fortran66のブログ

fortran について書きます。

Wybourne の Schur で Un → Un-1 計算二通り

ユニタリー群のブランチング計算をしてみます

U_n\rightarrow U_{n-1} の branching は、skew Young 図を用いれば \{\lambda/M\} で求められます。一方 Plethysm を用いると U_n\rightarrow U_{n-1} に対して \{1\}\rightarrow \{1\}+\{0\} なので (\{1\}+\{0\})\otimes\{\lambda\} とも表わされます。

証明


\begin{align}
  \{\lambda\}&\rightarrow (\{1\}+\{0\})\otimes\{\lambda\} \\
&\sum_\zeta(\{1\}\otimes\{\lambda/\zeta\})\cdot(\{0\}\otimes\{\zeta\}) \\
&=\sum_m\{\lambda/\{m\}\} \\
&=\{\lambda/M\}
\end{align}
ここで、

\{1\}\otimes\{\lambda\}=\{\lambda\}

\{0\}\otimes\{\zeta\}=\begin{cases}
                                        1 & ( \{\zeta\}=\{0\}, \{1\}, \{2\}, \cdots) \\
                                        0 & (otherwise)
                                   \end {cases}
を用いました。

プログラム

例として Schur のプログラム機能を用いて、U(8)\rightarrow U(7) の場合について、 Plethysm による方法を計算して、skew Young 図による方法と比較してみます。

\{\lambda\}\rightarrow\sum_\zeta(\{1\}\otimes\{\lambda/\zeta\})\cdot(\{0\}\otimes\{\zeta\})

gr u8
enter rv1
dim [rv1]
gr 4 u8 u8 u8 u8
rule [1*rv1*0*0] sum sk 2 eq 4
cont 3, 4        pl last
cont 1, 2        pl last 
sup false
gr u7
cont 1, 2        o  last
dim last
stop

Classical Groups for Physicists

Classical Groups for Physicists

Symmetry Principles and Atomic Spectroscopy

Symmetry Principles and Atomic Spectroscopy

The Theory of Group Representations (Phoenix Edition)

The Theory of Group Representations (Phoenix Edition)

計算結果

Plethysm による方法
DP> readfn 1 'unun1.fn'
=-
=-
=-
=-
=-
=-
=-
=-
=-
=-
=-
=-

DP> fn 1
Group is U(8)
enter rv10
Dimension = 1
Groups are U(8) * U(8) * U(8) * U(8)
Groups are U(8) * U(8) * U(8)
Groups are U(8) * U(8)
Group is U(7)
group not set
{0}
Dimension = 1

DP> fn 1
Group is U(8)
enter rv11
Dimension = 8
Groups are U(8) * U(8) * U(8) * U(8)
Groups are U(8) * U(8) * U(8)
Groups are U(8) * U(8)
Group is U(7)
group not set
{1} + {0}
Dimension = 8

DP> fn 1
Group is U(8)
enter rv12
Dimension = 36
Groups are U(8) * U(8) * U(8) * U(8)
Groups are U(8) * U(8) * U(8)
Groups are U(8) * U(8)
Group is U(7)
group not set
{2} + {1} + {0}
Dimension = 36
DP> fn 1
Group is U(8)
enter rv13
Dimension = 120
Groups are U(8) * U(8) * U(8) * U(8)
Groups are U(8) * U(8) * U(8)
Groups are U(8) * U(8)
Group is U(7)
group not set
{3} + {2} + {1} + {0}
Dimension = 120

DP> fn 1
Group is U(8)
enter rv121
Dimension = 168
Groups are U(8) * U(8) * U(8) * U(8)
Groups are U(8) * U(8) * U(8)
Groups are U(8) * U(8)
Group is U(7)
group not set
{21} + {2} + {1^2} + {1}
Dimension = 168

DP> fn 1
Group is U(8)
enter rv14331
Dimension = 199584
Groups are U(8) * U(8) * U(8) * U(8)
Groups are U(8) * U(8) * U(8)
Groups are U(8) * U(8)
Group is U(7)
group not set
{43^2 1} + {43^2} + {4321} + {432} + {431^2} + {431} + {3^3 1} + {3^3} + {3^2 21} + {3^2 2} + {3^2 1^2} + {3^2 1}
Dimension = 199584
DP>
skew Young 図を用いた計算
DP> sfn
Schur Function Mode
SFN> sk 0, m
{0}
SFN> sk 1, m
{1} + {0}
SFN> sk 2, m
{2} + {1} + {0}
SFN> sk 3, m
{3} + {2} + {1} + {0}
SFN> sk 21, m
{21} + {2} + {1^2} + {1}
SFN> sk 43^21, m
{43^2 1} + {43^2} + {4321} + {432} + {431^2} + {431} + {3^3 1} + {3^3} + {3^2 21} + {3^2 2} + {3^2 1^2} + {3^2 1}
SFN>

同じ結果が求まっています。