94 Partitions and symbols

The functions described below, used in various parts of the CHEVIE\ package, sometimes duplicate or have similar functions to some functions in other packages (like the SPECHT package). It is hoped that a review of this area will be done in the future.

The combinatorial objects dealt with here are partitions, beta numbers and symbols. A partition in CHEVIE is a decreasing list of strictly positive integers p1 ≥ p2 ≥ ... pn>0, represented as a GAP list. A beta-number list is a list of strictly increasing positive integers, up to the shift equivalence relation. A shift is the equivalence relation which is the transitive closure of the elementary equivalence of [s1,...,sn] and [0,1+s1,...,1+sn]. It is clear that an equivalence class has exactly one member which does not contain 0: it is called the normalized beta-list. To a partition is associated the list of its beta-numbers, which is the list whose normalized representative is pn,pn-1+1,...,p1+n-1. Conversely, to each beta-list is associated a partition, the one associated by the above formula to its normalized representative.

A symbol is a tuple S=[S1,..,Sn] of beta lists, taken modulo the equivalence relation generated by two elementary equivalences: the simultaneous shift of all lists, and the cyclic permutation of the tuple (in the particular case where n=2 it is thus an unordered pair of lists). This time there is a unique normalized symbol where 0 is not in the intersection of the Si. A basic invariant attached to symbols is the shape List(S,Length); when n=2 one can assume that S1 has at least the same length as S2 and the difference of cardinals Length(S[1])-Length(S[2]), called the defect, is then invariant by shift. Another invariant by shift in general is the rank, defined as

Sum(S,Sum)-QuoInt((Sum(S,Length)-1)*(Sum(S,Length)-Length(S)+1),2*Length(S))

Partitions and pairs of partitions are parameters for characters of the Weyl groups of classical types, and tuples of partitions are parameters for characters of imprimitive complex reflection groups. Symbols with two lines are parameters for the unipotent characters of classical Chevalley groups, and more general symbols for the unipotent characters of Spetses associated to complex reflection groups. The rank of the symbol is the semi-simple rank of the corresponding Chevalley group or Spetses.

Symbols of rank n and defect 0 parameterize characters of the Weyl group of type Dn, and symbols of rank n and defect divisible by 4 parameterize unipotent characters of split orthogonal groups of dimension 2n. Symbols of rank n and defect congruent to 2 (mod 4) parameterize unipotent characters of non-split orthogonal groups of dimension 2n. Symbols of rank n and defect 1 parameterize characters of the Weyl group of type Bn, and finally symbols of rank n and odd defect parameterize unipotent characters of symplectic groups of dimension 2n or orthogonal groups of dimension 2n+1.

Subsections

  1. BetaSet
  2. PartBeta
  3. ShiftBeta
  4. PartitionTupleToString
  5. Tableaux
  6. DefectSymbol
  7. RankSymbol
  8. Symbols
  9. SymbolsDefect
  10. CycPolGenericDegreeSymbol
  11. CycPolFakeDegreeSymbol
  12. LowestPowerGenericDegreeSymbol
  13. HighestPowerGenericDegreeSymbol

94.1 BetaSet

BetaSet( p )

Here p is a partition (a non-increasing list of positive integers). BetaSet the corresponding reduced list of Beta numbers (see the introduction of the section for definitions).

    gap> BetaSet([3,3,1]);
      [ 1, 4, 5 ]

This function does not require "chevie", it is part of the GAP library. However, as it is not documented elsewhere we document it here.

94.2 PartBeta

PartBeta( b )

Here b is a list of integers representing a list of beta-numbers. PartBeta returns the partition associated to the corresponding normalized beta-list (see the introduction of the section for definitions).

    gap> PartBeta([0,4,5]);
      [ 3, 3 ]

This function requires the package "chevie" (see RequirePackage).

94.3 ShiftBeta

ShiftBeta( b, n )

Here b is a list of integers representing a list of beta-numbers. ShiftBeta returns the list shifted by n (see the introduction of the section for definitions).

    gap> ShiftBeta([4,5],3);
      [ 0, 1, 2, 7, 8 ]

This function requires the package "chevie" (see RequirePackage).

