COMMENT:
	The HICCUP.C module deals with the generation of explicit irreducible
	matrix representations of the Hecke algebra H_n of type A.
	In addition, routines are provided which enable calculations in
	H_n to be carried out.

	This file gives a very brief introduction to the irreducible
	representations of H_n, describes aspects of their implementation in
	Symmetrica, describes the pertinent library functions, and finally
	describes the example programs.


	In the case when q is not a root of unity, the irreducible representations
	of H_n are labelled by the partitions of n, and are provided here by
	the generalisation of the Specht modules of the symmetric group
	described in:
	  R.C. King & B.G. Wybourne, "Representations and Traces of the Hecke
	  Algebras H_n(q) of type A_{n-1}", J. Math. Phys. 33(1), pp4-14 (1992).

	Although the Specht modules remain valid representations of H_n when
	q is a root of unity, they may be reducible. In fact, if q is a
	primitive pth root of unity, a full set of irreducibles are labelled
	by the p-regular partitions of n. They may be obtained by factoring
	the corresponding Specht module by its maximal submodule.
	For those representations labelled by partitions with two parts, this
	was first described in the conference paper:
	  T.A. Welsh, "Two-rowed Hecke algebra representations at roots of
	  unity", Czech J. Phys. 45, pp283--291 (1996).
	A more extensive version which details the algorithms required to
	produce the representation matrices is available from the proceedings
	of the 35th Seminaire Lotharingien de Combinatoire at the
	WorldWideWeb site http://cartan.u-strasbg.fr:80/~slc/
	(a more comprehensive paper is on the cards - maybe 1996). The
	algorithms in this work are implemented here. They are very much
	based on the Specht modules.

	The Hecke algebra H_n is the unital associative algebra over the
	complex numbers, generated by h_1,h_2,...,h_{n-1} and subject to:
	  1. (h_i+1)(h_i-q)=0   (1 <= i <= n-1);
	  2. h_i h_{i+1} h_i = h_{i+1} h_i h_{i+1}   (1 <= i <= n-2);
	  3. h_i h_j = h_j h_i   (1 <= i < j-1 <= n-2).
	Here q is a fixed complex number.

	These relations enable a map h:S_n -> H_n to be unambiguously defined
	as follows. Let s_i be the simple transposition (i,i+1) of S_n, and
	for w in S_m let
	     w = s_{i_1} s_{i_2} s_{i_3} ... s_{i_l}
	be a reduced expression for w in that l is the length of w. Then define
	     h(w) = h_{i_1} h_{i_2} h_{i_3} ... h_{i_l}.
	In fact, the images of all w in S_n provide a basis for H_n
	(moreover, if q is not a root of unity, then H_n and the group algebra
	of S_n over the complex numbers are isomorphic). This is made use of
	in the routines, where elements of H_n are stored as q-linear
	combinations of permutations: a permutation corresponds to the
	element of H_n obtained from the map h.

	In terms of the objects of Symmetrica, an element of H_n is a linked
	LIST of MONOM objects. The "self" part of each MONOM is the
	PERMUTATION, and the "koeff" part is its coefficient - a MONOPOLY
	object representing a polynomial in the single variable q.
	The MONOPOLY object is itself a linked list of MONOMs representing
	the individual terms in the polynomial. Here both the "self" and
	"koeff" parts of the MONOM are INTEGER objects; the former giving
	the power of the term, and the latter its coefficient.
	Schematically, this data structure is as follows:

	  LIST --> LIST (next)
	       --> MONOM   --> PERMUTATION
			   --> MONOPOLY (LIST) --> MONOPOLY (next)
					       --> MONOM   --> INTEGER (power)
							   --> INTEGER (coefficient).

