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3 Weights and Characters

This chapter documents additional functions available for computation of weights and characters.

3.1 Weights

3.1-1 Weight

Returns: a list of integers

The weight of the given element elt is calculated and returned.

gap> V:= WeylModule(3,[3,3],"A",2);
V[ 3, 3 ]
gap> m:= MaximalVectors(V);
[ 1*v0, y1*v0, y2*v0, y1^(2)*y2*v0, -1*y1*y2^(2)*v0+y2*y3*v0, 
  y1*y2*y3*v0+y1^(2)*y2^(2)*v0 ]
gap> Weight(m[2]);
[ 1, 4 ]
gap> List(m,Weight);
[ [ 3, 3 ], [ 1, 4 ], [ 4, 1 ], [ 0, 3 ], [ 3, 0 ], [ 1, 1 ] ]

NOTE. The above trick of applying the Weight function across an entire list lst of vectors, with the command List(lst, Weight), is very useful in many situations. This capability is built in to the List function in GAP.

3.2 Characters

We have already seen the function Character (2.3-2), that computes the (formal) character of a given Weyl module, quotient, submodule, or subquotient. We now consider some additional functions for computing characters.

3.2-1 DecomposeCharacter

Returns: a list (of simple highest weights and their multiplicities)

If ch is a given character (of some module) then this function computes the multiplicities of the simple characters in ch, thus obtaining the decomposition numbers of the module. Here it is necessary to specify the characteristic p and root system (of type typ and rank rk) for the simple characters. For instance, this can be used to decompose tensor products.

gap> V:= WeylModule(2,[2,0],"A",2);
V[ 2, 0 ]
gap> ch:= ProductCharacter(Character(V),Character(V));
[ [ 4, 0 ], 1, [ 2, 1 ], 2, [ 3, -1 ], 2, [ 0, 2 ], 3, [ 1, 0 ], 4, 
  [ 2, -2 ], 3, [ -2, 3 ], 2, [ -1, 1 ], 4, [ 0, -1 ], 4, [ 1, -3 ], 2, 
  [ -4, 4 ], 1, [ -3, 2 ], 2, [ -2, 0 ], 3, [ -1, -2 ], 2, [ 0, -4 ], 1 ]
gap> DecomposeCharacter(ch,2,"A",2);
[ [ 4, 0 ], 1, [ 2, 1 ], 2, [ 0, 2 ], 3, [ 1, 0 ], 2 ]

3.2-2 DifferenceCharacter

Returns: a list (a character)

If ch1 and ch2 are given characters, this function returns their formal difference character. It is used in the definition of the DecomposeCharacter function.

gap> DifferenceCharacter(Character(V),Character(V));
[  ]

The empty list here implements the zero character.

3.2-3 ProductCharacter

Returns: a list (a character)

Returns the product character of its inputs ch1 and ch2. If ch1 and ch2 are characters of modules then the output of this function is the character of the tensor product of the modules.

gap> V:= WeylModule(2,[2,0],"A",2);
V[ 2, 0 ]
gap> ch:= ProductCharacter(Character(V),Character(V));
[ [ 4, 0 ], 1, [ 2, 1 ], 2, [ 3, -1 ], 2, [ 0, 2 ], 3, [ 1, 0 ], 4, 
  [ 2, -2 ], 3, [ -2, 3 ], 2, [ -1, 1 ], 4, [ 0, -1 ], 4, [ 1, -3 ], 2, 
  [ -4, 4 ], 1, [ -3, 2 ], 2, [ -2, 0 ], 3, [ -1, -2 ], 2, [ 0, -4 ], 1 ]

By applying the function DecomposeCharacter (3.2-1) we can decompose tensor products in positive characteristic.

3.2-4 SimpleCharacter

Returns: a list (a character)

Computes the simple character of highest weight wt in characteristic p. The arguments typ and rk specify the type and rank of the underlying root system. The function uses Steinberg's tensor product theorem.

gap> SimpleCharacter(2,[2,0],"A",2);
[ [ 2, 0 ], 1, [ -2, 2 ], 1, [ 0, -2 ], 1 ]

Another way to compute the same result is to compute the Character of the output of SimpleQuotient(V), where V is the WeylModule in the same characteristic and root system with the same highest weight.

gap> V:= WeylModule(2,[2,0],"A",2);
V[ 2, 0 ]
gap> Character(SimpleQuotient(V));
[ [ 2, 0 ], 1, [ -2, 2 ], 1, [ 0, -2 ], 1 ]

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