 3{6431} + 8{643} + 3{642^2} + 7{6421} + 17{642}
 + 4{641^2} + 17{641} + 17{64} + {63^2 1} + 3{63^2} + 2{632^2} + 5{6321} + 13{632} + 3{631^2} + 14{631} + 16{63} + 3{62^3}
 + 5{62^2 1} + 12{62^2} + 2{621^2} + 12{621} + 16{62} + 3{61^2} + 9{61} + 7{6} + {5^2 31} + {5^2 3} + 2{5^2 21} + 3{5^2 2}
 + 2{5^2 1^2} + 4{5^2 1} + 2{5^2} + {54^2 2} + 2{54^2 1} + 4{54^2} + {5432} + 3{5431} + 6{543} + 2{542^2} + 6{5421} + 12{542}
 + 4{541^2} + 12{541} + 10{54} + {53^2 1} + 2{53^2} + {532^2} + 4{5321} + 9{532} + 3{531^2} + 10{531} + 10{53} + 2{52^3}
 + 4{52^2 1} + 9{52^2} + 2{521^2} + 9{521} + 11{52} + 2{51^2} + 6{51} + 5{5} + {4^4} + {4^3 3} + 2{4^3 2} + 3{4^3 1} + 5{4^3}
 + {4^2 32} + 2{4^2 31} + 5{4^2 3} + 2{4^2 2^2} + 4{4^2 21} + 9{4^2 2} + 2{4^2 1^2} + 8{4^2 1} + 8{4^2} + {43^2} + {432^2}
 + 2{4321} + 6{432} + {431^2} + 6{431} + 8{43} + 2{42^3} + 3{42^2 1} + 7{42^2} + {421^2} + 6{421} + 9{42} + {41^2} + 5{41} + 5{4}
 + {3^2 2} + {3^2 1} + 2{3^2} + {32^3} + {32^2 1} + 3{32^2} + 2{321} + 4{32} + 2{31} + 3{3} + {2^4} + {2^3 1} + 2{2^3} + {2^2 1}
 + 2{2^2} + {21} + 2{2} + {1} + {0}
SFN> wt 4, last
5{4} + 2{31} + 3{3} + 2{2^2} + {21} + 2{2} + {1} + {0}
SFN> 

問8

Just to remind ourselves that SCHUR handles lists of objects use SCHUR to calculate the
following: “o21+3,21+4”, “i21+3,21+4” and “pl21+3,21+4”. Think about the second result.

内積の場合、同じ対称群の中で演算が行われるので、weight が同じでなければならない。したがって、{4} はそもそも {21} や {3} と内積を取れない。

SFN> o21+3,21+4
{7} + 2{61} + 2{52} + {51^2} + {51} + {43} + {421} + 2{42} + 2{41^2} + {3^2} + 3{321} + {31^3} + {2^3} + {2^2 1^2}
SFN> i21+3,21+4
{3} + 2{21} + {1^3}
SFN> pl21+3,21+4
{12 } + {11 1} + 3{10 2} + {10 1^2} + 4{93} + 4{921} + {91^3} + 7{84} + 8{831} + 7{82^2} + 4{821^2} + {81^4} + 2{81} + 3{75}
 + 12{741} + 14{732} + 11{731^2} + 10{72^2 1} + 4{721^3} + 4{72} + {71^5} + 3{71^2} + 3{6^2} + 8{651} + 18{642} + 11{641^2}
 + 7{63^2} + 22{6321} + 9{631^3} + 5{63} + 12{62^3} + 9{62^2 1^2} + 4{621^4} + 10{621} + {61^6} + 3{61^3} + 5{5^2 2} + 7{5^2 1^2}
 + 10{543} + 22{5421} + 9{541^3} + 5{54} + 13{53^2 1} + 13{532^2} + 18{5321^2} + 5{531^4} + 12{531} + 10{52^3 1} + 5{52^2 1^3}
 + 7{52^2} + 2{521^5} + 9{521^2} + {51^7} + 2{51^4} + 4{4^3} + 7{4^2 31} + 10{4^2 2^2} + 9{4^2 21^2} + 4{4^2 1^4} + 5{4^2 1}
 + 6{43^2 2} + 8{43^2 1^2} + 10{432^2 1} + 7{4321^3} + 10{432} + 2{431^5} + 9{431^2} + 4{42^4} + 2{42^3 1^2} + {42^2 1^4}
 + 8{42^2 1} + 4{421^3} + {3^4} + 3{3^3 21} + {3^3 1^3} + {3^3} + {3^2 2^3} + 3{3^2 2^2 1^2} + 6{3^2 21} + 2{3^2 1^3} + 2{32^3}
 + 3{32^2 1^2} + {321^4} + {2^4 1}
SFN> 
SFN> i21+3,21+4
{3} + 2{21} + {1^3}
SFN> i21+3,21
{3} + 2{21} + {1^3}
SFN> i21+3,4
zero