94.4 PartitionTupleToString

PartitionTupleToString( tuple )

converts the partition tuple tuple to a string where the partitions are separated by a dot.

    gap> d:=PartitionTuples(3,2);
    [ [ [ 1, 1, 1 ], [  ] ], [ [ 1, 1 ], [ 1 ] ], [ [ 1 ], [ 1, 1 ] ],
      [ [  ], [ 1, 1, 1 ] ], [ [ 2, 1 ], [  ] ], [ [ 1 ], [ 2 ] ],
      [ [ 2 ], [ 1 ] ], [ [  ], [ 2, 1 ] ], [ [ 3 ], [  ] ],
      [ [  ], [ 3 ] ] ]
    gap>  for i in d do
    >      Print( PartitionTupleToString( i ),"   ");
    >  od; Print("\n");
    111.   11.1   1.11   .111   21.   1.2   2.1   .21   3.   .3

This function requires the package "chevie" (see RequirePackage).

94.5 Tableaux

Tableaux(tuple)

returns the list of standrad tableaux associated to the partition tuple tuple, that is a filling of the associated young diagrams with the numbers [1..Sum(tuple,Sum)] such that the numbers increase across the rows and down the columns.

   gap> Tableaux([[2,1],[1]]);
   [ [ [ [ 2, 4 ], [ 3 ] ], [ [ 1 ] ] ], 
     [ [ [ 1, 4 ], [ 3 ] ], [ [ 2 ] ] ],
     [ [ [ 1, 4 ], [ 2 ] ], [ [ 3 ] ] ], 
     [ [ [ 2, 3 ], [ 4 ] ], [ [ 1 ] ] ],
     [ [ [ 1, 3 ], [ 4 ] ], [ [ 2 ] ] ], 
     [ [ [ 1, 2 ], [ 4 ] ], [ [ 3 ] ] ],
     [ [ [ 1, 3 ], [ 2 ] ], [ [ 4 ] ] ], 
     [ [ [ 1, 2 ], [ 3 ] ], [ [ 4 ] ] ] ]

This function requires the package "chevie" (see RequirePackage).

94.6 DefectSymbol

DefectSymbol( s )

Let s=[S,T] be a symbol given as a pair of lists (see the introduction to the section). DefectSymbol returns the defect of s, equal to Length(S)-Length(T).

    gap> DefectSymbol([[1,2],[1,5,6]]);
      -1

This function requires the package "chevie" (see RequirePackage).

94.7 RankSymbol

RankSymbol( s )

Let s=[S1,..,Sn] be a symbol given as a tuple of lists (see the introduction to the section). RankSymbol returns the rank of s.

    gap> RankSymbol([[1,2],[1,5,6]]);
      11

This function requires the package "chevie" (see RequirePackage).

94.8 Symbols

Symbols( n, d )

Returns the list of all two-line symbols of defect d and rank n (see the introduction for definitions). If d=0 the symbols with equal parts are returned twice, so that Symbols(n, 0) returns a set of parameters for the characters of the Weyl group of type Dn.

    gap> Symbols(2,1);
     [ [ [ 1, 2 ], [ 0 ] ], [ [ 0, 2 ], [ 1 ] ], [ [ 0, 1, 2 ], [ 1, 2 ] ],
       [ [ 2 ], [  ] ], [ [ 0, 1 ], [ 2 ] ] ]
     gap> Symbols(4,0);
     [ [ [ 1, 2 ], [ 1, 2 ] ], [ [ 1, 2 ], [ 1, 2 ] ],
       [ [ 0, 1, 3 ], [ 1, 2, 3 ] ], [ [ 0, 1, 2, 3 ], [ 1, 2, 3, 4 ] ],
       [ [ 1, 2 ], [ 0, 3 ] ], [ [ 0, 2 ], [ 1, 3 ] ],
       [ [ 0, 1, 2 ], [ 1, 2, 4 ] ], [ [ 2 ], [ 2 ] ], [ [ 2 ], [ 2 ] ],
       [ [ 0, 1 ], [ 2, 3 ] ], [ [ 1 ], [ 3 ] ], [ [ 0, 1 ], [ 1, 4 ] ],
       [ [ 0 ], [ 4 ] ] ]

This function requires the package "chevie" (see RequirePackage).

