LEARN

@matrix
Matrix   A matrix can either be interpreted as a linear transformation
   (acting by right-multiplication on row vectors), or as a set of vectors,
   in which case each row of the matrix represents a vector in the set,
   or in a special way such as a character table. For instance, a matrix
   representing a set of roots will be termed a root matrix.  See also
   orbit matrix and restriction matrix.
@polynomial
Polynomial   A polynomial may either just represent itself, i.e., a Lau-
   rent polynomial (for instance in the case of the Kazhdan-Lusztig poly-
   nomials, which incidentally are always ordinary polynomials), or it
   may encode a set of vectors of equal size, with multiplicities.  In the
   latter case each term represents the occurrence of its exponent in the
   indicated set (such exponents are always interpreted as a weights),
   occurring with multiplicity equal to the coefficient of the term.  On
   their turn, sets of weights with multiplicities may have different in-
   terpretations, leading to a further distinction between polynomials.
   Three important such interpretations are that of decomposition poly-
   nomials, character polynomials, and dominant character polynomials.
@vector
Vector   A vector may represent an element of a vector space (or rather
   of a free Z-module, since its entries must be integral), such as the
   weight space, or it may just be interpreted as a set or sequence of
   integers.  In the former case it is always to be interpreted as a row
   vector, so that matrices are to be applied from the right.  In either
   case there are a further distinctions as to how the vector is to be
   interpreted. See also root vector, weight vector, Weyl word, partition
   and toral element.
@central torus
Central torus   The center of any reductive Lie group g is the direct
   product of a torus and a finite group, the former (which is clearly the
   connected component of the center) is called the central torus of g.
   For the groups LiE deals with this central torus is even a direct factor
   of g itself, the other factor being the semisimple part of g.
@diagram
Diagram   The (Dynkin) diagram of a semisimple Lie group is a graph
   indicating the isomorphism type of the group, the number of vertices
   is equal to the (semisimple) Lie rank, and the number of connected
   components of the diagram is equal to the number of simple factors
   of the group. The vertices are labeled with positive integer numbers,
   following the conventions of . The diagram represents the information
   contained in the Cartan matrix of the group in a compact form.
@fundamental lie subgroup
Fundamental Lie subgroup   A closed subgroup h of a Lie group g is
   called fundamental if it contains a maximal torus of g. If h contains T
   and is reductive, it is determined by the set of roots in the root system
    of g that are also roots of h, these form a closed subsystem of roots.
@general linear group
General  Linear  group   The group of all invertible linear transfor-
   mations of a vector space V  is called the general linear group of V ,
   written GL(V ). Up to isomorphism this depends only on n = dim V ,
   and this group is also written as GL(n, C) (assuming the vector space
   is over C).  This group is a Lie group, and any Lie group homomor-
   phism of some Lie group g to GL(V ) is called a representation of that
   Lie group on the vector space V . See also special linear group.
@lie algebra
Lie algebra   A finite-dimensional vector space V  supplied with a bi-
   linear operation [  ,  ]: V*V->V  satisfying [x, y]  =  - [y, x] and
   [[x, y], z]+[[y, z], x]+[[z, x], y] = 0 for all x, y, z element V 
   (anti-commutativity and the Jacobi identity, respectively) is called a 
   Lie algebra.  
   Every Lie group defines a Lie algebra structure on the tangent space to the
   group at the identity element. Although Lie algebras play no explicit
   role in this package, the representation theory of simply connected
   reductive complex Lie groups which LiE deals with coincides with the
   representation theory of reductive Lie algebras over C.
@lie group   
Lie  group   A group is called a Lie group if its underlying set is a
   differentiable variety, and the multiplication and inversion maps are
   differentiable.  The group is called complex, connected, simply con-
   nected, etc., if the variety is respectively complex, connected, simply
   connected, etc.  Each reductive complex Lie group is an algebraic
   group and the representation theory can be dealt with in an entirely
   algebraic manner.