	In the constructions introduced above, the module corresponding to a
	particular partition is spanned by tableaux of the shape of that
	partition. Thus a data structure similar to the above is used to
	store elements of the module:

	  LIST --> LIST (next)
	       --> MONOM   --> TABLEAUX
			   --> MONOPOLY (LIST) --> MONOPOLY (next)
					       --> MONOM   --> INTEGER (power)
							   --> INTEGER (coefficient).

	Each module has a basis which is a subset of the full set of tableaux.
	For the Specht module, the basis elements are known as standard tableaux
	(or S_n - standard tableaux). For the root of unity cases a subset
	of the standard tableaux, known as p-root standard tableaux, provide
	a basis. At the heart of the module constructions are algorithms that
	enable a tableau which is not in the basis to be rewritten in terms of
	those tableaux that are. The action of an arbitrary element of H_n on a
	tableau (see eq. (3) of [Welsh] which supercedes eqs. (3.8), (3.9) &
	(3.10) of [Ki & Wy]) then completes the description of the module.


	/********************************************************************
	 Here follow routines dealing with the generic irreducible modules.
	 ********************************************************************/


NAME:
	generate_standard_tableaux
SYNOPSIS:
	INT generate_standard_tableaux (OP partition, OP std)
DESCRIPTION:
	Generates the full set of standard tableaux for the
	       given partition. The tableaux are returned as a
	       lexicographically ordered list. If no error occurs,
	       their number is returned.
               (The number of standard tableaux may also be obtained
	       from the Symmetrica function
		 dimension_partition (OP pt, OP dim),
	       which is described in the section of user manual
	       concerning PARTITION objects.)
RETURN:
	ERROR or count of standard tableaux.


NAME:
	hecke_generator_reps
SYNOPSIS:
	INT hecke_generator_reps (OP partition, OP vector)
DESCRIPTION:
	For the given partition of n, calculates representation
	       matrices for each of the generators h_1,h_2,...,h_{n-1}.
	       The matrices are stored as MATRIX object elements of
	       a VECTOR of length n-1.
RETURN:
	OK or ERROR.


NAME:
	represent_hecke_element
SYNOPSIS:
	INT represent_hecke_element (OP partition, OP hecke, OP mat)
DESCRIPTION:
	For the given partition of n, calculates the matrix
	       representing the element of H_n given by hecke which
	       is a q-linear combination of PERMUTATIONS as described
	       in the preamble above.
RETURN:
	OK or ERROR.

NAME:
	hecke_dg
SYNOPSIS:
	INT hecke_dg(OP part, OP perm, OP mat)
DESCRIPTION:
	uses the routine represent_hecke_element to compute the matrix
	representing the permutation perm.


NAME:
	build_lc
SYNOPSIS:
	INT build_lc (OP schizo, OP list)
DESCRIPTION:
	This routine converts schizo, which is either a PERMUTATION
	       object or a TABLEAUX object into a linear combination LIST,
	       of the type described in the preamble above. The LIST has
	       one term (schizo) whose coefficient is a MONOPOLY
	       representing 1. schizo is incorporated into the list and
	       should not be subsequently freed.
	       This routine is sometimes useful before calling
	       hecke_action_lc_on_lc() or standardise_cold_tableaux_list().
RETURN:
	OK or ERROR.


NAME:
	hecke_action_lc_on_lc
SYNOPSIS:
	INT hecke_action_lc_on_lc (OP tableaux, OP hecke, OP result)
DESCRIPTION:
	The linear combination of Hecke algebra permutations given
	       in hecke, acts on the linear combination of tableaux. The
	       resultant linear combination of (in general, non-standard)
	       tableaux is in result. This result is not ordered and has
	       not had terms collected.
RETURN:
	OK or ERROR.


NAME:
	 standardise_cold_tableaux_list
SYNOPSIS:
	INT standardise_cold_tableaux_list (OP tableaux, OP result)
DESCRIPTION:
	The linear combination of tableaux is re-expressed in terms
	       of standard tableaux. The result is an ordered list in which
	       terms have been collected.
	       (This routine makes use of only Garnir & column relations).
RETURN:
	OK or ERROR.


