313 lines
12 KiB
C
313 lines
12 KiB
C
#include "coxeter.h"
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#include "queue.h"
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#include <memory.h>
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#include <gsl/gsl_linalg.h>
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#include <gsl/gsl_blas.h>
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static void applyGenerator(int i, double *x, double *y, int rank, double **schlaefliMatrix)
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{
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memcpy(y, x, rank*sizeof(double));
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for(int j = 0; j < rank; j++)
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y[i] -= schlaefliMatrix[i][j]*x[j];
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}
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static int check_equal_sector(gsl_matrix *x1, gsl_matrix *x2, gsl_matrix *yinv, gsl_matrix *schlaefli, gsl_vector *starting, gsl_vector *i1, gsl_vector *i2)
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{
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double scalarProduct;
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int rank = x1->size1;
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// calculate <y^{-1} * x1 * x2 * s, e_i> for all i, or identically the components of schlaefli*y^{-1}*x*s
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gsl_blas_dgemv(CblasNoTrans, 1.0, x2, starting, 0.0, i1);
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gsl_blas_dgemv(CblasNoTrans, 1.0, x1, i1, 0.0, i2);
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gsl_blas_dgemv(CblasNoTrans, 1.0, yinv, i2, 0.0, i1);
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gsl_blas_dgemv(CblasNoTrans, 1.0, schlaefli, i1, 0.0, i2);
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for(int i = 0; i < rank; i++)
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if(gsl_vector_get(i2, i) < 0)
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return 0;
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return 1;
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}
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static void invert(gsl_matrix *x, gsl_matrix *xinv)
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{
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int rank = x->size1;
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gsl_matrix *xLU = gsl_matrix_alloc(rank, rank);
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gsl_permutation *p = gsl_permutation_alloc(rank);
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int s;
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gsl_matrix_memcpy(xLU, x);
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gsl_linalg_LU_decomp(xLU, p, &s);
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gsl_linalg_LU_invert(xLU, p, xinv);
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gsl_permutation_free(p);
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gsl_matrix_free(xLU);
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}
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static void generate_schlaefli_matrix(semisimple_type_t type, gsl_matrix *mat)
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{
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gsl_matrix_set_zero(mat);
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int offset = 0;
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for(int k = 0; k < type.n; k++) {
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if(type.factors[k].series == 'A') {
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for(int i = 0; i < type.factors[k].rank; i++)
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for(int j = 0; j < type.factors[k].rank; j++)
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if(i == j)
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gsl_matrix_set(mat, offset + i, offset + j, 2.0);
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else if(i - j == 1 || i - j == -1)
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gsl_matrix_set(mat, offset + i, offset + j, -1.0);
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} else if(type.factors[k].series == 'B' || type.factors[k].series == 'C') {
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for(int i = 0; i < type.factors[k].rank; i++)
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for(int j = 0; j < type.factors[k].rank; j++)
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if(i == j)
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gsl_matrix_set(mat, offset + i, offset + j, 2.0);
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else if(i == 0 && j == 1 || i == 1 && j == 0)
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gsl_matrix_set(mat, offset + i, offset + j, -sqrt(2.0));
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else if(i - j == 1 || i - j == -1)
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gsl_matrix_set(mat, offset + i, offset + j, -1.0);
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} else if(type.factors[k].series == 'D') {
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for(int i = 0; i < type.factors[k].rank; i++)
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for(int j = 0; j < type.factors[k].rank; j++)
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if(i == j)
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gsl_matrix_set(mat, offset + i, offset + j, 2.0);
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else if(i > 0 && j > 0 && (i - j == 1 || i - j == -1) || i == 0 && j == 2 || i == 2 && j == 0)
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gsl_matrix_set(mat, offset + i, offset + j, -1.0);
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} else if(type.factors[k].series == 'F') {
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ERROR(type.factors[k].rank != 4, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[k].series, type.factors[k].rank);
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for(int i = 0; i < 4; i++)
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gsl_matrix_set(mat, offset + i, offset + i, 2.0);
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for(int i = 0; i < 3; i++) {
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gsl_matrix_set(mat, offset + i, offset + i + 1, i == 1 ? -sqrt(2.0) : -1.0);
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gsl_matrix_set(mat, offset + i + 1, offset + i, i == 1 ? -sqrt(2.0) : -1.0);
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}
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} else if(type.factors[k].series == 'G') {
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ERROR(type.factors[k].rank != 2, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[k].series, type.factors[k].rank);
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gsl_matrix_set(mat, offset + 0, offset + 0, 2.0);
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gsl_matrix_set(mat, offset + 0, offset + 1, -sqrt(3.0));
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gsl_matrix_set(mat, offset + 1, offset + 0, -sqrt(3.0));
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gsl_matrix_set(mat, offset + 1, offset + 1, 2.0);
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} else {
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ERROR(1, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[k].series, type.factors[k].rank);
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}
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offset += type.factors[k].rank;
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}
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}
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int coxeter_rank(semisimple_type_t type)
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{
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int rank = 0;
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for(int i = 0; i < type.n; i++)
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rank += type.factors[i].rank;
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return rank;
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}
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int coxeter_order(semisimple_type_t type)
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{
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int order = 1;
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for(int i = 0; i < type.n; i++) {
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if(type.factors[i].series == 'A') { // (rank+1)!