@lie rank
Lie rank   The dimension of a maximal torus of g is called the Lie rank
   of g.
@maximal torus
Maximal torus   A torus that is not properly contained in any other
   torus within g is called a maximal torus of g.  If g is a reductive Lie
   group, such tori exist and any two are conjugate.  In LiE, we always
   assume a fixed maximal torus T of g to be chosen, weights and roots
   are defined with respect to T .
@reductive group
Reductive group   A group is reductive if each of its finite dimensional
   representations decomposes into a direct sum of irreducible represen-
   tations. A connected reductive complex Lie group g is isomorphic to
   the quotient of the direct product of a simply connected semisimple
   group and a torus by a finite central subgroup.  An example is the
   General Linear group GL(n, C). The (images of) the semisimple fac-
   tor and the torus can be found as the commutator subgroup g0 of g
   and the central torus of g respectively. In LiE, the type group always
   refers to a group that itself is a direct product of a simply connected
   semisimple group and a torus (so no quotient is involved).
@semisimple element
Semisimple element   All conjugates of elements of the torus T  are
   called semisimple elements (not to be confused with the term semi-
   simple for groups), in any representation of g they correspond to diag-
   onalisable transformations. Hence each conjugacy class of semisimple
   elements has representatives in T , and some elements of T  namely
   those of finite order, can be represented in LiE,  see in Section 3.2
   under toral element.
@semisimple group
Semisimple group   A connected reductive Lie group is called semi-
   simple if it contains no non-trivial central torus, or equivalently if it is
   equal to its commutator subgroup. Note that a non-trivial semisimple
   group necessarily contains non-semisimple elements.
@semisimple rank
Semisimple rank   The semisimple rank of a group g is the Lie rank
   of its semisimple part, or stated differently, the Lie rank of g minus
   the dimension of its central torus.
@special linear group
Special Linear group   For a vector space M the Special Linear group
   SL(M ) is the closed Lie subgroup of the General Linear group GL(M )
   of all transformations with determinant equal to 1. It is the commu-
   tator subgroup of GL(M ).
@torus
Torus   A group which is isomorphic to (C*)^n  for some n is called a
   torus (plural: tori), it is a reductive Lie group of dimension n.  Any
   closed connected subgroup of a Lie group g all of whose elements are
   semisimple is a torus, called a torus of g. Every torus of g is contained
   in a maximal torus, and every maximal torus is conjugate to T , the
   fixed maximal torus.  See also semisimple element.  A fundamental
   property of a torus is that all of its irreducible representations are 1-
   dimensional. Since in such a representation of T each element acts as
   a scalar, the representation is essentially given by an algebraic group
   morphism T->C* , a so-called weight. Any representation of g may
   be restricted to a representation of T , as such it decomposes into 1-
   dimensional representations.  The resulting formal sum of weights is
   called the (formal) character of the representation with respect to T .
   This formal sum of weights can be represented by a polynomial, which
   is then called a character polynomial.
@cartan matrix
Cartan matrix   The matrix  (ai, aj) 1<=i,j<=s  is called the Cartan ma-
   trix (of the semisimple part) of g, its rows express the fundamental
   roots on the basis of fundamental weights.
@cartan type
Cartan type   The Cartan type of a closed subsystem  of roots of
   is the type of the fundamental Lie subgroup of g whose root system
   is .
@closed subsystem
Closed subsystem   Given a root system R, a closed subsystem S 1 is a
   subset  that is itself a root system, and has the property that when-
   ever ai + bi element   for ai, bi element R then ai + bi element S . 
   If R is the root system of g, then every closed subsystem corresponds 
   to a fundamental Lie subgroup of g.
@fundamental reflection
Fundamental reflection  For a chosen set of fundamental roots a1,...,as
   the reflections in the hyperplanes perpendicular to these roots are
   called fundamental reflections, they are often denoted by r1,...,rs.
   These reflections generate the Weyl group.
