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 * FIXME: summarize #5878
 1. '''Irreducible matrix representations of symmetric groups (Ticket #5878)'''. FrancoSaliola, based on the [[http://www-igm.univ-mlv.fr/~al|Alain Lascoux]] article
 [[http://phalanstere.univ-mlv.fr/~al/ARTICLES/ProcCrac.ps.gz|Young representations of the symmetric group]],
 added support for constructing irreducible representations of the symmetric group.

 Three types of representations have been implemented.

    * '''Specht representations'''. The matrices have integer entries.
    {{{
sage: chi = SymmetricGroupRepresentation([3,2])
Specht representation of the symmetric group corresponding to [3, 2]

sage: chi([5,4,3,2,1])
[ 1 -1 0 1 0]
[ 0 0 -1 0 1]
[ 0 0 0 -1 1]
[ 0 1 -1 -1 1]
[ 0 1 0 -1 1]
}}}

    * '''Young's seminormal representation'''. The matrices have rational entries.
    {{{
sage: snorm = SymmetricGroupRepresentation([2,1], "seminormal")
sage: snorm
Seminormal representation of the symmetric group corresponding to [2, 1]

sage: snorm([1,3,2])
[-1/2 3/2]
[ 1/2 1/2]
}}}

    * '''Young's orthogonal representation''' (the matrices are orthogonal). These matrices are defined over Sage's {{{Symbolic Ring}}}.
    {{{
sage: ortho = SymmetricGroupRepresentation([3,2], "orthogonal")
sage: ortho
Orthogonal representation of the symmetric group corresponding to [3, 2]

sage: ortho([1,3,2,4,5])
[ 1 0 0 0 0]
[ 0 -1/2 1/2*sqrt(3) 0 0]
[ 0 1/2*sqrt(3) 1/2 0 0]
[ 0 0 0 -1/2 1/2*sqrt(3)]
[ 0 0 0 1/2*sqrt(3) 1/2]
}}}

 One can also create the {{{CombinatorialClass}}} of all irreducible matrix representations of a given symmetric group.
 Then particular representations can be created by providing partitions. For example:
    {{{
sage: chi = SymmetricGroupRepresentations(5)
sage: chi
Specht representations of the symmetric group of order 5! over Integer Ring

sage: chi([5]) # the trivial representation
Specht representation of the symmetric group corresponding to [5]
sage: chi([5])([2,1,3,4,5])
[1]

sage: chi([1,1,1,1,1]) # the sign representation
Specht representation of the symmetric group corresponding to [1, 1, 1, 1, 1]
sage: chi([1,1,1,1,1])([2,1,3,4,5])
[-1]

sage: chi([3,2])
Specht representation of the symmetric group corresponding to [3, 2]
sage: chi([3,2])([5,4,3,2,1])
[ 1 -1 0 1 0]
[ 0 0 -1 0 1]
[ 0 0 0 -1 1]
[ 0 1 -1 -1 1]
[ 0 1 0 -1 1]
}}}

 See the documentation {{{SymmetricGroupRepresentation?}}} and
 {{{SymmetricGroupRepresentations?}}} for more information and examples.

 1. '''Yang-Baxter Graphs (Ticket #5878)'''.
 Ticket #5878 (irreducible matrix representations of the symmetric group) also
 introduced support for Yang-Baxter graphs. Besides being used for constructing
 those representations, they can also be used to construct the Cayley graph
 of a finite group:
    {{{
sage: def left_multiplication_by(g):
... return lambda h : h*g

sage: G = AlternatingGroup(4)
sage: operators = [ left_multiplication_by(gen) for gen in G.gens() ]
sage: Y = YangBaxterGraph(root=G.identity(), operators=operators); Y
Yang-Baxter graph with root vertex ()
sage: Y.plot(edge_labels=False)
}}}

 and to construct the permutahedron:
    {{{
sage: from sage.combinat.yang_baxter_graph import SwapIncreasingOperator
sage: operators = [SwapIncreasingOperator(i) for i in range(3)]
sage: Y = YangBaxterGraph(root=(1,2,3,4), operators=operators); Y
Yang-Baxter graph with root vertex (1, 2, 3, 4)
sage: Y.plot()
}}}

 See the documentation {{{YangBaxterGraph?}}} for more information and
 examples.

Sage 4.1 Release Tour

Sage 4.1 was released on FIXME. For the official, comprehensive release note, please refer to FIXME. A nicely formatted version of this release tour can be found at FIXME. The following points are some of the foci of this release:

Algebra

  • FIXME: summarize #6362

Algebraic Geometry

  • FIXME: summarize #4290

Basic Arithmetic

  • FIXME: summarize #6083

Combinatorics

  1. Irreducible matrix representations of symmetric groups (Ticket #5878). FrancoSaliola, based on the Alain Lascoux article Young representations of the symmetric group, added support for constructing irreducible representations of the symmetric group. Three types of representations have been implemented.

