Book: SymmetricGroupRepresentations

Challenge: Calculating and interpreting the irreducible representations of the symmetric groups

Literature

Theory

{$S_{n}$} is the symmetric group on n letters. The irreducible representations of {$S_{n}$} are indexed by the conjugacy classes, which is to say, the partitions λ. Given a partition λ of the numbers 1,...,n, which is to say, a Young diagram filled with numbers, define {$R_{ \lambda }$} to be the permutations {$e_{\sigma}$} which preserve the numbers in each row, and {$C_{ \lambda }$} to be the permutations {$e_{\tau}$} which preserve the numbers in each column. Define:

{$$s_{\lambda} = {\sum_{\sigma \in R_{ \lambda }}} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau}$$}

Then the subspaces {$\mathbb{C}S_{n}\cdot s_{\lambda}$} are the irreducible representations indexed by λ.

For example, for the partition 21 filled [12][3] we have that

{$$s_{\lambda} = e_\imath + e_{13} - e_{12} + e_{132}$$}

Defining {$A = e_\imath - e_{13}, B = e_{132} - e_{12}, C = e_{123} - e_{23}$} we have

{$s_{\lambda} = A-B$}

{$\imath \cdot s_{\lambda} = A-B$}

{$e_{12} \cdot s_{\lambda} = A-B$}

{$e_{13} \cdot s_{\lambda} = C-A$}

{$e_{23} \cdot s_{\lambda} = B-C$}

{$e_{123} \cdot s_{\lambda} = C-A$}

{$e_{132} \cdot s_{\lambda} = B-C$}

We have a three-cycle that can be written with 2x2 matrices acting on basis vectors A-B and B-C. Study the three-cycle!

Discussion

There is a natural numbering of the cells in a partition based on their intepretation as paths in Pascal's triangle. In this numbering we assign numbers part by part. Within the part there may be a secondary numbering. The assigning of numbers part by part can go in either direction and is dual in that sense.

It's important here to conceive how sign comes to play in going to different parts.

In working with standard tableaux we use a very different numbering based on the innermost corner and building out in both directions. This numbering can be thought of as an internal point of view, in conditions, in context, not from the top of Pascal's triangle. I should investigate how these relative and absolute perspectives relate.

• Does it make sense to analyze the combinatorics of the representations in terms of their complement spaces? For example, what happens when we multiply {$s_{\lambda}$} by an entire conjugacy class?

{$(e_12 + e_13 + e_23) \cdot s_{\lambda}=0$}

{$(e_123 + e_132) \cdot s_{\lambda} is not 0$}

{$(e_\imath + e_123 + e_132) \cdot s_{\lambda} = 0$}

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Puslapis paskutinį kartą pakeistas 2018 spalio 04 d., 09:49