@fundamental root
Fundamental root   It is often assumed that a subset of the roots has
   been chosen as the set of fundamental roots, which are then denoted
   by a1 ,..., as.  This set must form a basis of the root lattice such
   that any root can be expressed as a linear combination of them with
   either all positive or all negative integer coefficients. This is the basis
   on which root vectors are expressed.  The function inprod  gives a
   W-invariant inner product for weights on this basis.
@fundamental weight
Fundamental weight   For a chosen set of fundamental roots there
   is a basis of the weight lattice consisting of weights w1,..,wr such
   that <wi, aj> = delta(i,j) for all i,j element {1 ,.., s}. These weights are 
   called the fundamental weights. It is this basis on which weight vectors are
   expressed.
@highest root   
Highest root    This is the maximum of the set of roots with respect to the
   height partial ordering: lambda is higher than mu iff lambda-mu is a sum
   of positive roots. It is the highest weight of the adjoint representation.
@levi subgroup
Levi subgroup   Any subset of the set of fundamental roots determines
   a closed subsystem (of which it is a basis fundamental roots) of the
   root system, and the fundamental Lie subgroup corresponding to this
   subsystem is called a Levi subgroup of g.  The Dynkin diagrams of
   the Levi subgroups of g are therefore obtained by taking subsets of
   nodes of the diagram of g and retaining the edges between elements
   of the subset.

@one parameter subgroup
One  parameter subgroup   Any 1-dimensional subtorus h of T  is
   called a one parameter subgroup, there is an algebraic group isomor-
   phism phi: C* -> h. Such one parameters subgroups may be represented
   in the following way, which is very similar to the representation of toral
   elements. For 1<= i <= r we have a group homomorphism z -> phi(z)^wi
   from C* to C* ; this homomorphism is equal to the map z ^ ai for
   some ai element Z.  The one parameter subgroup h is now represented by
   the vector [a1,...,:ar, 0]. The final 0 serves to distinguish it from toral
   elements, which are valid in the same positions where one parameter
   subgroups may be used (e.g., as parameter to cent_roots). The integers
   a1 ... ar should not all have a non-trivial factor in common, because
   the morphism phi would then fail to be injective.  Any toral element
   obtained by substituting some number n for the final zero lies in h (it
   is phi(zeta_n) where zeta_2 = exp(2*Pi*I/n).  The restriction matrix of h 
   is obtained by arranging the ai (for i = 1, 2, : :,:r) vertically into a 
   one-column matrix.
@positive root
Positive root   A root that can be expressed as a linear combination of
   fundamental roots with non-negative coefficients is called a positive
   root. For every root a exactly one of { a, -a} is positive.
@root
Root   A non-zero weight for the adjoint representation of g is called a
   root of g.  For each root the orthogonal reflection in the hyperplane
   perpendicular to it preserves the weight lattice.
@root lattice
Root  lattice   The sublattice of the weight lattice generated by the
   roots of g is called the root lattice.  For semisimple groups the root
   lattice has finite index in the weight lattice, for simple groups of type
   An , Bn , Cn , Dn , En , F4 and G2 this index is n + 1, 2, 2, 4, 9  n,
   1 and 1 respectively. The fundamental roots form a basis of the root
   lattice, and the elements of the root lattice are root vectors. See also
   weight.
@root matrix
Root matrix   A root matrix is a matrix whose rows specify a set of
   roots, represented as root vectors.  Root matrices may be used to
   denote subsystems of the root system of g.
@root system
Root system   The set of all roots is called the root system of g. It is
   usually denoted by phi.
@root vector
Root vector   When an element of the root lattice is represented by its
   coefficients on the basis consisting of the fundamental roots a1 ,..., as,
   the result is called a root vector.  So a root vector has as size the
   semisimple rank of the group, and such a vector v = [v1 ,.., vs] is
   interpreted as the sum    s = a1*v1+ ... +as*vs.
