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数学

Discovery

Andrius Kulikauskas

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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Book.SymmetricGroupRepresentations istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2018 spalio 04 d., 09:49 atliko AndriusKulikauskas -
Pridėtos 48-55 eilutės:

[+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.
2018 spalio 04 d., 09:18 atliko AndriusKulikauskas -
Pridėtos 46-47 eilutės:

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!
2018 spalio 04 d., 09:14 atliko AndriusKulikauskas -
Pakeistos 29-32 eilutės iš
{$$s_{\lambda}$ = e_\imath + e_13_ - e_12 + e_132 $$}

Defining {$ A = e_\imath - e_13, B = e_132 - e_12, C = e_123 - e_23 $} we have
į:
{$$s_{\lambda} = e_\imath + e_{13} - e_{12} + e_{132} $$}

Defining {$ A = e_\imath - e_{13}, B = e_{132} - e_{12}, C = e_{123} - e_{23} $} we have
Pakeistos 37-45 eilutės iš
{$ 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 $}
į:
{$ 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 $}
2018 spalio 04 d., 09:13 atliko AndriusKulikauskas -
Pakeistos 21-22 eilutės iš
{$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 }$} to be the permutations which preserve the numbers in each row, and {$C_{ \lambda }$} to be the permutations which preserve the numbers in each column. Define:
į:
{$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:
Pridėtos 27-58 eilutės:
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 $}

>>bgcolor=#FFFFC0<<

* 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 $}


>><<
2018 spalio 04 d., 08:57 atliko AndriusKulikauskas -
Ištrintos 27-46 eilutės:






A numbering {$T$} or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n. Note that each letter appears only once.






* Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}.
* {$S^{\lambda}$} is a {$\mathbb{C}[S_{n}]$}-submodule of {$M^{\lambda}$}, which is a left {$\mathbb{C}[S_{n}]$}-module.
* As {$\mathbb{C}[S_{n}]$}-modules, both {$S^{\lambda}$} and {$M^{\lambda}$} are {$S_{n}$} representations.
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {$\left \{T \right \}$} of shape {${\lambda}$}.
* A tabloid {$\left \{T \right \}$} is a numbering of the Young diagram such that each row is in increasing order from left to right. The tabloid may be thought of as an equivalence class of Young tableau whose elements in any given row are the same.
* The action of {$S_{n}$} on tabloids is given by renumbering the cells and then resorting the numbers in the rows, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
* {$S^{\lambda}$} is the vector subspace of {$M^{\lambda}$} spanned by
2018 spalio 04 d., 08:50 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
Then the subspace {$\mathbb{C}$}
į:
Then the subspaces {$\mathbb{C}S_{n}\cdot s_{\lambda}$} are the irreducible representations indexed by λ.
2018 spalio 04 d., 08:47 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
Then the subspace {$mathbb{C}$}
į:
Then the subspace {$\mathbb{C}$}
2018 spalio 04 d., 08:47 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
Then the subspace mathbb{C}
į:
Then the subspace {$mathbb{C}$}
2018 spalio 04 d., 08:47 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
į:
Then the subspace mathbb{C}
2018 spalio 04 d., 08:44 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
{$ s_{\lambda} = {\sum_{\sigma \in R_{ \lambda }}} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $}
į:
{$$ s_{\lambda} = {\sum_{\sigma \in R_{ \lambda }}} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $$}
2018 spalio 04 d., 08:42 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
{$ s_{\lambda} = \sum_{\sigma \in R_{ \lambda }} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $}
į:
{$ s_{\lambda} = {\sum_{\sigma \in R_{ \lambda }}} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $}
2018 spalio 04 d., 08:41 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
{$ s_{\lambda} = \sum_ {\sigma \in R_{ \lambda }} e_{\sigma} \sum_ { \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $}
į:
{$ s_{\lambda} = \sum_{\sigma \in R_{ \lambda }} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $}
2018 spalio 04 d., 08:39 atliko AndriusKulikauskas -
Pridėta 11 eilutė:
* [[http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Kao.pdf | Representations of the Symmetric Group]] Daphne Kao - my favorite
Ištrinta 13 eilutė:
* [[http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Kao.pdf | Representations of the Symmetric Group]] Daphne Kao
Pakeista 23 eilutė iš:
{$s_{ \lambda } =
į:
{$ s_{\lambda} = \sum_ {\sigma \in R_{ \lambda }} e_{\sigma} \sum_ { \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $}
2018 spalio 04 d., 08:32 atliko AndriusKulikauskas -
Pakeistos 21-39 eilutės iš
{$S_{n}$} is the symmetric group on n letters. A numbering {$T$} or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n. Note that each letter appears only once.
į:
{$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 }$} to be the permutations which preserve the numbers in each row, and {$C_{ \lambda }$} to be the permutations which preserve the numbers in each column. Define:

{$s_{ \lambda } =










A numbering {$T$} or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n. Note that each letter appears only once.



