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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( `

in the current `n` )**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 4.12). We have `Unknown( `

if and only if `n` ) >= Unknown( `m` )

holds, unknowns are larger than all cyclotomics that are not unknowns.`n` >= `m`

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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