    • Specht representations. The matrices have integer entries.

      sage: chi = SymmetricGroupRepresentation([3,2])
      Specht representation of the symmetric group corresponding to [3, 2]
      
      sage: chi([5,4,3,2,1])
      [ 1 -1  0  1  0]
      [ 0  0 -1  0  1]
      [ 0  0  0 -1  1]
      [ 0  1 -1 -1  1]
      [ 0  1  0 -1  1]
    • Young's seminormal representation. The matrices have rational entries.

      sage: snorm = SymmetricGroupRepresentation([2,1], "seminormal")
      sage: snorm
      Seminormal representation of the symmetric group corresponding to [2, 1]
      
      sage: snorm([1,3,2])
      [-1/2  3/2]
      [ 1/2  1/2]
    • Young's orthogonal representation (the matrices are orthogonal). These matrices are defined over Sage's Symbolic Ring.

      sage: ortho = SymmetricGroupRepresentation([3,2], "orthogonal")
      sage: ortho
      Orthogonal representation of the symmetric group corresponding to [3, 2]
      
      sage: ortho([1,3,2,4,5])
      [          1           0           0           0           0]
      [          0        -1/2 1/2*sqrt(3)           0           0]
      [          0 1/2*sqrt(3)         1/2           0           0]
      [          0           0           0        -1/2 1/2*sqrt(3)]
      [          0           0           0 1/2*sqrt(3)         1/2]

    One can also create the CombinatorialClass of all irreducible matrix representations of a given symmetric group. Then particular representations can be created by providing partitions. For example:

    • sage: chi = SymmetricGroupRepresentations(5)
      sage: chi
      Specht representations of the symmetric group of order 5! over Integer Ring
      
      sage: chi([5]) # the trivial representation
      Specht representation of the symmetric group corresponding to [5]
      sage: chi([5])([2,1,3,4,5])
      [1]
      
      sage: chi([1,1,1,1,1]) # the sign representation
      Specht representation of the symmetric group corresponding to [1, 1, 1, 1, 1]
      sage: chi([1,1,1,1,1])([2,1,3,4,5])
      [-1]
      
      sage: chi([3,2])
      Specht representation of the symmetric group corresponding to [3, 2]
      sage: chi([3,2])([5,4,3,2,1])
      [ 1 -1  0  1  0]
      [ 0  0 -1  0  1]
      [ 0  0  0 -1  1]
      [ 0  1 -1 -1  1]
      [ 0  1  0 -1  1]

    See the documentation SymmetricGroupRepresentation? and SymmetricGroupRepresentations? for more information and examples.

  2. Yang-Baxter Graphs (Ticket #5878). Ticket #5878 (irreducible matrix representations of the symmetric group) also introduced support for Yang-Baxter graphs. Besides being used for constructing those representations, they can also be used to construct the Cayley graph of a finite group:

    • sage: def left_multiplication_by(g):
      ...       return lambda h : h*g
      
      sage: G = AlternatingGroup(4)
      sage: operators = [ left_multiplication_by(gen) for gen in G.gens() ]
      sage: Y = YangBaxterGraph(root=G.identity(), operators=operators); Y
      Yang-Baxter graph with root vertex ()
      sage: Y.plot(edge_labels=False)
    and to construct the permutahedron:
    • sage: from sage.combinat.yang_baxter_graph import SwapIncreasingOperator
      sage: operators = [SwapIncreasingOperator(i) for i in range(3)]
      sage: Y = YangBaxterGraph(root=(1,2,3,4), operators=operators); Y
      Yang-Baxter graph with root vertex (1, 2, 3, 4)
      sage: Y.plot()

    See the documentation YangBaxterGraph? for more information and examples.

Commutative Algebra

Cryptography

  • FIXME: summarize #6164

Geometry

Graph Theory

  • FIXME: summarize #6085
  • FIXME: summarize #6258

Graphics

  • FIXME: summarize #6162

Group Theory

Interfaces

  • FIXME: summarize #4313

Linear Algebra

  • FIXME: summarize #6261
  • FIXME: summarize #5882

Miscellaneous

  • FIXME: summarize #3084
  • FIXME: summarize #6097
  • FIXME: summarize #6417

Modular Forms

Notebook

  • FIXME: summarize #5637

Number Theory

  • FIXME: summarize #6273
  • FIXME: summarize #5854
  • FIXME: summarize #6386

Numerical

  • FIXME: summarize #6200

Packages

  • FIXME: summarize #6359
  • FIXME: summarize #6196
  • FIXME: summarize #6276
  • FIXME: summarize #5517
  • FIXME: summarize #5854
  • FIXME: summarize #5866
  • FIXME: summarize #5867
  • FIXME: summarize #5868
  • FIXME: summarize #5869
  • FIXME: summarize #5870
  • FIXME: summarize #5872
  • FIXME: summarize #5874
  • FIXME: summarize #5875
  • FIXME: summarize #6281
  • FIXME: summarize #6470
  • FIXME: summarize #6470
  • FIXME: summarize #6492
  • FIXME: summarize #6408

P-adics

Quadratic Forms

Symbolics

  • FIXME: summarize #6421

Topology

ReleaseTours/sage-4.1 (last edited 2009-12-26 15:00:52 by Minh Nguyen)