SFN> 

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> 

Schur マニュアルの問題を解く その3

6-6節 Exercises 4-6

問4

4. In the simple SU_3 quark model of baryons and mesons the  (u,d,s) quarks span the threedimensional SU_3 irreducible representation \{1\} while the corresponding antiquarks (u,d,s) span the three-dimensional SU_3 irreducible representation \{1^2\}. The SU_3 group is sometimes referred to as the flavour group SU^{fl}_3.

(a) Show that combining quarks with antiquarks leads to an octet and singlet of mesons.

(b) Show that combining a triple product of quarks leads to a baryon singlet and two baryon octets.

REP> gr su3
Group is SU(3)

REP> prod 1, 1^2
{21} + {0}
REP> dim 21
dimension = 8
REP> dim 0
dimension = 1

REP> prod prod 1,1,1
{3} + 2{21} + {0}
REP> std last 
{3} + 2{21} + {0}
REP> dim 3
dimension = 10
REP> 

(b) で {3} も出てきてよく分からない。 

問5

5. A student tries to unite the lowest lying baryons into a single irreducible representation of a Lie group G. Noting that the baryon octet has spin J_p = {1\over2}^+ and the decuplet spin J_p = {3\over2}^+ identify G and the relevant irreducible representation she found.

  • よく分からんが・・・

spin SU(2) * quark SU(3) なので Lie 群的に SU(6)→SU(2)*SU(3) か?
spin3/2, decuplet → 4*10=40 次元, spin1/2, octet → 2*8=16 次元となる。
問4で SU(3) の {3} は10重項、{21} は八重項と分かっているので、求める規約表現は {3}{3} + {1}{21} となるはず。

以下の計算で、SU(2) \{\lambda_1, \lambda_2\}=\{\lambda_1-\lambda_2,0\}であることを考えると、SU(6) の規約表現 {3} が SU(2)xSU(3) では {3}{3}+{21}{21} = {3}{3}+{1}{21} と見なせるので、これが求める答えだと思ふ。

BRM> stop
enter branching & rule numbers> 5 2,3
U(6) to U(2) * U(3) 
BRM> 0
{0}{0}
BRM> dim 0
Irrep modifies to null
BRM> 0
{0}{0}
BRM> 1
{1}{1}
BRM> 2
{2}{2} + {1^2}{1^2}
BRM> 3
{3}{3} + {21}{21}
BRM> 1^2
{2}{1^2} + {1^2}{2}
BRM> 1^3
{3}{1^3} + {21}{21}
BRM> quit

問6

6. Assume that you have the group SU^{fl}_3 , as in exercise 5. Your quarks are now also endowed with spin (SU^s_2) and color (SU^c_3). The total of 18 single quark states span the vector irreducible representation \{1\} of SU_{18} and you have the chain of groups

SU_{18}\supset SU^{fl}_3\times(SU^{cs}_6\supset SU^{s}_2\times SU^{c}_3)

Assume that each quark is in an S-state and that the states of the six quark configuration
q^6, span the totally antisymmetric irreducible representation \{1^6\} of SU_{18}. In terms of QCD (Quantum ChromoDynamics) physical states correspond to color singlets i.e. states transforming as \{0\} under SU^c_3. With respect to the color-spin group SU^{cs}_6 this can only happen if the weight w_\lambda of the partition \lambda labelling the irreducible representation \lambda of SU^{cs}_6 is divisible by three. That is for irreducible representations of null triality.

(a) Determine the irreducible representations of SU^{fl}_3\times SU^{cs}_6 contained in \{1^6\} of SU_{18} that have null triality.

(b) Determine the spin content of each SU_6^{cs} found in (a).

(c) Which of the multi-quark configurations q^7, q^8 and q^9 might you expect to contain color singlet states?

(d) In the MIT bag model of multi-quark states there is an energy term proportional to the
eigenvalues of the second-order Casimir operator of the color-spin group. Show that
the eigenvalues for the irreducible representations \{21^4\}, \{2^3\} and \{3^2\} are in the ratio 1:2:3.

