Macaulay2 » Documentation
Packages » SpechtModule :: SpechtModuleElement
next | previous | forward | backward | up | index | toc

SpechtModuleElement -- the class of Specht Module elements

Description

Polytabloids of shape $p$ are elements of the module of tabloids of the form $\sum_{\tau \in C(T)}\sum_{\sigma \in R(T)}sgn(\tau) \tau\sigma(T)$ where T is a tabloid of shape $p$.

The set of polytabloids generates the Specht Module of shape $p$.

In other words the element in a SpechtModule are linear combinations of polytabloids. This is the way such elements are implemented in this package.

The constructor takes just one polytabloid and a coefficient

i1 : p = new Partition from {3,2,1}

o1 = Partition{3, 2, 1}

o1 : Partition
 
i2 : y = youngTableau(p,{2,0,3,4,5,1})

o2 = | 2 0 3 |
     | 4 5 |
     | 1 |

o2 : YoungTableau
 
i3 : e = spechtModuleElement(y,-2)

o3 = -2 | 2 0 3 |
        | 4 5 |
        | 1 |

o3 : SpechtModuleElement

More complex elements can be made by adding or subtracting previously build elements and multiplying by any element of the base field (which is assumed to be \mathbb{Q}).

i4 : y2 = youngTableau(p,{5,0,2,4,1,3})

o4 = | 5 0 2 |
     | 4 1 |
     | 3 |

o4 : YoungTableau
 
i5 : e2 = spechtModuleElement(y2)

o5 = | 5 0 2 |
     | 4 1 |
     | 3 |

o5 : SpechtModuleElement
 
i6 : e+e2

o6 = -2 | 2 0 3 | + | 5 0 2 |
        | 4 5 |     | 4 1 |
        | 1 |       | 3 |

o6 : SpechtModuleElement
 
i7 : e-e2

o7 = -2 | 2 0 3 | - | 5 0 2 |
        | 4 5 |     | 4 1 |
        | 1 |       | 3 |

o7 : SpechtModuleElement
 
i8 : 3*oo

o8 = -6 | 2 0 3 | - 3 | 5 0 2 |
        | 4 5 |       | 4 1 |
        | 1 |         | 3 |

o8 : SpechtModuleElement

The element SpechtModuleElement is implemented as a MutableHashTable. The keys are the filling of the tableaux that label the polytabloids and they point to their respective coefficients

i9 : peek oo

o9 = SpechtModuleElement{partition => Partition{3, 2, 1}    }
                         values => MutableHashTable{...2...}
 
i10 : peek ooo#values

o10 = MutableHashTable{{2, 0, 3, 4, 5, 1} => -6}
                       {5, 0, 2, 4, 1, 3} => -3

The method terms is used to retrieve the polytabloid with their respective coefficient. This is given as a list of pairs of tableaux and coefficients.

i11 : terms (3*(e-e2))

o11 = {(| 2 0 3 |, -6), (| 5 0 2 |, -3)}
        | 4 5 |          | 4 1 |
        | 1 |            | 3 |

o11 : List

A method was implemented to apply a permutation to a SpechtModuleElement. The action is defined by permuting the entries of the tableaux that label the polytabloids.

i12 : {0,1,2,3,4,5} (3*(e-e2))

o12 = -6 | 2 0 3 | - 3 | 5 0 2 |
         | 4 5 |       | 4 1 |
         | 1 |         | 3 |

o12 : SpechtModuleElement
 
i13 : {1,0,2,3,4,5} (3*(e-e2))

o13 = -6 | 2 1 3 | - 3 | 5 1 2 |
         | 4 5 |       | 4 0 |
         | 0 |         | 3 |

o13 : SpechtModuleElement

See also

Functions and methods returning an object of class SpechtModuleElement :

Methods that use an object of class SpechtModuleElement :

For the programmer

The object SpechtModuleElement is a type, with ancestor classes HashTable < Thing.