Sometimes the result of an operation does not allow further computations with it. In many cases, then an error is signalled, and the computation is stopped.
This is not appropriate for some applications in character theory. For example, if one wants to induce a character of a group to a supergroup (see InducedClassFunction (72.9-3)) but the class fusion is only a parametrized map (see Chapter 73), there may be values of the induced character which are determined by the fusion map, whereas other values are not known.
For this and other situations, GAP provides the data type unknown. An object of this type, further on called an unknown, may stand for any cyclotomic (see Chapter 18), in particular its family (see 13.1) is CyclotomicsFamily.
Unknowns are parametrized by positive integers. When a GAP session is started, no unknowns exist.
The only ways to create unknowns are to call the function Unknown (74.1-1) or a function that calls it, or to do arithmetical operations with unknowns.
GAP objects containing unknowns will contain fixed unknowns when they are printed to files, i.e., function calls Unknown(n) instead of Unknown(). So be careful to read files printed in different GAP sessions, since there may be the same unknown at different places.
The rest of this chapter contains information about the unknown constructor, the category, and comparison of and arithmetical operations for unknowns. More is not known about unknowns in GAP.
‣ Unknown( [n] ) | ( operation ) |
Called without argument, Unknown returns a new unknown value, i.e., the first one that is larger than all unknowns which exist in the current GAP session.
Called with a positive integer n, Unknown returns the n-th unknown; if this did not exist yet, it is created.
‣ LargestUnknown | ( global variable ) |
LargestUnknown is the largest n that is used in any Unknown( n ) in the current GAP session. This is used in Unknown (74.1-1) which increments this value when asked to make a new unknown.
‣ IsUnknown( obj ) | ( category ) |
is the category of unknowns in GAP.
gap> Unknown(); List( [ 1 .. 20 ], i -> Unknown() );; Unknown(1) gap> Unknown(); # note that we have already created 21 unknowns. Unknown(22) gap> Unknown(2000); Unknown(); Unknown(2000) Unknown(2001) gap> LargestUnknown; 2001 gap> IsUnknown( Unknown ); IsUnknown( Unknown() ); false true
Unknowns can be compared via = and < with all cyclotomics and with certain other GAP objects (see 4.13). We have Unknown( n ) >= Unknown( m ) if and only if n >= m holds, unknowns are larger than all cyclotomics that are not unknowns.
gap> Unknown() >= Unknown(); Unknown(2) < Unknown(3); false true gap> Unknown() > 3; Unknown() > E(3); true true gap> Unknown() > Z(8); Unknown() > []; false false
The usual arithmetic operations +, -, * and / are defined for addition, subtraction, multiplication and division of unknowns and cyclotomics. The result will be a new unknown except in one of the following cases.
Multiplication with zero yields zero, and multiplication with one or addition of zero yields the old unknown. Note that division by an unknown causes an error, since an unknown might stand for zero.
As unknowns are cyclotomics, dense lists of unknowns and other cyclotomics are row vectors and they can be added and multiplied in the usual way. Consequently, lists of such row vectors of equal length are (ordinary) matrices (see IsOrdinaryMatrix (24.2-2)).
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