  • 謎が多いが・・・

(a)
U(18) ⊃ U(3) * U(6) で、U(6) の λi がみな3の倍数になっていて null triality の条件をみたしているのは、{2^3}{3^2} の項だけ。

DP> brm
Branch Mode
enter branching & rule numbers> 5 3,6
U(18) to U(3) * U(6) 
BRM> 1^6
{6}{1^6} + {51}{21^4} + {42}{2^2 1^2} + {41^2}{31^3} + {3^2}{2^3} + {321}{321} + {2^3}{3^2}

(b)
SU(6) ⊃ SU(2) * SU(3) で、U(2) のスピン成分は \{\lambda_1, \lambda_2\}=\{\lambda_1-\lambda_2,0\} であるから、{6}={6}, {51}={4}, {42}={2}, {3^2}={0} となる。

BRM> stop
enter branching & rule numbers> 5 2, 3
U(6) to U(2) * U(3) 
BRM> 3^2
{6}{3^2} + {51}{42} + {51}{321} + {42}{51} + {42}{41^2} + {42}{3^2} + {42}{321} + {3^2}{6} + {3^2}{42} + {3^2}{2^3}

SU(3) の方は {321}={21}, {41^2}={3}, {2^3}={0} となるので、{3^2}{2^3}={0}{0} が color singlet と思われる。(が、よく分からんw 3の倍数でアホになります)

(c)
素朴に3の倍数である q^9 と思われる。以下の計算で 確かに {1^9} から {3^3}{3^3} にブランチングする。

BRM> stop
enter branching & rule numbers> 5 3 6
U(18) to U(3) * U(6) 
BRM> 1^7
{61}{21^5} + {52}{2^2 1^3} + {51^2}{31^4} + {43}{2^3 1} + {421}{321^2} + {3^2 1}{32^2} + {32^2}{3^2 1}
BRM> 1^8
{62}{2^2 1^4} + {61^2}{31^5} + {53}{2^3 1^2} + {521}{321^3} + {4^2}{2^4} + {431}{32^2 1} + {42^2}{3^2 1^2} + {3^2 2}{3^2 2}
BRM> 1^9
{63}{2^3 1^3} + {621}{321^4} + {54}{2^4 1} + {531}{32^2 1^2} + {52^2}{3^2 1^3} + {4^2 1}{32^3} + {432}{3^2 21} + {3^3}{3^3}
BRM> exit

(d)
SU(6)群で規約表現 {21^4}, {2^3}, {3^2} の性質を書かせて、2次のカシミール演算子のところを見ると、72:144:216=1:2:3 になっている。

DP> gr su6
Group is SU(6)
DP> rep
REP mode       
Group is SU(6)
REP> prop 21^4
<dynkin label> (10001)
dimension = 35   72*2nd-casimir=72
2nd-dynkin = 12

REP> prop 2^3
<dynkin label> (00200)
dimension = 175   72*2nd-casimir=144
2nd-dynkin = 120

REP> prop 3^2
<dynkin label> (03000)
dimension = 490   72*2nd-casimir=216
2nd-dynkin = 504

REP> 

Schur マニュアルの問題を解く その2

6-6節 Exercises 1-3

問1

1. The states of the d^n electron configuration may be classified by the following chain of groups
leading, finally, to the spectroscopic terms {}^{2S+1}L. Use SCHUR to establish the results of Table
5.1 for the spectroscopic terms of the d^5 configuration

U_{10} \supset Sp_{10}\supset SU_{2}\times SO_{5}\supset SU_{2}\times SO_{3} \sim SO^S_3\times SO^L_3\equiv
{}^{2S+1}L

f:id:fortran66:20171210124749p:plain

  • d 電子の分類

SU10 での {1^5} 表現は、d電子の10個の軌道に5個の電子が反対称に詰まっているということ。Sp10 での分裂は seniority 数に対応 (または j-j coupling での軌道の詰め方)、SU2 はスピン角運動量

ブランチングを求める

SU10→Sp10
Sp10→SU2×SO5
SO5→SO3

DP> ? tab
 TABleOfBranchingRules
      Format:-
        Modes:- DP
 Description:- Table A.2 : Table of branching rules