NAME:
	input_tableau
SYNOPSIS:
	INT input_tableau (OP partit, OP tab)
DESCRIPTION:
	Asks the user to input a tableau of the shape specified
	       by the PARTITION object partit. An ERROR is returned if
	       the entries are not distinct elements of {1,2,...,n},
	       where n is the weight of partit.
RETURN:
	OK or ERROR.


NAME:
	input_lc_permutations
SYNOPSIS:
	INT input_lc_permutations (OP save)
DESCRIPTION:
	Asks the user to input a linear combination of permutations.
RETURN:
	OK.


NAME:
	substitute_one_matrix
SYNOPSIS:
	INT substitute_one_matrix (OP matrix)
DESCRIPTION:
	Every entry of the matrix that is a MONOPOLY polynomial has
	       q=1 substituted. Using this function, the Specht module
	       representations of the Hecke algebra are converted into
	       those of the symmetric group.
RETURN:
	OK or ERROR.


NAME:
	substitute_one_monopoly
SYNOPSIS:
	INT substitute_one_monopoly (OP monopoly)
DESCRIPTION:
	The MONOPOLY polynomial has q=1 substituted (it is
	       converted into an INTEGER object).
RETURN:
	OK or ERROR.


COMMENT:
/********************************************************************
 Here follow routines dealing with the 2-rowed non-generic modules.
 ********************************************************************/


NAME:
	root_dimension
SYNOPSIS:
	INT root_dimension (OP partition, OP p_root, OP dim)
DESCRIPTION:
	Calculates the dimension of irreducible representation
	       labelled by partition at primitive p_root of unity.
	       The result is in the INTEGER object dim.
	       (Calculated using eq. (22) of [Welsh].)
RETURN:
	OK or ERROR.


NAME:
	generate_root_tableaux
SYNOPSIS:
	INT generate_root_tableaux (OP partition, OP p_root, OP std)
DESCRIPTION:
	Generates the full set of p-root standard tableaux for
	       the given partition. The tableaux are returned as a
	       lexicographically ordered list. If no error occurs,
	       their number is returned. This number should be equal
	       to that obtained from root_dimension().
	       (This routine simply generates all standard tableaux and
	       discards those that are not p-root standard.)
RETURN:
	ERROR or the number of p-root standard tableaux.


NAME:
	hecke_root_generator_reps
SYNOPSIS:
	INT hecke_root_generator_reps (OP partition, OP p_root, OP vector)
DESCRIPTION:
	For the given partition of n, and primitive p_root of
	       unity, calculates representation matrices for each of
	       the generators h_1,h_2,...,h_{n-1}.
	       The matrices are stored as MATRIX object elements of
	       a VECTOR of length n-1.
RETURN:
	OK or ERROR


NAME:
	root_represent_hecke_element
SYNOPSIS:
	INT root_represent_hecke_element (OP partition, OP p_root, OP hecke, OP mat)
DESCRIPTION:
	For the given partition of n, and primitive p_root of
	       unity, calculates the matrix representing the element
	       of H_n given by hecke which is a q-linear combination
	       of PERMUTATIONS as described in the preamble above.
RETURN:
	OK or ERROR


NAME:
	root_standardise_cold_tableaux_list
SYNOPSIS:
	INT root_standardise_cold_tableaux_list (OP tableaux, OP p_root, OP result)
DESCRIPTION:
	The linear combination of tableaux is re-expressed in
	       terms of the p_root standard tableaux. The result is
	       an ordered list in which terms have been collected.
	       (This routine makes use of Garnir & column relations
	       and some new relations known as strip relations - yet
	       to be documented).
RETURN:
	OK or ERROR.

