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

Algebraic Geometry

Basic Arithmetic

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

Geometry

Graph Theory

  1. FIXME: summarize #6085

  2. Improve accuracy of graph eigenvalues (Ticket #6258), Rob Beezer. New routines compute eigenvalues and eigenvectors of integer matrices more precisely than before. Rather than convert adjacency matrices of graphs to computations over the reals or complexes, this patch retains adjacency matrices as matrices over the integers, yielding more accurate and informative results for eigenvalues, eigenvectors, and eigenspaces.

    • Examples follow for a circuit on 8 vertices: 
      g = graphs.CycleGraph(8)
    • Integer eigenvalues are exact, irrational eigenvalues are more precise, making multiplicities easier to determine. 
      sage: g.spectrum()

[2, 1.414213562373095?, 1.414213562373095?, 0, 0, -1.414213562373095?, -1.414213562373095?, -2]
    • Similar comments apply to eigenvectors. 
      sage: g.eigenvectors()

[(2, [
(1, 1, 1, 1, 1, 1, 1, 1)
], 1),
 (-2, [
(1, -1, 1, -1, 1, -1, 1, -1)
], 1),
 (0, [
(1, 0, -1, 0, 1, 0, -1, 0),
(0, 1, 0, -1, 0, 1, 0, -1)
], 2),
 (-1.414213562373095?,
  [(1, 0, -1, 1.414213562373095?, -1, 0, 1, -1.414213562373095?),
   (0, 1, -1.414213562373095?, 1, 0, -1, 1.414213562373095?, -1)],
  2),
 (1.414213562373095?,
  [(1, 0, -1, -1.414213562373095?, -1, 0, 1, 1.414213562373095?),
   (0, 1, 1.414213562373095?, 1, 0, -1, -1.414213562373095?, -1)],
  2)]

    • Eigenspaces are exact, in that they can be expressed as vector spaces over number fields. When the defining polynomial has several roots, the eigenspaces are not repeated. Previously eigenspaces were "fractured" owing to slight computational differences in identical eigenvalues. In concert with eigenvectors() this command illuminates the structure of a graph's eigenspaces more than purely numerical results. 

      sage: g.eigenspaces()

[
(2, Vector space of degree 8 and dimension 1 over Rational Field
User basis matrix:
[1 1 1 1 1 1 1 1]),
(-2, Vector space of degree 8 and dimension 1 over Rational Field
User basis matrix:
[ 1 -1  1 -1  1 -1  1 -1]),
(0, Vector space of degree 8 and dimension 2 over Rational Field
User basis matrix:
[ 1  0 -1  0  1  0 -1  0]
[ 0  1  0 -1  0  1  0 -1]),
(a3, Vector space of degree 8 and dimension 2 over Number Field in a3 with defining polynomial x^2 - 2
User basis matrix:
[  1   0  -1 -a3  -1   0   1  a3]
[  0   1  a3   1   0  -1 -a3  -1])
    • Complex eigenvalues (of digraphs) previously were missing their imaginary parts. This bug has been fixed as part of this ticket.

Graphics

Group Theory

Interfaces

Linear Algebra

Miscellaneous

Modular Forms

Notebook

Number Theory

Numerical

Packages

P-adics

Quadratic Forms

Symbolics

Topology