NOTES:
Rule No. Group         Subgroup             Rule and number(s) in BRM 
1 :-  U(n)            ->  O(n)                       1,n   
2 :-  U(n)            ->  Sp(n)                      2,n
3 :-  U(n)            ->  U(n-1)                     3,n
4 :-  U(m+n)          ->  U(m) x U(n)                4,m,n
5 :-  U(mn)           ->  U(m) x U(n)                5,m,n
6 :-  U(2k)           ->  U(k)                       6,2k
7 :-  U(n)            ->  SO(3)                      7,n
8 :-  SU(m+n)         ->  U(1) x SU(m) x SU(n)       8,m,n
9 :-  Sp(2k)          ->  SO(3)                      9,2k
10:-  Sp(2k)          ->  U(1) x SU(k)              10,2k
11:-  Sp(2k)          ->  SU(2) x SO(k)             11,2k
12:-  Sp(2k)          ->  U(2k)                     12,2k
13:-  Sp(2k)          ->  U(k)                      13,2k
14:-  Sp(2m+2n)       ->  Sp(2m) x Sp(2n)           14,2m,2n
15:-  Sp(2mn)         ->  Sp(2m) x O(n)             15,2m,n
16:-  S(m+n)          ->  S(m) x S(n)               16,m,n
17:-  S(n)            ->  A(n)                      17,n
18:-  O(n)            ->  S(n)                      18,n
19:-  O(n)            ->  S(n+1)                    19,n
20:-  O(n)            ->  U(n)                      20,n
21:-  O(2k) or O(2k+1)->  U(k)                      21,2k (or 2k+1)    
22:-  O(m+n)          ->  O(m) x O(n)               22,m,n
23:-  O(mn)           ->  O(m) x O(n)               23,m,n
24:-  O(4mn)          ->  Sp(2m) x Sp(2n)           24,2m,2n
25:-  SO(2k+1)        ->  SO(3)                     25,2k+1

DP> brm 
Branch Mode
enter branching & rule numbers> 2 10
U(10) to Sp(10) 
BRM> 1^5
<1^5> + <1^3> + <1>
BRM> stop

enter branching & rule numbers> 11 10
Sp(10) to SU(2) * SO(5) 
BRM> 1^5
{5}[0] + {3}[2] + {1}[2^2]
BRM> 1^3
{3}[1^2] + {1}[21]
BRM> 1
{1}[1]
BRM> stop

enter branching & rule numbers> 25 5
O(5) to SO(3) 
BRM> 0
[0]
BRM> 2
[4] + [2]
BRM> 2^2
[6] + [4] + [3] + [2] + [0]
BRM> 1^2
[3] + [1]
BRM> 21
[5] + [4] + [3] + [2] + [1]
BRM> 1
[2]
BRM> exit

U2表現からスピン角運動量

SU2xSO3 →SO3xSO3 の自己同型 

DPrep Mode (with function)
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [5*0]
Groups are SO(3) * SO(3)
[s;2][0]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [3*4]
Groups are SO(3) * SO(3)
[s;1][4]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [3*2]
Groups are SO(3) * SO(3)
[s;1][2]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*6]
Groups are SO(3) * SO(3)
[s;0][6]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*4]
Groups are SO(3) * SO(3)
[s;0][4]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*3]
Groups are SO(3) * SO(3)
[s;0][3]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*2]
Groups are SO(3) * SO(3)
[s;0][2]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*0]
Groups are SO(3) * SO(3)
[s;0][0]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [3*3]
Groups are SO(3) * SO(3)
[s;1][3]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [3*1]
Groups are SO(3) * SO(3)
[s;1][1]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*5]
Groups are SO(3) * SO(3)
[s;0][5]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*4]
Groups are SO(3) * SO(3)
[s;0][4]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*3]
Groups are SO(3) * SO(3)
[s;0][3]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*2]
Groups are SO(3) * SO(3)
[s;0][2]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*1]
Groups are SO(3) * SO(3)
[s;0][1]
DP> gr 2 su2,so3
Groups are SU(2) * SO(3)
DP> au gr1so3, [1*2]
Groups are SO(3) * SO(3)
[s;0][2]