COMMENT:
/********************************************************************
 The following routines check the representation matrices
 ********************************************************************/


NAME:
	check_hecke_generators
SYNOPSIS:
	INT check_hecke_generators (OP vector, OP p_root, INT flag)
DESCRIPTION:
	This routine checks that the MATRIX object elements of
	       the VECTOR object vector, satisfy the defining
	       relations 1, 2, 3, of the Hecke algebra given in
	       the preamble above. n is deduced from the length
	       of the vector. If p_root=0, then the relations
	       are checked for general q. Otherwise they are
	       checked for q a primitive p_root of unity.
	       For each identity checked, a message is printed
	       indicating whether the identity is OK for general q,
	       whether is it OK provided q is a root of unity with
	       the given primitive index, or whether it is not OK.
	       If flag is non-zero and the relation is not OK then the
	       difference between the two sides is output (untidily!).
RETURN:
	        OK or ERROR


NAME:
	        check_hecke_quadratic
SYNOPSIS:
	      INT check_hecke_quadratic (OP mat, OP p_root, INT flag)
DESCRIPTION:
	  Checks that the matrix satisfies the first Hecke algebra
	       relation. ( (m+1)(m-q) == 0 )
	       If p_root=0, then the relations are checked for general q.
	       Otherwise they are checked for q a primitive p_root of unity.
	       If there is no ERROR, returns 0 if true for all q,
	       1 if true at primitive p_root of unity, 2 otherwise.
	       If flag is non-zero and the relation is not OK then the
	       left side is output.
RETURN:
	        ERROR, 0, 1, or 2 as described above.


NAME:
	        check_braid
SYNOPSIS:
	      INT check_braid (OP mat1, OP mat2, OP p_root, INT flag)
DESCRIPTION:
	  Checks that the matrices satisfy the second Hecke algebra
	       relation. ( m1*m2*m1 == m2*m1*m2 )
	       If p_root=0, then the relations are checked for general q.
	       Otherwise they are checked for q a primitive p_root of unity.
	       If there is no ERROR, returns 0 if true for all q,
	       1 if true at primitive p_root of unity, 2 otherwise.
	       If flag is non-zero and the relation is not OK then the
	       difference between the two sides is output.
RETURN:
	        ERROR, 0, 1, or 2 as described above.


NAME:
	        check_commute
SYNOPSIS:
	      INT check_commute (OP mat1, OP mat2, OP p_root, INT flag)
DESCRIPTION:
	   Checks that the matrices satisfy the third Hecke algebra
	       relation. ( m1*m2 == m2*m1 )
	       If p_root=0, then the relations are checked for general q.
	       Otherwise they are checked for q a primitive p_root of unity.
	       If there is no ERROR, returns 0 if true for all q,
	       1 if true at primitive p_root of unity, 2 otherwise.
	       If flag is non-zero and the relation is not OK then the
	       difference between the two sides is output.
RETURN:
	        ERROR, 0, 1, or 2 as described above.


NAME:
	        check_zero_matrix
SYNOPSIS:
	     INT check_zero_matrix (OP mat, OP p_root)
DESCRIPTION:
	  Checks that the matrix is zero.
	       If p_root=0, then the matrix is checked for general q.
	       Otherwise it is checked for q a primitive p_root of unity.
	       If there is no ERROR, returns 0 if zero for all q,
	       1 if zero at primitive p_root of unity, 2 if non-zero.
RETURN:
	        ERROR, 0, 1, or 2 as described above.

COMMENT:
/********************************************************************
 Here follow routines to add or multiply hecke algebra elements
 ********************************************************************/


NAME:
         hecke_add
SYNOPSIS:
      INT hecke_add (OP hecke1, OP hecke2, OP result)
DESCRIPTION:
  The hecke algebra elements hecke1 and hecke2,
	       which are linear combinations of permutations
	       as described in the preamble, are added to
	       give result. hecke1 and hecke2 are unchanged.
RETURN:
       ERROR or OK.