@toral element
Toral element   To describe elements of T we can use the fundamental
   weights wi. Recall that weights are in fact mappings T->C* , and a
   weight  can therefore be evaluated at an element t element T, the resulting
   value be written t^lambda, the set of fundamental weights form a complete
   set of coordinates in the sense that any element t element T in uniquely
   determined by the values t^wi for i = 1,...,r.  Although LiE cannot
   represent arbitrary complex numbers, it can represent torus elements
   of finite order, i.e., elements for which all t^wi are roots of unity.  To
   this end, a vector [a1 ,..., ar, n] in LiE may represent the element t 
   for which t^wi = exp(2*Pi*I/n) = zeta_n^ai where zeta_n = exp(2*Pi*I/n) is a
   canonical n-th root of unity. See also one parameter subgroup. Since
   this is not the usual presentation of a toral element in a classical Lie
   group like GL(n, C) (namely by the diagonal entries occurring when
   the element is diagonalised), an example is given in Chapter 5 of how
   to transform from one presentation to another.
@weight
Weight   A weight with respect to a torus T is an algebraic group morphism 
   T->C* ; it describes a 1-dimensional representation of T.
   These arise in the decomposition of the restriction to T  of repre-
   sentations of g, in which case they are called the weights of the 
   g-representation with respect to T. The set Lambda(T) of weights is an
   Abelian group, where the group operation is pointwise multiplication of 
   weights as C -valued functions (which corresponds to the tensor product 
   of 1-dimensional T-representations), this is written additively.  
   We consequently use the exponential notation t  to indicate application of 
   a weight  to t element T, so that we have t^(lambda+mu)= t^lambda*t^mu. 
   The fundamental weights span the weight lattice as a free Z-module;
   expressing a weight on this basis we obtain a so-called weight vector.
@weight lattice
Weight lattice   The set Lambda(T) of all weights of g with respect to T is
   called the weight lattice. The addition defined for weights turns Lambda(T)
   into an Abelian group isomorphic to Z^r.
@weight vector
Weight vector   When a vector is represented by its coefficients on the
   basis consisting of the fundamental weights w1 ,.., wr, it is called a
   weight vector. So a weight vector v = [v1 ,..,vs] is interpreted as the
   sum s = v1*w1+ ... + vs*ws.
@bruhat descendent
Bruhat descendent   For any element w element W the Bruhat descendents
   are those elements x element W for which x < w in the Bruhat order 
   and the length l(x) is exactly one less than l(w). Any element y element W 
   with y <= w can be obtained starting from w by repeatedly moving from
   an element to one of its Bruhat descendents.
@bruhat order
Bruhat order   The Bruhat order is a partial ordering defined on any
   Coxeter group. It can be determined by the following recursive defini-
   tion. For the identity element e element W we have x <= e  <-> x = e. For
   any other element y element W there is some simple reflection ri such that
   l(yri)<l(y), then for any x element W we have x<= y <->  x0<=yri, where
   x0 is the element with the smaller length from {x, xri}. This definition
   is independent of the choice of ri. Incidentally, the definition implies
   that the condition l(yri)<l(y) used above is equivalent to yri<y.
   It is tempting to omit the function l from similar conditions (as occur
   for instance in the description of the Kazhdan-Lusztig polynomials),
   but we have not done this to avoid the suggestion that the full Bruhat
   order is involved in cases where a simple length comparison suffices.
@canonical weyl word
Canonical Weyl word   When representing Weyl group elements by
   Weyl words it is sometimes useful (for instance when testing for equal-
   ity) to select for each w element W a particular expression among the many
   possibilities, this is then called the canonical Weyl word for W . In LiE
   we choose for this purpose the lexicographically first reduced expres-
   sion for w.
@coxeter group
Coxeter group   A Coxeter group is a finitely presented group, where
   the presentation is determined by its Coxeter matrix m = (mi,mj)1<=i,j<=s;
   the presentation of the Coxeter group is <g1 ,...,gs | (gi*gj)^m[i,j] = 1>.
   Every Weyl group is a Coxeter group, with Coxeter matrix given by
   mi,j= order(ri*rj).