規約表現の次元を求めて検算

SU10->Sp10
252=132+110+10

Sp10->SU2xSO5
132=6*1+4*14+2*35
110=4*10+2*35
10 = 2*5

SO5->SO3
1=1
14=9+5
35=13+9+7+5+1
10=7+3
35=11+9+7+5+3
5=5

DP> gr sp10
Group is Sp(10)
DP> dim 1^5
missing brackets or group not set?
Dimension = 0
DP> rep
REP mode       
Group is Sp(10)
REP> dim 1^5
dimension = 132
REP> gr u10
Group is U(10)
REP> dim 1^5
dimension = 252
REP> gr sp10
Group is Sp(10)
REP> dim 1^5
dimension = 132
REP> dim 1^3
dimension = 110
REP> dim 1^1
dimension = 10
REP> dim 1^5+1^3+1
dimension = 252
REP> gr su2
Group is SU(2)
REP> dim 5
dimension = 6
REP> dim3
dimension = 4
REP> dim 1
dimension = 2
REP> gr so5
Group is SO(5)
REP> dim 0
dimension = 1
REP> dim 2
dimension = 14
REP> dim 2^2
dimension = 35
REP> dim 1^2
dimension = 10
REP> dim 21
dimension = 35
REP> dim 1
dimension = 5
REP> gr so3
Group is SO(3)
REP> dim0
dimension = 1
REP> dim 4
dimension = 9
REP> dim 2
dimension = 5
REP> dim 6
dimension = 13
REP> dim 4
dimension = 9
REP> dim 3
dimension = 7
REP> dim 2
dimension = 5
REP> dim 0
dimension = 1
REP> dim 3
dimension = 7
REP> dim 1
dimension = 3
REP> dim 5
dimension = 11
REP> dim 4
dimension = 9
REP> dim 3
dimension = 7
REP> dim 2
dimension = 5
REP> dim 1
dimension = 3
REP> dim 2
dimension = 5
REP> gr so3
Group is SO(3)
REP> dim s2
dimension = 6
REP> dim s1
dimension = 4
REP> dim s0
dimension = 2
REP> 

問2

2. If two sets of states of a d^n configuration transform under SO_5 as the irreducible representations [\lambda] and [\mu] respectively their matrix elements of an operator transforming as [\nu] will certainly vanish unless

[\lambda]\times[\mu]\supset[\nu]

The Coulomb interaction within the d^n configuration can be expanded in terms of operators symmetrised with respect to the same groups used to classify the states. One of the relevant operators transforms as [2^2] under SO_5. Let c([\lambda][\mu][2^2]) be the number of times [2^2] occurs in the SO_5 Kronecker product [\lambda]\times[\mu]. Use SCHUR to construct the entries given in Table 5.2 for the numbers c([\lambda][\mu][2^2]).

Table 5.2 とは、Table 6.3 のことか? 
f:id:fortran66:20171210131733p:plain

  • rep mode で積を取る

積に関しては可換なので半分だけ計算すればよい。
rep mode での外積は prod

DPrep Mode (with function)
DP> rep
REP mode       
REP> gr so5
Group is SO(5)
REP> compare 22 prod 0,0
zero
REP> compare 22 prod 0,1
zero
REP> compare 22 prod 0,1^2
zero
REP> compare 22 prod 0,2
zero
REP> compare 22 prod 0,21
zero
REP> compare 22 prod 0,2^2
[2^2]
REP> compare 22 prod 1,1
zero
REP> compare 22 prod 1,1^2
zero
REP> compare 22 prod 1,2
zero
REP> compare 22 prod 1,21
[2^2]
REP> compare 22 prod 1,2^2
[2^2]
REP> compare 22 prod 1^2,1^2
[2^2]
REP> compare 22 prod 1^2,2
zero
REP> compare 22 prod 1^2,21
[2^2]
REP> compare 22 prod 1^2,2^2
[2^2]
REP> compare 22 prod 2,2
[2^2]
REP> compare 22 prod 2,21
[2^2]
REP> compare 22 prod 2,2^2
[2^2]
REP> compare 22 prod 21,21
2[2^2]
REP> compare 22 prod 21,2^2
[2^2]
REP> compare 22 prod 2^2,2^2
[2^2]
REP> 

問3

3. Show that for SO_5 the irreducible representation [2^2] occurs once in the symmetric part of the Kronecker square of the irreducible representation [2^2] and once in the antisymmetric part.

d 電子に関わる問題が続いているとして、[2^2] が d^5 のが問1で求めた {1}[2^2] ~[s;0][2^2] のことであるとする。この二乗をとってスピンで分類する。

DP> gr 2 so3,so5
Groups are SO(3) * SO(5)
DP> prod [s0*2^2], [s0,2^2]
mistake.
missing brackets or group not set?
zero
DP> prod [s0*2^2], [s0*2^2]
[1][4^2] + [1][43] + [1][42] + [1][41] + [1][4] + [1][3^2] + [1][32] + [1][31] + [1][3] + [1][2^2] + [1][21] + [1][2] + [1][1^2]
 + [1][1] + [1][0] + [0][4^2] + [0][43] + [0][42] + [0][41] + [0][4] + [0][3^2] + [0][32] + [0][31] + [0][3] + [0][2^2] + [0][21]
 + [0][2] + [0][1^2] + [0][1] + [0][0]
DP> 

[1][2^2] spin triplet
[0][2^2] spin singlet が各1個づつ。これでいいのか?

