EXPLICIT MATRIX MODELS FOR IRREPS OF WEYL GROUPS For Use with Maple Contents of this directory A8_data - models for A1,...,A8 B8_data - models for B2,...,B8 D8_data - models for D4,...,D8 E6_data - models for E6 E7_data - models for E6, E7 E8_data - models for E6, E7, E8 E8_data.m - models for E6, E7, E8 in Maple's '.m' format F4_data - models for F4 G2_data - models for G2 unpack - Maple programs for unpacking the data md5sums.txt - if you don't know what this is, don't worry about it READ_ME - this file The data files are in Maple text format, except for `E8_data.m`. This '.m' file should be readable by Maple 6 or later versions. Be sure to read all of the WARNINGS below. For more information about the notation and terminology, see . ------------------------------------------------------- I. Introduction The files *_data contain -- in a highly compressed format -- explicit orthogonal and rational seminormal models for all of the irreducible representations of these Weyl groups, and recursively, all smaller groups in a parabolic chain obtained by repeatedly deleting the last node in the the following numberings of the Dynkin diagrams: A8: 1---2---3---4---5---6---7---8 B8: 1=<=2---3---4---5---6---7---8 2 | D8: 1---3---4---5---6---7---8 2 | E8: 1---3---4---5---6---7---8 F4: 1---2=<=3---4 So in particular, the E7 file includes everything in the E6 file, and the E8 file includes everything in the E7 file. To use the data, you need to "unpack" it. In a Maple session, try read unpack; read E7_data; At this point, several procedures from the 'unpack' file, and three tables from the data file named 'Rep', 'Semi', and 'Branch', will have been loaded. Understanding the internal structure of these tables is not necessary for use of the database, but a description is provided in Sec. IV below for completeness. II. The Orthogonal Models Matrices are represented in a sparse format consisting of lists of linear polynomials in the variables e1,e2,.... (an orthonormal basis). Each linear form represents a row. Example: the 2x2 matrix [ 5 6 ] is [5*e1 + 6*e2, 7*e1 + 8*e2]. [ 7 8 ] Continuing the above Maple session, the command a:=unpack(12,7); will assign to the variable 'a' a list of 7 orthogonal matrices for the generators in the 12th irrep of E7. (See IV for information about how the irreps have been numbered. For the exceptional groups, it is in order of increasing dimension.) So a[5]; is the representing matrix for the action of s[5], and a[5][16] is the 16th row of that matrix. The tables for a given group also contain complete information about all previous groups in the parabolic chain, so unpack(15,6); will unpack the 15th irrep of E6, and so on. NOTE: in these models, the square of every matrix entry is rational. NOTE^2: The plain text `E8_data` file loads slowly, and forces allocation of a large chunk of memory (10-15MB). It is preferable to use the '.m' version of the E8 file--it will load faster and use significantly less memory. WARNING: Versions of Maple older than Maple 6 use an incompatible '.m' format. To create an `E8_data.m` file that is compatible with your version of Maple, do the following in a fresh Maple session: read E8_data; save Rep, Semi, Branch, `E8_data.m`; quit; III. The Seminormal Models Each of the irreps also has a *rational* matrix model that is "seminormal" in the sense that the positive definite quadratic form preserved by the model is a diagonal (rational) matrix D (so A * D * A^T = D for each matrix A in the representation). Continuing the above Maple session, the command a:=unpack(12,7,'semi'); will unpack and assign to 'a' a list of *8* matrices. The first 7 are rational seminormal matrices for the generators in the 12th irrep of E7, and the 8th is the diagonal matrix D defining the quadratic form. The third argument to 'unpack' can be anything. The mere presence of any third argument suffices to specify the rational seminormal form. NOTE: The positive square root of D is the change of basis between the orthogonal and seminormal models; i.e., D^{1/2} * Ortho * D^{-1/2} = Seminormal. NOTE^2: The largest E8 models take up to 1-2 GHz-minutes to unpack, and require allocation of ~10MB of memory. The seminormal models unpack more rapidly and in less space. WARNING: If you later switch to the models from a second data file during the