2018 spalio 03 d., 10:47 atliko AndriusKulikauskas -
Pridėta 14 eilutė:
* VGTU biblioteka: Young Tableaux : With Applications to Representation Theory and Geometry. William Fulton.
2018 spalio 02 d., 11:33 atliko AndriusKulikauskas -
Pridėta 13 eilutė:
* [[http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Kao.pdf | Representations of the Symmetric Group]] Daphne Kao
2018 rugsėjo 30 d., 23:30 atliko AndriusKulikauskas -
Pakeista 12 eilutė iš:
* [[http://math.uchicago.edu/~may/REU2013/REUPapers/McNamara.pdf | Irreducible Representations of the Symmetric Group]]
į:
* [[http://math.uchicago.edu/~may/REU2013/REUPapers/McNamara.pdf | Irreducible Representations of the Symmetric Group]] Redmond McNamara
2018 rugsėjo 30 d., 23:29 atliko AndriusKulikauskas -
Pridėta 12 eilutė:
* [[http://math.uchicago.edu/~may/REU2013/REUPapers/McNamara.pdf | Irreducible Representations of the Symmetric Group]]
2018 rugsėjo 30 d., 22:31 atliko AndriusKulikauskas -
Pakeista 18 eilutė iš:
{$S_{n}$} is the symmetric group on n letters. A numbering ''T'' or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n. Note that each letter appears only once.
į:
{$S_{n}$} is the symmetric group on n letters. A numbering {$T$} or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n. Note that each letter appears only once.
2018 rugsėjo 30 d., 22:27 atliko AndriusKulikauskas -
Pakeistos 18-19 eilutės iš
{$S_{n}$} is the symmetric group on n letters. A numbering ''T'' or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n.
į:
{$S_{n}$} is the symmetric group on n letters. A numbering ''T'' or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n. Note that each letter appears only once.
Pakeistos 24-25 eilutės iš
* A tabloid {$\left \{T \right \}$} is a numbering of the Young diagram such that each row is in nondecreasing order from left to right. The tabloid may be thought of as an equivalence class of Young tableau whose elements in any given row are the same.
* The action of {$S_{n}$} on tabloids is given by renumbering the cells and then resorting the numbers in the rows, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
į:
* A tabloid {$\left \{T \right \}$} is a numbering of the Young diagram such that each row is in increasing order from left to right. The tabloid may be thought of as an equivalence class of Young tableau whose elements in any given row are the same.
* The action of {$S_{n}$} on tabloids is given by renumbering the cells and then resorting the numbers in the rows, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
* {$S^{\lambda}$} is the vector subspace of {$M^{\lambda}$} spanned by
2018 rugsėjo 30 d., 21:59 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
* The action of {$S_{n}$} on tabloids is given by renumbering the cells and resorting the rows, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
į:
* The action of {$S_{n}$} on tabloids is given by renumbering the cells and then resorting the numbers in the rows, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
2018 rugsėjo 30 d., 21:59 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
* The action of {$S_{n}$} on tabloids is given by renumbering, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
į:
* The action of {$S_{n}$} on tabloids is given by renumbering the cells and resorting the rows, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}.
2018 rugsėjo 30 d., 21:58 atliko AndriusKulikauskas -
Pakeistos 24-25 eilutės iš
* A tabloid {$\left \{T \right \}$} is a numbering of the Young diagram such that each row is in nondecreasing order from left to right. The tabloid may be thought of as an equivalence class of Young tableau whose elements in any given row are the same.
į:
* A tabloid {$\left \{T \right \}$} is a numbering of the Young diagram such that each row is in nondecreasing order from left to right. The tabloid may be thought of as an equivalence class of Young tableau whose elements in any given row are the same.
* The action of {$S_{n}$} on tabloids is given by renumbering, so that {$\sigma \cdot \left \{ T \right \} = \left \{\sigma \cdot T \right \}$}
.
2018 rugsėjo 30 d., 21:47 atliko AndriusKulikauskas -
Pakeistos 18-19 eilutės iš
{$S_{n}$} is the symmetric group on n letters.
į:
{$S_{n}$} is the symmetric group on n letters. A numbering ''T'' or {${T}'$} of a Young diagram with ''n'' cells fills it with the letters 1 through n.
Pakeistos 23-24 eilutės iš
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {$\left \{T \right \}$} of shape {${\lambda}$}.
į:
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {$\left \{T \right \}$} of shape {${\lambda}$}.
* A tabloid {$\left \{T \right \}$} is a numbering of the Young diagram such that each row is in nondecreasing order from left to right. The tabloid may be thought of as an equivalence class of Young tableau whose elements in any given row are the same
.
2018 rugsėjo 30 d., 21:40 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {${{T}}$} of shape {${\lambda|$}}.
į:
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {$\left \{T \right \}$} of shape {${\lambda}$}.
2018 rugsėjo 30 d., 21:38 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {${T}$} of shape {${\lambda|$}.
į:
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {${{T}}$} of shape {${\lambda|$}}.
2018 rugsėjo 30 d., 21:38 atliko AndriusKulikauskas -
Pakeistos 22-23 eilutės iš
* As {$\mathbb{C}[S_{n}]$}-modules, both {$S^{\lambda}$} and {$M^{\lambda}$} are {$S_{n}$} representations.
į:
* As {$\mathbb{C}[S_{n}]$}-modules, both {$S^{\lambda}$} and {$M^{\lambda}$} are {$S_{n}$} representations.
* {$M^{\lambda}$} is defined as the complex vector space with basis the tabloids {${T}$} of shape {${\lambda|$}
.
2018 rugsėjo 30 d., 21:36 atliko AndriusKulikauskas -
Pakeista 22 eilutė iš:
* As {$\mathbb{C}[S_{n}]$}-modules, {$S^{\lambda}$} and {$M^{\lambda}$} are both {$S_{n}$} representations.
į:
* As {$\mathbb{C}[S_{n}]$}-modules, both {$S^{\lambda}$} and {$M^{\lambda}$} are {$S_{n}$} representations.
2018 rugsėjo 30 d., 21:35 atliko AndriusKulikauskas -
Pakeistos 20-22 eilutės iš
Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}. {$S^{\lambda}$} is a {$\mathbb{C}[S_{n}]$}-submodule of {$M^{\lambda}$}, which is a left {$\mathbb{C}[S_{n}]$}-module, and thus a {$S_{n}$} representation.
į:
* Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}.
* {$S^{\lambda}$} is a {$\mathbb{C}[S_{n}]$}-submodule of {$M^{\lambda}$}, which is a left {$\mathbb{C}[S_{n}]$}-module.
* As
{$\mathbb{C}[S_{n}]$}-modules, {$S^{\lambda}$} and {$M^{\lambda}$} are both {$S_{n}$} representations.
2018 rugsėjo 30 d., 21:34 atliko AndriusKulikauskas -
Pakeista 20 eilutė iš:
Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}. {$S^{\lambda}$} is a -submodule of {$M^{\lambda}$}, which is a left -module, and thus a {$S_{n}$} representation.
į:
Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}. {$S^{\lambda}$} is a {$\mathbb{C}[S_{n}]$}-submodule of {$M^{\lambda}$}, which is a left {$\mathbb{C}[S_{n}]$}-module, and thus a {$S_{n}$} representation.
2018 rugsėjo 30 d., 21:33 atliko AndriusKulikauskas -
Pakeista 20 eilutė iš:
Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}.
į:
Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}. {$S^{\lambda}$} is a -submodule of {$M^{\lambda}$}, which is a left -module, and thus a {$S_{n}$} representation.
2018 rugsėjo 30 d., 21:28 atliko AndriusKulikauskas -
Pakeistos 13-20 eilutės iš
* [[https://en.wikipedia.org/wiki/Young_symmetrizer | Wikipedia: Young symmetrizer]]
į:
* [[https://en.wikipedia.org/wiki/Young_symmetrizer | Wikipedia: Young symmetrizer]]


