Lines Matching refs:of
3 ## This file is part of GAP, a system for computational discrete algebra.
6 ## Copyright of GAP belongs to its developers, whose names are too numerous
11 ## This file contains the declaration of those functions that are needed to
22 #T (check whether <pc> can be a permutation character of <tbl>;
23 #T use also the kernel of <pc>, i.e., check whether the kernel factor
24 #T of <pc> can be a permutation character of the factor of <tbl> by the
25 #T kernel; one example where this helps is the sum of characters of S3
39 ## <E>permutation character</E> of the operation of <M>G</M>
40 ## on the right cosets of <M>H</M>.
41 ## If only the character table of <M>G</M> is available and not the group
43 ## one can try to get information about possible subgroups of <M>G</M>
44 ## by inspection of those <M>G</M>-class functions that might be
53 ## <M>\pi</M> is a character of <M>G</M>,
73 ## the multiplicity of any rational irreducible <M>G</M>-character
74 ## <M>\psi</M> as a constituent of <M>\pi</M> is at most
79 ## <M>\pi(g) = 0</M> if the order of <M>g</M> does not divide
91 ## where <M>|Gal_G(g)|</M> denotes the number of conjugacy classes
92 ## of <M>G</M> that contain generators of the group
100 ## where <M>s</M> is the number of elements of order <M>p</M> in the
103 ## (Note that <M>s/(p-1)</M> equals the number of Sylow <M>p</M>
112 ## the character table of <M>G</M>;
113 ## clearly (d) holds for all integers if it holds for all prime divisors of
138 ## Let <A>tbl</A> be the ordinary character table of the group <M>G</M>,
140 ## for a subgroup <M>U</M> of <M>G</M>, or a list of such permutation
149 ## which equals the number of those elements of <M>U</M>
150 ## that are contained in class <M>i</M> of <A>tbl</A>,
158 ## which divides the class length in <M>U</M> of an element in class <M>i</M>
159 ## of <A>tbl</A>,
163 ## a record that can be used as second argument of <Ref Oper="Display"/>
166 ## for those classes where at least one character of <A>permchars</A> is
171 ## a list of strings describing the decomposition of the permutation
172 ## characters in <A>permchars</A> into the irreducible characters of
177 ## which indicate the successive irreducible characters of <A>tbl</A>
178 ## of that degree,
180 ## A sequence of small letters (not necessarily distinct) after a single
181 ## number indicates a sum of irreducible constituents all of the same
243 ## Let <A>tbl</A> and <A>tbl2</A> be the ordinary character tables of two
245 ## where <M>H</M> is of index two in <M>G</M>,
247 ## for a subgroup <M>U</M> of <M>G</M>,
248 ## or a list of such permutation characters.
251 ## entries of the <C>ATLAS</C> component are names relative to <A>tbl</A>.
253 ## More precisely, the <M>i</M>-th entry of the <C>ATLAS</C> component is a
254 ## string describing the decomposition of the <M>i</M>-th entry in
256 ## The degrees and distinguishing letters of the constituents refer to
257 ## the irreducibles of <A>tbl</A>, as follows.
258 ## The two irreducible characters of <A>tbl2</A> of degree <M>N</M>, say,
259 ## that extend the irreducible character <M>N</M> <C>a</C> of <A>tbl</A>
261 ## The irreducible character of <A>tbl2</A> of degree <M>2N</M>, say, whose
262 ## restriction to <A>tbl</A> is the sum of the irreducible characters
264 ## Multiplicities larger than <M>1</M> of constituents are denoted by
313 ## The first three of these functions implement tests of the properties of
316 ## The other two implement test of additional properties.
317 ## Let <A>tbl</A> be the ordinary character table of a group <M>G</M>, say,
318 ## <A>char</A> a rational character of <A>tbl</A>,
319 ## and <A>chars</A> a list of rational characters of <A>tbl</A>.
320 ## For applying <Ref Func="TestPerm5"/>, the knowledge of a <M>p</M>-modular
321 ## Brauer table <A>modtbl</A> of <M>G</M> is required.
326 ## The return values of the functions were chosen parallel to the tests
330 ## because of (T1) or (T2), respectively;
336 ## <Ref Func="TestPerm2"/> returns <C>1</C> if <A>char</A> fails because of
339 ## of (T3), (T4), or (T5), respectively;
340 ## these tests correspond to (g), a weaker form of (h), and (j).
343 ## <Ref Func="TestPerm3"/> returns the list of all those class functions in
345 ## this is a stronger version of (T6).
347 ## <Ref Func="TestPerm4"/> returns the list of all those class functions in
349 ## <M>p</M> of the order of <M>G</M>;
351 ## knowledge of decomposition matrices
354 ## (T8) implements the test of the fact that in the case that <M>p</M>
355 ## divides <M>|G|</M> and the degree of a transitive permutation character
357 ## the projective cover of the trivial character is a summand of <M>\pi</M>.
360 ## Given a permutation character <M>\pi</M> of a group <M>G</M> and a prime
362 ## the restriction <M>\pi_B</M> to a <M>p</M>-block <M>B</M> of <M>G</M> has
364 ## For each <M>g \in G</M> such that <M>g^n</M> is a <M>p</M>-element of
370 ## <A>modtbl</A> of <M>G</M>, for some prime <M>p</M> dividing the order of
373 ## is divisible by the <M>p</M>-part of the order of <M>G</M> can be
375 ## <Ref Func="TestPerm5"/> returns the sublist of all those characters in
380 ## <!-- This is the check whether the cycle structure of elements is well-defined;-->
381 ## <!-- the check is superfluous (at least) for elements of prime power order-->
382 ## <!-- or order equal to the product of two primes (see <Cite Key="NPP84"/>);-->
383 ## <!-- note that by construction, the numbers of <Q>cycles</Q> are always integral,-->
446 ## The algorithm is selected from the choice of the optional argument
451 ## Regardless of the algorithm used in a specific case,
452 ## <Ref Func="PermChars"/> returns a list of <E>all</E>
455 ## There is no guarantee that a character of this list is in fact
458 ## with these properties (e.g., of a certain degree).