Schur マニュアルの問題を解く その1

Schur 付属のマニュアルの問題を解きます

Schur Group Theory Software

6-3節

問1

1. The branching rule for the maximal embedding of Sp_{2k} in U_{2k} is given as \{\lambda\}\downarrow\{\lambda/B\}. Inversely, for Sp_{2k}\uparrow U_{2k} we have <\lambda >\uparrow\{\lambda/A\}.
(a) Use SCHUR to show that for 2k = 8
< 21>\uparrow\{21\}-\{1\} and \{21\}\downarrow< 21> + < 1 >
(b) Use the knowledge gained in (a.) to write a single sequence of commands to evaluate
the Sp_8 Kronecker product < 21>\times< 21>. Check that your result is dimensionally
correct and compare your result with that obtained by simply setting the group as Sp_8
and using the command “prod21,21”.

  • Schur function mode で {21/A}, {21/B} の確認
REP> sfn
Schur Function Mode    
SFN> sk 21, a
{21} - {1}
SFN> sk 21, b
{21} + {1}
SFN> 
  • Representation mode と Branching Rule mode で解けという事でしょうか?

6-3節はまだ BRMode の説明をしていなので、違うかもしれませんw
まず branching のテーブルを表示させて欲しい branching の番号を知ります。

DP> ?tab
 TABleOfBranchingRules
      Format:-
        Modes:- DP
 Description:- Table A.2 : Table of branching rules

NOTES:
Rule No. Group         Subgroup             Rule and number(s) in BRM 
1 :-  U(n)            ->  O(n)                       1,n   
2 :-  U(n)            ->  Sp(n)                      2,n
3 :-  U(n)            ->  U(n-1)                     3,n
4 :-  U(m+n)          ->  U(m) x U(n)                4,m,n
5 :-  U(mn)           ->  U(m) x U(n)                5,m,n
6 :-  U(2k)           ->  U(k)                       6,2k
7 :-  U(n)            ->  SO(3)                      7,n
8 :-  SU(m+n)         ->  U(1) x SU(m) x SU(n)       8,m,n
9 :-  Sp(2k)          ->  SO(3)                      9,2k
10:-  Sp(2k)          ->  U(1) x SU(k)              10,2k
11:-  Sp(2k)          ->  SU(2) x SO(k)             11,2k
12:-  Sp(2k)          ->  U(2k)                     12,2k
13:-  Sp(2k)          ->  U(k)                      13,2k

DP> brm  
Branch Mode
enter branching & rule numbers> 12 8
Sp(8) to SU(8) 
BRM> 21
{21} - {1}
BRM> stop
enter branching & rule numbers> 2 8
U(8) to Sp(8) 
BRM> 21
<21> + <1>
BRM> exit

DPrep Mode (with function)
DP> rep 
REP mode       
REP> gr sp8
Group is Sp(8)
REP> prop 21
<dynkin label> (1100)
dimension = 160   20*2nd-casimir=27
2nd-dynkin = 60

REP> prod 21, 21                                                                                                                   
<42> + <41^2> + <4> + <3^2> + 2<321> + <31^3> + 3<31> + <2^3> + <2^2 1^2> + 2<2^2> + 3<21^2> + 2<2> + <1^4> + 2<1^2> + <0>
REP> dim last
dimension = 25600

REP> conv_s_to_rep sk o, sk 21,a, sk 21,a, b                                                                                       
<42> + <41^2> + <4> + <3^2> + 2<321> + <31^3> + 3<31> + <2^3> + <2^2 1^2> + 2<2^2> + 3<21^2> + 2<2> + <1^4> + 2<1^2> + <0>

確認:160x160 = 25600

問2

2. Read the information in Appendix A about the command au and use the command to show
that under the SO_8 automorphism [1]\to[s;0]- \rightarrow[s;0]+. Use that observation to obtain
the content of the Kronecker products [s;0]-\times[s;0]- and [s;0]+\times[s;0]+ from the result
obtained from the command “prod1,1”.

  • SO8 の 三回対称なルート図に対応した自己同型

prod 1,1 → prod s0-,s0- → prod s0+,s0+ → prod 1,1

REP> rep
REP> gr so8
Group is SO(8)
REP> au so8, 1
Group is SO(8)
[s;0]-
REP> au so8, last
Group is SO(8)
[s;0]+
REP> au so8, last
Group is SO(8)
[1]
REP> prod 1,1
[2] + [1^2] + [0]
REP> prod s0+,s0+
[1^4]+ + [1^2] + [0]
REP> prod s0-,s0-
[1^4]- + [1^2] + [0]
REP> au so8, prod 1, 1                                                                                                             
Group is SO(8)
[1^4]- + [1^2] + [0]
REP> au so8, last
Group is SO(8)
[1^4]+ + [1^2] + [0]
REP> au so8, last
Group is SO(8)
[2] + [1^2] + [0]
REP> 