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for(int j = 1; j <= type.factors[i].rank + 1; j++)
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order *= j;
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} else if(type.factors[i].series == 'B' || type.factors[i].series == 'C') { // 2^rank * rank!
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for(int j = 1; j <= type.factors[i].rank; j++)
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order *= 2*j;
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} else if(type.factors[i].series == 'D') { // 2^(rank-1) * rank!
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for(int j = 2; j <= type.factors[i].rank; j++)
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order *= 2*j;
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} else if(type.factors[i].series == 'E') {
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if(type.factors[i].rank == 6)
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order *= 51840;
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else if(type.factors[i].rank == 7)
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order *= 2903040;
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else if(type.factors[i].rank == 8)
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order *= 696729600;
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else
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ERROR(1, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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} else if(type.factors[i].series == 'F') {
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ERROR(type.factors[i].rank != 4, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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order *= 1152;
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} else if(type.factors[i].series == 'G') {
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ERROR(type.factors[i].rank != 2, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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order *= 12;
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} else if(type.factors[i].series == 'H') {
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if(type.factors[i].rank == 2)
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order *= 10;
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else if(type.factors[i].rank == 3)
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order *= 120;
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else if(type.factors[i].rank == 4)
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order *= 14400;
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else
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ERROR(1, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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} else {
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ERROR(1, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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}
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}
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return order;
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}
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int coxeter_hyperplanes(semisimple_type_t type)
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{
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int hyperplanes = 0;
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for(int i = 0; i < type.n; i++) {
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if(type.factors[i].series == 'A') // rank*(rank+1)/2
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hyperplanes += (type.factors[i].rank * (type.factors[i].rank + 1)) / 2;
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else if(type.factors[i].series == 'B' || type.factors[i].series == 'C') // rank * rank
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hyperplanes += type.factors[i].rank * type.factors[i].rank;
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else if(type.factors[i].series == 'D') // rank * (rank - 1)
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hyperplanes += type.factors[i].rank * (type.factors[i].rank - 1);
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else if(type.factors[i].series == 'E') {
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if(type.factors[i].rank == 6)
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hyperplanes += 36;
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else if(type.factors[i].rank == 7)
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hyperplanes += 63;
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else if(type.factors[i].rank == 8)
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hyperplanes += 120;
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else
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ERROR(1, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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} else if(type.factors[i].series == 'F') {
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ERROR(type.factors[i].rank != 4, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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hyperplanes += 24;
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} else if(type.factors[i].series == 'G') {
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ERROR(type.factors[i].rank != 2, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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hyperplanes += 6;
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} else {
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ERROR(1, "A Coxeter group of type %c%d does not exist or is not implemented!\n", type.factors[i].series, type.factors[i].rank);
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}
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}
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return hyperplanes;
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}
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unsigned long opposition_involution(semisimple_type_t type, unsigned long theta)
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{
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int offset = 0;
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unsigned long result = 0;
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for(int i = 0; i < type.n; i++) {
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unsigned long current = (theta >> offset) & ((1 << type.factors[i].rank) - 1);
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unsigned long iota_current;
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if(type.factors[i].series == 'B' || type.factors[i].series == 'C' || type.factors[i].series == 'F' || type.factors[i].series == 'G') {
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iota_current = current;
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} else if(type.factors[i].series == 'A') {
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iota_current = 0;
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for(int j = 0; j < type.factors[i].rank; j++)
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iota_current += ((current >> j) & 1) << (type.factors[i].rank - 1 - j);
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} else if(type.factors[i].series == 'D') {
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if(type.factors[i].rank % 2 == 0) {
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iota_current = current;
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} else {
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ERROR(1, "The opposition involution for type %c%d is not yet implemented!\n", type.factors[i].series, type.factors[i].rank);
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}
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} else if(type.factors[i].series == 'E') {
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ERROR(1, "The opposition involution for En is not yet implemented!\n");
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}
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result += iota_current << offset;
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offset += type.factors[i].rank;
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}
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return result;
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}