NAME:
         hecke_mult
SYNOPSIS:
      INT hecke_mult (OP hecke1, OP hecke2, OP result)
DESCRIPTION:
   The hecke algebra elements hecke1 and hecke2,
	       which are linear combinations of permutations
	       as described in the preamble, are multiplied to
	       give result. hecke1 and hecke2 are unchanged.
RETURN:
        ERROR or OK.

NAME:
        hecke_scale
SYNOPSIS:
     INT hecke_scale (OP hecke, OP power, OP coeff)
DESCRIPTION:
   The hecke algebra element hecke which is a linear
	       combinations of permutations as described in the
	       preamble, is multiplied by coeff*q^power where
	       coeff & power are both INTEGER objects.
RETURN:
      ERROR or OK.

COMMENT:
/********************************************************************
 Here follows a list of prototypes for the routines described above.
 ********************************************************************/


/* function prototypes for generic representation routines */

INT generate_standard_tableaux (OP partition, OP std);
INT hecke_generator_reps (OP partition, OP vector);
INT represent_hecke_element (OP partition, OP hecke, OP mat);
INT build_lc (OP schizo, OP list);
INT hecke_action_lc_on_lc (OP tableaux, OP hecke, OP result);
INT standardise_cold_tableaux_list (OP tableaux, OP result);
INT input_tableau (OP partit, OP tab);
INT input_lc_permutations (OP save);
INT substitute_one_matrix (OP mat);
INT substitute_one_monopoly (OP mp);


/* function prototypes for non-generic representation routines */

INT root_dimension (OP partition, OP p_root, OP dim);
INT generate_root_tableaux (OP partition, OP p_root, OP std);
INT hecke_root_generator_reps (OP partition, OP p_root, OP vector);
INT root_represent_hecke_action (OP partition, OP p_root, OP hecke, OP mat);
INT root_standardise_cold_tableaux_list (OP tableaux, OP p_root, OP result);


/* function prototypes for matrix representation checking routines */

INT check_hecke_generators (OP vector, OP p_root, INT flag);
INT check_hecke_quadratic (OP mat, OP p_root, INT flag);
INT check_braid (OP mat1, OP mat2, OP p_root, INT flag);
INT check_commute (OP mat1, OP mat2, OP p_root, INT flag);
INT check_zero_matrix (OP mat, OP p_root);


/* function prototypes to add or multiply hecke algebra elements */

INT hecke_add (OP hecke1, OP hecke2, OP result);
INT hecke_mult (OP hecke1, OP hecke2, OP result);
INT hecke_scale (OP hecke, OP power, OP coeff);


/********************************************************************
 Here follow brief descriptions of the example programs.
 ********************************************************************/

The following programs are each written in the C language, using the
Symmetrica object oriented approach.

EX1.C
  This program first requests a partition from the user. It then
  calculates the dimension of the corresponding irreducible representation
  of H_n in the generic case (using the function dimension_partition()),
  and outputs it. It then calculates representation matrices for each
  of the generators h_1,h_2,...h_{n-1} of H_n. These are stored as
  elements of the VECTOR object v. They are output. It is then
  checked that they satisfy the Hecke algebra defining relations.
  Note that this check may take considerably more time than the
  generation of the matrices themselves.

EX2.C
  As in EX1.C, a partition is requested, and the dimension of the
  corresponding irreducible representation calculated and output.
  The user is then requested for a permutation (which should be
  input with length not exceeding the weight of the original
  partition). The matrix representing the element of H_n
  corresponding to the permutation is calculated and output.
  Note that in the call to represent_hecke_element(), a linear
  combination of permutations is required. This is the reason
  that the call to build_lc() is necessary: it converts the
  permutation into the required linear combination.
  A program more sophisticated than EX2.C would enable a linear
  combination of permutations to be input, and thus the matrix
  representing an arbitrary element of H_n to be obtained
  (using e.g., input_lc_permutations(); c.f. EX2X.C & EX4.C).
  Also note that the object w is not freed, since it is inside
  l, and is freed when l is freed.