B.R.Judd が Nieson & Koster の cfp の表に関して、SO(7) の対称性では説明のつかない 0 要素が多くあるから隠れた対称性がまだあるはず、と言っていた正体が、この SO(8) ⊃ SO(7),SO(7)',SO(7)'' の対称性に基づくものだったようです。ここで三つの SO(7) は線形従属で独立なのは2つだそうです。

問3

3. Set the group as Sp_8 and use SCHUR to calculate the Kronecker product < 321>\times< 421>.
Now save the result by using the command “setr1 last” . Use the command ”zero” to eliminate
“last” and then issue the command “rv1” and you should once again see the output of the
Kronecker product. Now save the “rvar” to a diskfile using the command “save rvar ’filename’
” as described in Appendix A. If you now issue the command “setr1 zero” you will find on
saying “rvar 1” the data for the Kronecker product has been lost and “rvar 1” simply reports
zero. All is not lost as we can reload, even days later, the saved “rvar1” using the command
load as described in Appendix A. Thus issuing the command “load rvar’filename’ ” followed
by the command “rv1” will put to screen once again the Kronecker product. Note however,
we must set the group as appropriate to that of the saved rvar.

  • file への保存・読み出し

マニュアルには、"setr1 xxx" と書いてありますが、"setrv1 xxx" と「v」を付け加えないとうまくゆかないです。

注:setrv 1 xxx はいいですが set rv1 xxx は許されません。

REP> gr sp8
Group is Sp(8)
REP> prod 321,421
<742> + <741^2> + <74> + <73^2> + 2<7321> + 3<731> + <72^3> + 2<72^2> + 3<721^2> + 2<72> + 2<71^2> + <7> + <652> + <651^2> + <65>
 + 2<643> + 4<6421> + 6<641> + 3<63^2 1> + 3<632^2> + 9<632> + 9<631^2> + 6<63> + 8<62^2 1> + 12<621> + 5<61^3> + 5<61> + <5^2 3>
 + 2<5^2 21> + 3<5^2 1> + <54^2> + 4<5431> + 3<542^2> + 10<542> + 9<541^2> + 6<54> + 3<53^2 2> + 7<53^2> + 18<5321> + 21<531>
 + 6<52^3> + 15<52^2> + 18<521^2> + 12<52> + 12<51^2> + 3<5> + <4^3 1> + 2<4^2 32> + 5<4^2 3> + 10<4^2 21> + 11<4^2 1> + <43^3>
 + 9<43^2 1> + 9<432^2> + 21<432> + 20<431^2> + 12<43> + 19<42^2 1> + 26<421> + 11<41^3> + 10<41> + 3<3^3 2> + 5<3^3> + 14<3^2 21>
 + 15<3^2 1> + 6<32^3> + 14<32^2> + 17<321^2> + 11<32> + 12<31^2> + 3<3> + 6<2^3 1> + 9<2^2 1> + 5<21^3> + 5<21> + 2<1^3> + <1>
REP> setrv1 last
REP> zero
zero
REP> last
zero
REP> rv1
<742> + <741^2> + <74> + <73^2> + 2<7321> + 3<731> + <72^3> + 2<72^2> + 3<721^2> + 2<72> + 2<71^2> + <7> + <652> + <651^2> + <65>
 + 2<643> + 4<6421> + 6<641> + 3<63^2 1> + 3<632^2> + 9<632> + 9<631^2> + 6<63> + 8<62^2 1> + 12<621> + 5<61^3> + 5<61> + <5^2 3>
 + 2<5^2 21> + 3<5^2 1> + <54^2> + 4<5431> + 3<542^2> + 10<542> + 9<541^2> + 6<54> + 3<53^2 2> + 7<53^2> + 18<5321> + 21<531>
 + 6<52^3> + 15<52^2> + 18<521^2> + 12<52> + 12<51^2> + 3<5> + <4^3 1> + 2<4^2 32> + 5<4^2 3> + 10<4^2 21> + 11<4^2 1> + <43^3>
 + 9<43^2 1> + 9<432^2> + 21<432> + 20<431^2> + 12<43> + 19<42^2 1> + 26<421> + 11<41^3> + 10<41> + 3<3^3 2> + 5<3^3> + 14<3^2 21>
 + 15<3^2 1> + 6<32^3> + 14<32^2> + 17<321^2> + 11<32> + 12<31^2> + 3<3> + 6<2^3 1> + 9<2^2 1> + 5<21^3> + 5<21> + 2<1^3> + <1>
REP> save rvar '6-3-3.rvar'
save[1]
REP> setrv1 zero
REP> rvar 1
zero
REP> rv1
zero
REP> load rvar '6-3-3.rvar'
load[1]
REP> rvar 1
<742> + <741^2> + <74> + <73^2> + 2<7321> + 3<731> + <72^3> + 2<72^2> + 3<721^2> + 2<72> + 2<71^2> + <7> + <652> + <651^2> + <65>
 + 2<643> + 4<6421> + 6<641> + 3<63^2 1> + 3<632^2> + 9<632> + 9<631^2> + 6<63> + 8<62^2 1> + 12<621> + 5<61^3> + 5<61> + <5^2 3>
 + 2<5^2 21> + 3<5^2 1> + <54^2> + 4<5431> + 3<542^2> + 10<542> + 9<541^2> + 6<54> + 3<53^2 2> + 7<53^2> + 18<5321> + 21<531>
 + 6<52^3> + 15<52^2> + 18<521^2> + 12<52> + 12<51^2> + 3<5> + <4^3 1> + 2<4^2 32> + 5<4^2 3> + 10<4^2 21> + 11<4^2 1> + <43^3>
 + 9<43^2 1> + 9<432^2> + 21<432> + 20<431^2> + 12<43> + 19<42^2 1> + 26<421> + 11<41^3> + 10<41> + 3<3^3 2> + 5<3^3> + 14<3^2 21>
 + 15<3^2 1> + 6<32^3> + 14<32^2> + 17<321^2> + 11<32> + 12<31^2> + 3<3> + 6<2^3 1> + 9<2^2 1> + 5<21^3> + 5<21> + 2<1^3> + <1>
REP> 