[+Theory+]

{$S_{n}$} is the symmetric group on n letters.

Every irreducible representation of {$S_{n}$} is isomorphic to exactly one Specht module {$S^{\lambda}$}.
2018 rugsėjo 30 d., 21:24 atliko AndriusKulikauskas -
Pakeistos 12-13 eilutės iš
* [[https://en.wikipedia.org/wiki/Specht_module | Specht module]]
į:
* [[https://en.wikipedia.org/wiki/Specht_module | Wikipedia: Specht module]]
* [[https://en.wikipedia.org/wiki/Young_symmetrizer | Wikipedia: Young symmetrizer
]]
2018 rugsėjo 30 d., 21:23 atliko AndriusKulikauskas -
Pakeistos 11-12 eilutės iš
* [[http://math.iisc.ernet.in/~library/msp/MathMSP9.pdf | Irreducible Representations of the Symmetric Group and the General Linear Group]] Abhinav Verma
į:
* [[http://math.iisc.ernet.in/~library/msp/MathMSP9.pdf | Irreducible Representations of the Symmetric Group and the General Linear Group]] Abhinav Verma
* [[https://en.wikipedia.org/wiki/Specht_module | Specht module]]
2018 rugsėjo 30 d., 21:21 atliko AndriusKulikauskas -
Pakeista 5 eilutė iš:
'''Challenge: Calculating and interpreting the irreducible representations of the symmetric groups?'''
į:
'''Challenge: Calculating and interpreting the irreducible representations of the symmetric groups'''
2018 rugsėjo 30 d., 21:21 atliko AndriusKulikauskas -
Pakeistos 3-5 eilutės iš
į:
See: [[Math notebook]], [[Representation theory]]

'''Challenge: Calculating and interpreting the irreducible representations of the symmetric groups?'''
2018 rugsėjo 30 d., 21:20 atliko AndriusKulikauskas -
Pridėtos 1-9 eilutės:
>>bgcolor=#E9F5FC<<
--------------


--------------
>><<

Literature
* [[http://math.iisc.ernet.in/~library/msp/MathMSP9.pdf | Irreducible Representations of the Symmetric Group and the General Linear Group]] Abhinav Verma

SymmetricGroupRepresentations


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