@coxeter matrix
Coxeter matrix   A Coxeter matrix is a symmetric matrix m = (m[i,j]) 1<=i,j<=s
   with positive integer entries, such that m[i,j]= 1 if and only if i = j.
   Such a matrix is used to define a Coxeter group.
@distinguished coset representative
Distinguished coset representative   Within the Weyl group W we
   may consider left-, right-, and double cosets with respect to a sub-
   group (or in the case of double cosets, two subgroups) generated by
   fundamental reflections,  in each case the unique element of small-
   est length in its coset is called the distinguished coset representative.
   Note that this term refers to a Weyl group element as representative
   for a coset, not to a Weyl word representing that Weyl group element.
@dominant weight
Dominant  weight   A weight whose inner products with all funda-
   mental roots are non-negative is called dominant. Equivalently, if the
   weight is written on the basis of the fundamental weights w1 ,..., wr,
   then the first s coefficients (corresponding to the semisimple part of
   the weight lattice Lambda(T)) are non-negative.
@exponents
Exponents e1 ,..., er   The algebra of polynomial functions invariant under
   the action of the Weyl group of g in its standard reflection representa-
   tion is generated by r homogeneous polynomials of respective degrees
   e1 + 1, e2 + 1 ,..., er + 1.  Usually the exponents of g are given in
   weakly increasing order.
@kazhdan-Lusztig polynomial
Kazhdan-Lusztig polynomial   For any pair x, y of elements of a Cox-
   eter group W  a polynomial Px,y is defined, called Kazhdan-Lusztig
   polynomial.  Further information in manual.
@lenght
Length   The length of a Weyl group element w is the smallest number
   l such that w is a product of l fundamental reflections.  Hence, it is
   the size of a reduced Weyl word representing w.
@longest element
Longest element   In every finite Coxeter group W there is a unique
   element of maximal length.  It is an involution (but in general not a
   reflection), and is called the longest element of W .
@orbit
Orbit   When a group W acts (from the right) on a set X, any x element X
   has an orbit,which is the set of all distinct values of x.w for w element W.
@orbit matrix
Orbit  matrix   When a finite group acts on any lattice by integral
   matrices, an orbit may be represented by an orbit matrix, each row
   of which represents one element of the orbit.
@reduced weyl word
Reduced Weyl word   If an element w of the Weyl group is expressed
   as a product ri1    rim  of fundamental reflections, and no product
   of fewer than m fundamental reflections yields w, then the sequence
   [i1, : :,:im ] is a reduced Weyl word for w. In general such a reduced
   Weyl word is not uniquely determined by w, but see canonical Weyl
   word.
@reflection
Reflection   A Weyl group element that acts on the weight lattice, fix-
   ing a sublattice of rank r  1, is an orthogonal reflection in the hyper-
   plane perpendicular to some root.  The reflections are precisely the
   conjugates of the simple reflections, and the latter description makes
   sense for arbitrary Coxeter groups.
@r-polynomial
R-polynomial   For any pair x, y of elements of a Coxeter group W a
   polynomial R_x_y in one indeterminate is defined, called R-polynomial.
   These polynomials are related to the Kazhdan-Lusztig polynomials.
@weyl group
Weyl group   The Weyl group W is defined as the quotient of the nor-
   maliser N_g(T) of the maximal torus T in g by the centraliser of T in g
   (which is T itself). W is a finite group, and has a faithful linear rep-
   resentation on the weight lattice Lambda(T). The elements of W are often
   identified with their images in this representation. The fundamental
   reflections r1 ,..., rs in this representation are canonical generators
   of W.
@weyl word
Weyl word   An element of the Weyl group W may be presented as a
   product of the fundamental reflections ri (1<=i<=s).  If ri_1,...,ri_l
   is such a product, the corresponding Weyl group element may be
   represented by the so-called Weyl word [ri_1 ,..., ri_l].  It is allowed to
   include entries equal to 0 in a Weyl word, which are ignored by LiE,
   no function that returns Weyl words will include such zeros in the
   result, except possibly as a padding at the right end when that Weyl
   word forms a row in a matrix of Weyl words of different lengths.