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static void generate_starting_vector(int rank, gsl_matrix *schlaefli, gsl_vector *result)
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{
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gsl_matrix *schlaefliLU = gsl_matrix_alloc(rank, rank);
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gsl_vector *diagonal = gsl_vector_alloc(rank);
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gsl_permutation *p = gsl_permutation_alloc(rank);
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int s;
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for(int i = 0; i < rank; i++)
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gsl_vector_set(diagonal, i, 1.0);
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gsl_matrix_memcpy(schlaefliLU, schlaefli);
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gsl_linalg_LU_decomp(schlaefliLU, p, &s);
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gsl_linalg_LU_solve(schlaefliLU, p, diagonal, result);
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gsl_permutation_free(p);
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gsl_vector_free(diagonal);
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gsl_matrix_free(schlaefliLU);
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}
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void generate_coxeter_graph(semisimple_type_t type, int *result)
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{
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int rank = coxeter_rank(type);
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int order = coxeter_order(type);
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int element_count;
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gsl_matrix *schlaefliMatrix = gsl_matrix_alloc(rank, rank);
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gsl_vector *startingVector = gsl_vector_alloc(rank);
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gsl_matrix *generators = (gsl_matrix*)malloc(rank*sizeof(gsl_matrix));
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double *generators_data = (double*)malloc(rank*rank*rank*sizeof(double));
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gsl_matrix *elements = (gsl_matrix*)malloc(order*sizeof(gsl_matrix));
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double *elements_data = (double*)malloc(rank*rank*order*sizeof(double));
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gsl_matrix *inverses = (gsl_matrix*)malloc(order*sizeof(gsl_matrix));
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double *inverses_data = (double*)malloc(rank*rank*order*sizeof(double));
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gsl_vector *i1 = gsl_vector_alloc(rank);
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gsl_vector *i2 = gsl_vector_alloc(rank);
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gsl_matrix *current_element = gsl_matrix_alloc(rank, rank);
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queue_t queue;
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int current;
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for(int i = 0; i < rank; i++)
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generators[i] = gsl_matrix_view_array(generators_data + i*rank*rank, rank, rank).matrix;
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for(int i = 0; i < order; i++)
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elements[i] = gsl_matrix_view_array(elements_data + i*rank*rank, rank, rank).matrix;
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for(int i = 0; i < order; i++)
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inverses[i] = gsl_matrix_view_array(inverses_data + i*rank*rank, rank, rank).matrix;
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generate_schlaefli_matrix(type, schlaefliMatrix);
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generate_starting_vector(rank, schlaefliMatrix, startingVector);
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for(int i = 0; i < rank; i++) {
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gsl_matrix_set_identity(&generators[i]);
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for(int j = 0; j < rank; j++)
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if(i == j)
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gsl_matrix_set(&generators[i], i, j, -1.0);
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else
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gsl_matrix_set(&generators[i], i, j, -gsl_matrix_get(schlaefliMatrix, i, j));
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// gsl_matrix_fprintf(stdout, &generators[i], "%f");
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// printf("\n");
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}
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queue_init(&queue);
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queue_put(&queue, 0);
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element_count = 1;
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gsl_matrix_set_identity(&elements[0]);
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gsl_matrix_set_identity(&inverses[0]);
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while((current = queue_get(&queue)) != -1) {
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for(int i = 0; i < rank; i++) {
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int j;
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for(j = 0; j < element_count; j++) {
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if(check_equal_sector(&generators[i], &elements[current], &inverses[j], schlaefliMatrix, startingVector, i1, i2)) { // generators[i] * elements[current] = elements[j]
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result[rank*current + i] = j;
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break;
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}
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}
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// if no existing element equals generators[i] * elements[current], create one
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if(j == element_count) {
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ERROR(element_count >= order, "Got more elements than the order of the group should be!\n");
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gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, &generators[i], &elements[current], 0.0, &elements[element_count]);
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invert(&elements[element_count], &inverses[element_count]);
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result[rank*current + i] = element_count;
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queue_put(&queue, element_count);
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element_count++;
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}
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}
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}
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ERROR(element_count != order, "Something went wrong building the Coxeter group. Found %d elements, %d expected\n", element_count, order);
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/*
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for(int i = 0; i < order; i++) {
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printf("%d: ", i);
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for(int j = 0; j < rank; j++)
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printf("%d ", result[rank*i + j]);
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printf("\n");
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}
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*/
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gsl_vector_free(startingVector);
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gsl_vector_free(i1);
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gsl_vector_free(i2);
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gsl_matrix_free(schlaefliMatrix);
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gsl_matrix_free(current_element);
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free(generators);
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free(elements);
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free(generators_data);
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free(elements_data);
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}
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