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| = 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 [[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. == 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 == |