@character polynomial
Character polynomial (symmetric group). For the symmetric group on n
   letters, the conjugacy classes are parametrised by partitions of n,
   where the parts of the partition correspond to the disjoint cycles of the
   permutation.  Therefore a character X_lambda of the symmetric group may
   be represented by a character polynomial, which is a polynomial in n
   indeterminates, in which each exponent represents a partition mu of n
   (padded with trailing zeros) and its coefficient is the (integral)
   value X_lambda(mu) of the character X_lambda on the conjugacy class
   corresponding to mu.
@partition
Partition   A partition of a natural number n is a weakly decreasing
   sequence of numbers whose sum is n; adding or removing trailing zeros
   does not alter the partition. Any partition of n can be represented as
   a vector v = [v1 ,..., vn ] of length n.  The LiE function partitions(n)
   produces a matrix whose rows represent the partitions of n. Partitions
   of n parametrise the conjugacy classes of the symmetric group on
   n letters and also their irreducible characters, they also parametrise
   dominant weights of SL_n or GL_n .
@partition coordinates
Partition coordinates   A weight x for a group of type A_(n-1)  can be
   expressed in partition coordinates by forming a vector of length n
   whose i-th entry is the sum of the coefficients in x of the j-th fun-
   damental weights for j >=i (note that the final entry is always 0).
   Conversely the coefficient of the i-th fundamental weight can be ob-
   tained as the difference between the i-th and the i + 1-st partition
   coordinate.  In LiE these conversions can be performed by the func-
   tions to_part and from_part.  Partition coordinates are used for the
   function LR_tensor.
@robinson-schensted correspondence
Robinson-Schensted correspondence   The Robinson-Schensted cor-
   respondence is an algorithmically defined bijection between the el-
   ements of the Symmetric group Sn  and the set of pairs of Young
   tableaux of equal shape with n entries.
@shape
Shape   The shape of a Young tableau is a partition describing the
   length of the rows of the tableau.
@symmetric group
Symmetric group   The set of permutations of {1,...,n} is called the
   Symmetric group on n letters, often denoted by S_n.  Its conjugacy
   classes are described by partitions,|as well as its characters.
@tableau
Tableau   A (Young) tableau is an arrangement of a set {1,2,...,n}
   of numbers into rows of weakly decreasing length,such that the num-
   bers increase along rows and columns.The shape of a tableau is the
   sequence of its row lengths,which is a partition. A typical example is

                                 1  2  4  6  11
                                 3  5  8 
                                 7  10 
                                 9  12

   which has shape [5, 3, 2, 2].  In LiE, tableaux are represented linearly
   by vectors of size n.  If t is such a vector, then t[i] indicates the
   row number of the entry i in the 2-dimensional form.  For instance,
   the tableau above would be encoded as [1, 1, 2, 1, 2, 1, 3, 2, 4, 3, 1, 4]. A
   function print_tab is provided to display the 2-dimensional form of a
   tableau. Young tableaux have many applications in the theory of the
   symmetric group, for instance the number of tableaux of shape  is
   equal to the dimension of the irreducible representation of S_n corre-
   sponding to .
@adams operator
Adams' operator   For each n > 1 there is an operator, called the
   n-th Adams' operator, defined on the set of virtual g-modules, which
   has the effect on the characters of scaling each occurring weight by a
   factor n (while retaining its multiplicity). In general the result is a vir-
   tual module even if the original module was actual. The n-th Adams'
   operator is the `weight analog' of the operator that, given a character
   O of a finite group g, computes the decomposition of the class function
   gamma->X(gamma^n) as an integral linear combination of irreducible charac-
   ters.  The operator is useful for computing symmetrised tensor powers.
@adjoint representation
Adjoint representation   Each Lie group g acts on its Lie algebra by
   conjugation, which defines a representation of the group, the so-called
   adjoint representation.  The non-zero weights of this representation
   (all of which occur with multiplicity 1) are called the roots of g.