461 ## <Ref Func="PermChars"/> returns the list of all possible permutation
462 ## characters of the group with this character table.
468 ## a system of inequalities that guides the search
470 ## So the following commands compute the list of 39 possible permutation
471 ## characters of the Mathieu group <M>M_{11}</M>.
484 ## is a divisor of the order of the group.
486 ## whether a character of a certain degree can lie
487 ## in the currently searched part of the search tree.
488 ## To choose this strategy, enter a record as the second argument of
490 ## and set its component <C>degree</C> to the range of degrees
491 ## (which might also be a range containing all divisors of the group order)
497 ## <Ref Func="PermChars"/> returns the list of all
498 ## possible permutation characters of <A>tbl</A> that have degree
503 ## in which the possible permutation characters of the given degree
505 ## The algorithm is described at the end of
507 ## Note that inverting big integer matrices needs a lot of time and space.
519 ## <Ref Func="PermChars"/> returns the list of all
521 ## the components of this record.
523 ## If <A>cond</A> contains a degree as value of the record component
533 ## For the meaning of additional components of <A>cond</A> besides
536 ## Instead of <C>degree</C>, <A>cond</A> may have the component <C>torso</C>
537 ## bound to a list that contains some known values of the required
542 ## The component <C>chars</C>, if present, holds a list of all those
543 ## <E>rational</E> irreducible characters of <A>tbl</A> that might be
544 ## constituents of the required characters.
547 ## <E>all</E> rational irreducible characters of <A>tbl</A>,
548 ## &GAP; checks whether the scalar products of all class functions in the
549 ## result list with the omitted rational irreducible characters of
552 ## that is known to be not a constituent of the desired possible permutation
567 ## with values a list of class positions of a normal subgroup <M>N</M> of
568 ## the group <M>G</M> of <A>tbl</A> and a possible permutation character
569 ## <M>\pi</M> of <M>G</M>, respectively, such that <M>N</M> is contained in
570 ## the kernel of <M>\pi</M>.
571 ## In this case, <Ref Func="PermChars"/> returns the list of those possible
572 ## permutation characters <M>\psi</M> of <A>tbl</A> coinciding with
574 ## and having the property that no irreducible constituent of
578 ## An interpretation of the computed characters is the following.
579 ## Suppose there exists a subgroup <M>V</M> of <M>G</M> such that
581 ## Then <M>N \leq V</M>, and if a computed character is of the form
582 ## <M>(1_U)^G</M>, for a subgroup <M>U</M> of <M>G</M>, then <M>V = UN</M>.
621 ## Let <A>tbl</A> be the ordinary character table of a group <M>G</M>.
623 ## character <M>\pi</M> of <M>G</M> places restrictions on the
624 ## multiplicities <M>a_i</M> of the irreducible constituents <M>\chi_i</M>
625 ## of <M>\pi = \sum_{{i = 1}}^r a_i \chi_i</M>.
630 ## This system of inequalities is kind of diagonalized,
631 ## resulting in a system of inequalities restricting <M>a_i</M>
632 ## in terms of <M>a_j</M>, <M>j < i</M>.
636 ## this information from the <C>ineq</C> component of its argument record.
638 ## The number of inequalities arising in the process of diagonalization may
647 ## third argument <A>option</A>, tries to keep the number of intermediate
648 ## inequalities small by eventually changing the order of characters.
681 ## <C>Permut</C> computes possible permutation characters of the character table
682 ## <A>tbl</A> by the algorithm that solves a system of inequalities.
690 ## the result of <Ref Func="Inequalities"/>,
693 ## the list of degrees for which the possible permutation characters
695 ## this will lead to a speedup only if the range of degrees is
714 ## Let <A>tbl</A> be the ordinary character table of the group <M>G</M>.
719 ## <Ref Func="PermBounds"/> computes the boundary points of this simplex for
722 ## (Some conditions from the power maps of <A>tbl</A> are also involved.)
723 ## For this purpose, a matrix similar to the rational character table of
727 ## (see <Ref Sect="Possible Permutation Characters"/>) of a given
730 ## this information from the <C>bounds</C> component of its argument record.
747 ## <Ref Func="PermComb"/> computes possible permutation characters of the
749 ## described at the end of <Cite Key="BP98" Where="Section 3.2"/>.
752 ## better bounds (i.e., when the computation of bounds shall be suppressed),
761 ## A list of upper bounds on the multiplicities of the rational irreducibles
779 ## <C>PermCandidates</C> computes possible permutation characters of the
785 ## constituents, and that they are all completions of <A>torso</A>.
786 ## (Note that scalar products with rational irreducible characters of
790 ## of <A>tbl</A>.)
792 ## Known values of the candidates must be nonnegative integers in
793 ## <A>torso</A>, the other positions of <A>torso</A> are unbound;
812 ## computes certain possible permutation characters of the character table
813 ## <A>tbl</A> with a generalization of the strategy
823 ## they are completions of <A>torso</A>, and
831 ## Known values of the candidates must be nonnegative integers in
832 ## <A>torso</A>, the other positions of <A>torso</A> are unbound;