94.9 SymbolsDefect

SymbolsDefect( e, r, def , inh)

Returns the list of symbols defined by Malle for Unipotent characters of imprimitive Spetses. Returns e-symbols of rank r, defect def (equal to 0 or 1) and content equal to inh modulo e. Thus the symbols for unipotent characters of G(d,1,r) are given by SymbolsDefect(d,r,0,1) and those for unipotent characters of G(e,e,r) by SymbolsDefect(e,r,0,0).

   gap> SymbolsDefect(3,2,0,1);
   [ [ [ 1, 2 ], [ 0 ], [ 0 ] ], [ [ 0, 2 ], [ 1 ], [ 0 ] ],
     [ [ 0, 2 ], [ 0 ], [ 1 ] ], [ [ 0, 1, 2 ], [ 1, 2 ], [ 0, 1 ] ],
     [ [ 0, 1 ], [ 1 ], [ 1 ] ], [ [ 0, 1, 2 ], [ 0, 1 ], [ 1, 2 ] ],
     [ [ 2 ], [  ], [  ] ], [ [ 0, 1 ], [ 2 ], [ 0 ] ],
     [ [ 0, 1 ], [ 0 ], [ 2 ] ], [ [ 1 ], [ 0, 1, 2 ], [ 0, 1, 2 ] ],
     [ [  ], [ 0, 2 ], [ 0, 1 ] ], [ [  ], [ 0, 1 ], [ 0, 2 ] ],
     [ [ 0 ], [  ], [ 0, 1, 2 ] ], [ [ 0 ], [ 0, 1, 2 ], [  ] ] ]
   gap> List(last,StringSymbol);
   [ "(12,0,0)", "(02,1,0)", "(02,0,1)", "(012,12,01)", "(01,1,1)",
     "(012,01,12)", "(2,,)", "(01,2,0)", "(01,0,2)", "(1,012,012)", "(,02,01)",
     "(,01,02)", "(0,,012)", "(0,012,)" ]
   gap> SymbolsDefect(3,3,0,0);
   [ [ [ 1 ], [ 1 ], [ 1 ] ], [ [ 0, 1 ], [ 1, 2 ], [ 0, 2 ] ],
     [ [ 0, 1 ], [ 0, 2 ], [ 1, 2 ] ], [ [ 0, 1, 2 ], [ 0, 1, 2 ], [ 1, 2, 3 ] ],
     [ [ 0 ], [ 1 ], [ 2 ] ], [ [ 0 ], [ 2 ], [ 1 ] ],
     [ [ 0, 1 ], [ 0, 1 ], [ 1, 3 ] ], [ [ 0 ], [ 0 ], [ 3 ] ],
     [ [ 0, 1, 2 ], [  ], [  ] ], [ [ 0, 1, 2 ], [ 0, 1, 2 ], [  ] ] ]
   gap> List(last,StringSymbol);
   [ "(1,1,1)", "(01,12,02)", "(01,02,12)", "(012,012,123)", "(0,1,2)",
     "(0,2,1)", "(01,01,13)", "(0,0,3)", "(012,,)", "(012,012,)" ]

This function requires the package "chevie" (see RequirePackage).

94.10 CycPolGenericDegreeSymbol

CycPolGenericDegreeSymbol( s )

Let s=[S1,..,Sn] be a symbol given as a tuple of lists (see the introduction to the section). CycPolGenericDegreeSymbol returns as a CycPol the generic degree of the unipotent character parameterized by s.

    gap> CycPolGenericDegreeSymbol([[1,2],[1,5,6]]);
     (1/2)q^13P5P6P7P8^2P9P10P11P14P16P18P20P22

This function requires the package "chevie" (see RequirePackage).

94.11 CycPolFakeDegreeSymbol

CycPolFakeDegreeSymbol( s )

Let s=[S1,..,Sn] be a symbol given as a tuple of lists (see the introduction to the section). CycPolFakeDegreeSymbol returns as a CycPol the fake degree of the unipotent character parameterized by s.

    gap> CycPolFakeDegreeSymbol([[1,5,6],[1,2]]);
      q^16P5P7P8P9P10P11P14P16P18P20P22

This function requires the package "chevie" (see RequirePackage).

94.12 LowestPowerGenericDegreeSymbol

LowestPowerGenericDegreeSymbol( s )

Let s=[S1,..,Sn] be a symbol given as a pair of lists (see the introduction to the section). LowestPowerGenericDegreeSymbol returns the valuation of the generic degree of the unipotent character parameterized by s.

    gap> LowestPowerGenericDegreeSymbol([[1,2],[1,5,6]]);
     13

This function requires the package "chevie" (see RequirePackage).

94.13 HighestPowerGenericDegreeSymbol

HighestPowerGenericDegreeSymbol( s )

Let s=[S1,..,Sn] be a symbol given as a pair of lists (see the introduction to the section). HighestPowerGenericDegreeSymbol returns the degree of the generic degree of the unipotent character parameterized by s.

    gap> HighestPowerGenericDegreeSymbol([[1,5,6],[1,2]]);
      91

This function requires the package "chevie" (see RequirePackage). Previous Up Next
Index

GAP 3.4.4
April 1997