@alternating Weyl sum
Alternating Weyl sum The Alternating Weyl has the interesting property that 
   it gives the same result when applied to a decomposition polynomial as 
   when applied to the corresponding character polynomial.  
   Note that the expression above suggests an alternative action of W on 
   polynomials, where the i-th generator
   of W (as a Coxeter group) does not act on exponents by reflection in
   the hyperplane perpendicular to ai, but rather in that plane shifted
   by wi (or equivalently by rho), and meanwhile also changes the sign
   of the coefficients.  For this action J (p) is just the sum of the W -
   images of p.  However, since this "shifted alternating action" plays
   no role except via the operator J , we will not introduce any further
   terminology or notation relating to it.
@branching
Branching   Branching is another word for restricting a g-module M
   to another reductive group h.  Suppose h is a closed reductive Lie
   subgroup of g. The branching problem concerns finding the decompo-
   sition into highest weight modules of M when viewed as an h-module.
   Since the maximal torus T_g of g is unique up to conjugacy, and sim-
   ilarly for h, the maximal torus T_h  of h may be chosen within T_g.
   Consequently, each weight with respect to T_g determines by restric-
   tion a weight with respect to T_h, which defines a linear transformation
   Lambda(T_g)->Lambda(T_h). In fact we can define such a restiction transforma-
   tion in the more general setting of an arbitrary Lie group morphism
   f: h->g (not just for embeddings), consequently we can consider
   branching for such situations as well. The matrix m which describes
   this transformation on the respective bases of fundamental weights, is
   called the restriction matrix for h in g, and plays a crucial role in the
   function branch.  The function res_mat helps to find the restriction
   matrix in cases where h is a fundamental Lie subgroup, LiE has also
   access to a table of precomputed restriction matrices for cases where
   h is a maximal subgroup in g but not a fundamental Lie subgroup.
@character
Character   For a representation of a group on a finite dimensional vec-
   tor space we may define a function on the group by assigning to each
   group element the trace of the corresponding transformation of the
   vector space.  This function, which is constant on conjugacy classes,
   is called the character of the representation.  For reductive complex
   Lie groups the character determines the representation up to isomor-
   phism, and this is already true for the restriction of the character to
   the maximal torus T . Now the restriction to T of the representation
   decomposes into a direct sum of 1-dimensional representations, and
   the character of such a 1-dimensional representation is just a weight.
   Hence the restriction to T of the character of the whole representa-
   tion can be correspondingly written as a formal sum of weights (formal
   because we don't use the Abelian group structure of Lambda(T) here, but
   just count the occurring weights with multiplicities, in other words,
   the sum is taken in the group algebra of Lambda(T) ) and this is called the
   formal character of the representation. This character can be conve-
   niently encoded as a character polynomial. 
@character polynomial
Character  polynomial   The (formal) character of a representation
   of g can be expressed as a polynomial, which records each weight w
   occurring with multiplicity m in the character as a term m*X^w of the
   polynomial.
@contragredient representation
Contragredient representation   For each representation of g on a
   vector space V there is a corresponding representation, called its con-
   tragredient representation, on the dual vector space V^*  . Here a group
   element a acts on an element f: V->C of V^*  by mapping it to
   f.a: v->f (v.a^(-1)).  As an example where the contragredient occurs,
   consider the space of homogeneous polynomial functions of degree n
   on V , this is a finite dimensional space on which g acts (by the same
   formula as above, but with for a polynomial function replacing f ).
   This representation of g is isomorphic to the n-th symmetric tensor
   power of the contragredient representation of the original representa-
   tion.
@decomposition  polynomial
Decomposition  polynomial   The decomposition of a g-module M
   into irreducible modules may be represented by a decomposition poly-
   nomial d. Each term m*X^lambda  of d represents a dominant weight such
   that the highest weight module V  occurs in M with multiplicity m.
