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Root Systems






At present there is no satisfactory theory of root systems for
unitary reflection groups comparable to the theory for finite
Coxeter groups.  However, if we choose a representative r_i for
each conjugacy class of reflections in the reflection group G
and if we choose a root a_i of r_i, then the union of the
orbits of the a_i form a suitable set on which G acts as
a group of permutations.


A standard theorem states that for any set {r_1, r_2, ..., r_n} 
of reflections that generate G, every reflection in G is
conjugate to a power of some r_i.  Thus the magma function
Class can be used to find all reflections in G.


Even though there is no satisfactory notion of "set of
fundamental roots", a reflection group can nevertheless be
described by specifying the set of roots corresponding to a set of
reflection generators together with a root of unity attached to
each root. Moreover, the inner products between the roots can be
described by means of a diagram similar to the Dynkin diagram of a
Coxeter group.  This notation was suggested by Coxeter and used by
Cohen in his thesis on the classification of the groups.  (There
is a different type of diagram used by Brou'e, Malle and others.)


In this package we use Cohen's naming scheme for the diagrams.
This extends the standard notation A_n, B_n, ..., H_3,
H_4 used for Coxeter groups.



RootSystemMatrix(t, n) : MonStgElt, RngIntElt -> AlgMatElt



Given a string t defining the type and an integer n specifying
the rank, this function returns the matrix of (modified) inner 
products of roots corresponding to generating reflections of a 
reflection group of type t and rank n. The rank is the 
dimension of the space on which the group acts; it is not always 
the number of generators.
The function constructs root system matrices for the types A,
B, C, D, E, F, G, H, J_3(4), J_3(5),
K, L, M, N, and O.  (The function accepts the 
abbreviations J4 and J5 for the types J_3(4) and J_3(5).)






The (i, j)-th entry of the root system matrix for the roots
a_1, a_2, ..., a_k is delta_(ij) + (alpha_j -
1)(a_i, a_j), where alpha_j is an m-th root of unity, for
some m.  The effect of the reflection r_j with root a_j on
the root a_i is given by


  a_i r_j = a_i + (alpha_j - 1)(a_i, a_j) a_j.




In the associated diagram (given below) there is a node for each
root.  The (i, i)-th entry of the root system matrix is
alpha_i and if this is -1, the node is shown as a circle,
otherwise it is represented by alpha_i itself.


This construction includes all finite irreducible Coxeter
groups.


The ordering of the fundamental root vectors is given in the
following diagrams.  A pair of nodes not joined by an edge
corresponds to a matrix entry of 0.  A single bond corresponds
to 1 and all other bonds are labelled by the matrix entry
(reading from left to right, from lower numbered node to higher).


Thus an unlabelled edge between nodes of reflections of order 2
corresponds to an inner product of -1/2.








beginschema{A_n}
       1   2   3       n
       o---o---o- ... -o


endschema


beginschema{B_n = C_n}
       1   2   3       n
       o===o---o- ... -o
        sqrt2


endschema


beginschema{D_n}
      1 o
          3  4       n
          o---o- ... -o
         /
      2 o


endschema




beginschema{E_6}


      2   3   4   5   6
      o---o---o---o---o
              |
            1 o


endschema


beginschema{E_7}
      2   3   4   5   6   7
      o---o---o---o---o---o
              |
            1 o


endschema


beginschema{E_8}
      2   3   4   5   6   7   8
      o---o---o---o---o---o---o
              |
            1 o


endschema


beginschema{F_4}
      1   2   3   4
      o---o===o---o
           sqrt2


endschema


beginschema{G_2}
      1   2
      o===o
       sqrt3


endschema


beginschema{H_3}
      1   2   3
      o===o---o        tau^2 = tau + 1
        tau


endschema


beginschema{H_4}
      1   2   3   4
      o===o---o---o
        tau


endschema






beginschema{J_3(4)}
            2
            o
           /            c^2 = c - 2
        1 o===o 3
           -c


endschema


beginschema{J_3(5)}
            2
            o
           /        omega^2 + omega+ 1 = 0
        1 o===o 3
           omegatau


endschema


beginschema{K_4}
            3
            o
           / \
      o---o===o                 W(K_4) = G(3, 3, 4)
      1   2 omega4


endschema


beginschema{K_5}
            3
            o
           / \
      o---o===o---o
      1   2 omega4   5


endschema


beginschema{K_6}
            3
            o
           / \
      o---o===o---o---o
      1   2 omega4   5   6


endschema


beginschema{L_3}
      1     2     3
      omega=== omega=== omega        -omega^2    omega^2


endschema






beginschema{L_4}
      1     2     3     4
      omega=== omega=== omega=== omega        -omega^2    omega^2   -omega^2


endschema


beginschema{M_3}
      1     2     3
      o === omega=== omega        sqrt2   -omega^2


endschema


beginschema{N_4}
          2
          o
         /  3   4
      1 o===o---o
         i - 1


endschema


beginschema{O_4}
              3
      2 o --- o --- o 4     |W(O_4) : W(N_4)| = 6
           // \  /
          /scriptstyle i - 1 \
           o === o
           1 scriptstyle i + 1  5


endschema
hrule



ReflectionGroup(M) : AlgMatElt -> GrpMat, Fld



Given a root system matrix M the function returns the corresponding 
unitary reflection group.  In addition, the field of definition is 
returned.
We assume that M corresponds to a positive semidefinite
inner product and that the first n - 1 columns of M - I
are linearly independent.
The reflection generators are created as matrices with respect to
the standard basis of the reflection representation.  The matrices
represent the action on row vectors.
The k-th reflection matrix is obtained from the identity
matrix by replacing its k-th column with the k-th column
of the root system matrix.
If the determinant of M - I is 0, the matrices can be
thought of as arising from transformations constructed as
just described, but acting on the quotient of the space
modulo the null space of M - I.



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