EX3.C
  As in EX1.C, a partition is requested, and the dimension of the
  corresponding irreducible representation calculated and output.
  The user is then requested for a tableau of the corresponding
  shape. This should be input row by row, starting from the top
  row (or the bottom row if you are French!). The tableau is
  then expressed in terms of the standard tableaux, and this
  result output. The earlier declaration
    english_tableau=TRUE;
  ensures that the tableaux are output in the conventional way
  for non-French people.
  Note that the function standardise_cold_tableaux_list() requires
  a linear combination of tableaux: hence the preceding call
  to build_lc().
  Here, account is taken of the possibility that the user enters
  an inappropriate tableau (i.e. not one whose entries are distinct
  and from {1,2,...,weight_of_partition} ). In such a case, the
  function input_tableau() will return an ERROR, and the program
  will end gracefully.

EX4.C
  As in EX1.C, a partition is requested, and the dimension of the
  corresponding irreducible representation calculated and output.
  The user is then requested for a tableau, and further for a linear
  combination of permutations. The action of the latter on the former
  is calculated and output. This list is then standardised.
  In essence, the action of an arbitrary element of H_n on a
  particular vector in the module is calculated.
  Error checking is rife: if one should occur, the program will
  exit gracefully.
  The way in which input_lc_permutations() requests for input is
  as follows: first a permutation is requested, then its one
  variable polynomial coefficient is built up a term at a time;
  an exponent is input followed by its coefficient. The program
  asks whether there are more terms to be added to the polynomial.
  When this polynomial coefficient is completed, the program asks
  whether there are further permutations. If the answer is 'y', then
  they and their coefficients will be requested.

EX5.C
  As EX4.C except that the result of the action of the Hecke algebra
  element on the tableau can be subsequently acted on by another
  Hecke algebra element. And that subsequent result also, and so on.
  This program is more conveniently used with input redirected from
  a file (using e.g., a.out <input). A sample input file IN5 is provided.

EX1X.C
  This program is the same as EX1.C, except that q=1 is substituted
  into the final matrices (using the function substitute_one_matrix()
  on each matrix). This yields explicit matrix representations
  of the generators of the symmetric group. Note that when they are
  checked using the function check_hecke_generators(), it is necessary
  to specify that we are using a 1st root of unity!

EX2X.C
  As for EX2.C except that an arbitrary Hecke algebra element
  may be represented. The q-linear combination of permutations
  is input using input_lc_permutations() as in EX4.C. The program
  finally asks whether there are more elements to represent.
  If not, the program finishes; but if so, the entering phase is
  repeated. As with EX5.C, the input is more conveniently achieved
  using redirected input: a sample input file IN2X is provided.

EX6.C
  This program is similar to EX1.C, except that irreducible matrix
  representations at roots of unity are generated (but only for
  partitions with 2 parts). The user is first prompted for the
  primitive index of the root of unity. The program then proceeds
  as in EX1.C, except that here the appropriate root-standard
  tableaux are calculated (using generate_root_tableaux()) and
  displayed. The integer value of the INTEGER object r is obtained
  via s_i_i(r).

EX7.C
  This program performs the same function as EX2.C except that an
  irreducible matrix representation at root of unity is calculated
  for a partition with two parts.

EX8.C
  This program performs the same function as EX3.C except that the
  result is in terms of the root-standard tableaux appropriate to
  the input (index of the) primitive root of unity.
  A little more error checking is performed here and in one instance
  t has to be freed since it would not be freed at the end.

EX9.C
  This program performs the same function as EX4.C except that,
  once more, the result is in terms of the root-standard tableaux
  appropriate to the input (index of the) primitive root of unity.

EX10.C
  This program obtains two elements of the Hecke algebra from
  the user, and calculates and outputs their product.

                                  Trevor Welsh, Bayreuth, November 1995.
                                                Modified 12/1/96.