   In certain circumstances we allow m to be negative, in which case
   there is no module corresponding to d, but we may think of M as a
   formal sum (with integral scalar coefficients) of irreducible modules.
   In this case M is called a virtual module, and the polynomial a virtual
   decomposition polynomial.
@demazure operator
Demazure operator   For each simple root ai a linear operator M_ai,
   called Demazure operator, is defined on the set of polynomials with
   exponents in Lambda(T).  A discussion of the mathematical
   significance of this operator is beyond the scope of this manual, as it
   involves the representation theory of parabolic subgroups, which are
   not reductive. 
@dominant character polynomial
Dominant character polynomial   Since character polynomials are
   invariant under W , and each W-orbit of weights contains a unique
   dominant element, the information of a character polynomial can be
   more compactly represented by omitting all terms whose exponents
   are not dominant.  The polynomial obtained in this way from the
   character polynomial of a g-module M is called the dominant charac-
   ter polynomial of M .  LiE  provides functions filter_dom and W_orbit
   for going from the character polynomial to the dominant character
   polynomial and back again.
@highest weight
Highest  weight   The maximum of the set of weights of some irre-
   ducible representation of g with respect to the partial ordering < is
   called the highest weight, it always exists and is a dominant weight
   that occurs with multiplicity 1. Conversely, every dominant weight lambda
   occurs as the highest weight of a unique irreducible representation V_lambda
   of g.  By definition,  lambda0<lambda  holds if and only if 
   lambda - lambda0 is a sum of positive roots, in this case  lambda is called 
   higher than lambda0.
@highest weight module
Highest weight module   For a dominant weight  the unique irre-
   ducible representation of g with lambda as highest weight, is called the
   highest weight module (or representation) of g for lambda, and is denoted
   by V_lambda.
@irreducible representation
Irreducible representation   A representation of a group g is called
   irreducible if the representation space has no proper non-zero subspace
   that is stable under g.  In case g is a reductive group it suffices that
   the representation space cannot be decomposed as a direct sum of two
   non-trivial g-stable subspaces.
@module
Module   See representation.
@plethysm
Plethysm   A representation of a group g on a vector space M corre-
   sponds to a group morphism g ->GL(M ). As such it can be composed
   with any representation of the group GL(M ) on a vector space N , giv-
   ing rise to a representation of g on the space N .  Now if we take for
   the representation of GL(M ) the irreducible one parametrised by the
   partition lambda (in partition coordinates), then the resulting representa-
   tion of g is called the plethysm, or symmetrised tensor power, of M
   with respect to lambda. 
@representation
Representation   An action by linear transformations of a group g on
   a finite dimensional vector space V  (where the representing matrices
   depend in a polynomial way on the coordinates of the group element),
   is called a (rational) representation of the group, the space V  is then
   called a module for g. This is equivalent to giving a (Lie) group mor-
   phism g -> GL(V ).  The irreducible representations of finite groups,
   as well as of reductive Lie groups, are determined (up to equivalence)
   by their characters.  For reductive Lie groups, the irreducible repre-
   sentations are parametrised by their highest weights. 
@restiction matrix
Restriction matrix   If h is a reductive subgroup of g, and a maximal
   torus of h is chosen within the maximal torus T of g, then any weight
   of g with respect to T (which is a function on T ) becomes by restriction
   to the maximal torus of h a weight of h. Consequently there is a map
   from the weight lattice of g to that of h, and this map is linear, it
   can therefore be given by a matrix, called the restriction matrix for
   the subgroup h.  A similar matrix can be defined for an arbitrary
   Lie group morphism f : h->g.  Each row of this matrix represents
   the restriction to the maximal torus of h of a fundamental weight
   of g, viewed as a weight of h.  The restriction matrix plays a role in
   branching.
@virtual module
virtual  module   A formal sum of irreducible g-modules with inte-
   ger coefficients corresponds to an actual g-module only if all the co-
   efficients are non-negative (the module can then be constructed by
@finish