メモ帳:Schur 群論プログラム

ubuntu gui でないダム端末からのインストール。

Schur Group Theory Software
Brian G. Wybourne による Schur 関数など対称関数と組み合わせ論的な Lie 群論プログラム Schur。そのバイナリは、windows 版は ver.6.09 で止まっているが、Linux 版は一歩進んで ver. 6.10 となっている。Ver.6.10 は TeX 出力ができるなど少しバグがとれている模様。しかし、Windows 版より不安定ですぐにコアをダンプで死んでしまう。

なお本人死亡のため、sourceforge 全盛時代の面影を残しつつ放置。合掌。

sourceforge からなのでよく分からず試行錯誤の上、以下で成就。

wget https://sourceforge.net/projects/schur/files/Debian%20package/6.10/schur_6.10-2_amd64.deb
sudo apt install ./schur_6.10-2_amd64.deb

Cloud9 の bash コンソールから実行 5次の Kostka 行列を表示。 

ubuntu@xxxxxx:~$ schur
+-----------------------------------------------------------------------------+
|                            SCHUR 6.10                                       |
| Copyright (C) 1996 Brian G. Wybourne,                                       |
|               2006 Franck BUTELLE, Steven M. Christensen,                   |
|                    Ronald C. KING & Frederic TOUMAZET                       |
| SCHUR comes with ABSOLUTELY NO WARRANTY. This is free software, and you are |
| welcome to redistribute it under certain conditions; type ?LICENSE for      |
| details.  - If you wish to exit, enter END or QUIT                          |
|           - If you wish to obtain help: HELP or ?cmd or APROPOS search      |
| Please report bugs to                                                       |
|                                Franck.Butelle@lipn.fr & toumazet@univ-mlv.fr|
+-----------------------------------------------------------------------------+

                              *** Status ***
digits:T reverse:F more:F TeX:F debug:F setlimit:12 pwt:201 logging:F
     .........10........20........30........40........50
 fns:_________._________._________._________._________.
 var:_________._________._________._________._________. Direct Product mode
                              *------------*

DPrep Mode (with function)
DP> sfn
Schur Function Mode    
SFN> km 5
1 1 1 1 1 1 1 
0 1 1 2 2 3 4 
0 0 1 1 2 3 5 
0 0 0 1 1 3 6 
0 0 0 0 1 2 5 
0 0 0 0 0 1 4 
0 0 0 0 0 0 1 
SFN> 

The Theory of Group Characters and Matrix Representations of Groups (AMS Chelsea Publishing)

The Theory of Group Characters and Matrix Representations of Groups (AMS Chelsea Publishing)

The Theory of Group Representations (Phoenix Edition)

The Theory of Group Representations (Phoenix Edition)

Classical Groups for Physicists

Classical Groups for Physicists

Symmetry Principles and Atomic Spectroscopy

Symmetry Principles and Atomic Spectroscopy