same Maple session, then you will need to erase a remember table that is used by 'unpack'. This can be accomplished either by using the command `unpack/erase`(); or simply by re-reading the 'unpack' file. These measures can be taken either before or after reading the second data file. IV. Low-level Information About The Data Files Each of the data files consists of three Maple tables named 'Rep', 'Branch', and 'Semi'. ---------------- Rep: low-level data structures encoding orthogonal matrices For each k from 1 up to the top rank n, Rep[k] is the character table of W[k] = . It is a list of lists. The i-th list is the i-th row of the character table. The ordering within each row follows the ordering of conjugacy classes in the coxeter package . In particular, the trivial conjugacy class is always first. So in the file E8_data, Rep[6] is the character table of E6 Rep[6][12] is the 12th irreducible character of E6 Rep[6][12][1] is the degree of the 12th irrep of E6 In addition to the character table, Rep[0,k] is the largest r <= k such that s[k] centralizes W[r]. Rep[i,k][j] is a list indexing all of the copies of the j-th irrep of W[r] in the i-th irrep of W[k]. Each item in the list represents a chain in the Bratteli diagram. Example: Rep[17,7][12] = [[12,3,17], [12,11,17], [12,16,17]] means that there are 3 copies of the 12th irrep of E5 = W[r] in the 17th irrep of E7, and these copies are found in the 3rd, 11th and 16th irreps of E6. If the j-th irrep of W[r] does not occur in the i-th irrep of W[k], then Rep[i,k][j] is not assigned. In a hereditary model, s[k] may be represented as a direct sum of many copies of a set of operators indexed by W[r]-irreps. In the i-th irrep of W[k], the operator indexed by the j-th irrep of W[r] has rows and columns indexed by the list Rep[i,k][j], and for each item L in this list, Rep[L,k] represents a row of the matrix for this operator. ---------------- Branch: information about branching of irreps from one group in the parabolic chain to the next. For each k from 1 up to the top rank n, Branch[k] is a branching matrix for restriction from W[k] to W[k-1]. If ch is a (column) vector of character values on W[k]-conjugacy classes, then the matrix-vector product Branch[k]*ch is the vector of irreducible multiplicities for ch as a W[k-1]-character. Branch[i,k] is a list that indicates which irreps appear (with repetitions for multiplicity > 1) when the i-th irrep of W[k] is restricted to W[k-1]. Example: Branch[31,7] = [10, 13, 17, 21] means that the 31st irrep of E7 decomposes as the sum of the 10th, 13th, 17th, and 21st irreps of E6. NOTE: if an irrep j occurs with multiplicity 2, then the second occurrence of the irrep is listed as -i. (None of the models provided involve branching multiplicities greater than 2.) Branch[0,k] is a list of representatives from the conjugacy classes of W[k] of para-Coxeter type (i.e., products of distinct simple reflections) that include s[k] at least once. The format is a list of items of the form [i,w], where w is a class representative, and i is the position of w within the listing of conjugacy classes used by the coxeter package. If no irreps of W[k] have clones, Branch[0,k] is not assigned. ---------------- Semi: data encoding the seminormal forms for each irrep. For each k from 1 up to the top rank n, and for each irrep j of W[k-1] that occurs in the i-th irrep of W[k], (i.e., for each j in Branch[i,k]), Semi[j,i,k] is a rescaling factor for rationalizing the i-th irrep of W[k-1] in the j-th irrep of W[k]. Each scaling factor is the (positive) square root of a rational. The rows and columns of the matrices for the i-th irrep of W[n] may be represented by chains in the Bratteli diagram; i.e., (n+1)-tuples [i[0],...,i[n]], where i[0]=1, i[n]=i, and i[k-1] is an item of Branch[i[k],k] for each k=1,...,n. If D is the diagonal matrix whose entry in position [i[0],...,i[n]] is Semi[i[0],i[1],1] * ... * Semi[i[n-1],i[n],n], then D is the change of basis needed to convert the orthogonal model to the rational seminormal model; i.e., D * Ortho * D^{-1} = Seminormal. Semi[k] is a set of indices that indicates for which irreps of W[k] a seminormal has been constructed. This is useful only during the process of building the models; all of the data files provided have seminormal data for all irreps.