818 lines
24 KiB
C
818 lines
24 KiB
C
#include "weyl.h"
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#include "queue.h"
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#include <stdio.h>
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#include <memory.h>
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#include <stdlib.h>
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#define BIT(n) ((uint64_t)1 << (n))
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typedef struct {
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weylid_t id;
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int position;
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} weylid_lookup_t;
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static void generate_left_and_ids(semisimple_type_t type, weylgroup_element_t *group);
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static int search(const void *key, const void *base, size_t nmem, size_t size, int (*compar) (const void *, const void *, void *), void *arg);
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static int compare_root_vectors(int rank, const int *x, const int *y);
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static int compare_root_vectors_qsort(const void *x, const void *y, void *arg);
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static int compare_weylid(const void *x, const void *y);
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static int compare_weylid_lookup(const void *x, const void *y);
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static int lookup_id(weylid_t id, weylid_lookup_t *list, int len);
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static weylid_t multiply_generator(int s, weylid_t w, const int *simple, const int *mapping, int rank, int positive);
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static void reflect_root_vector(const int *cartan, int rank, int i, int *old, int *new);
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static weylgroup_element_t* apply_word(int *word, int len, weylgroup_element_t *current);
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static weylgroup_element_t* apply_word_reverse(int *word, int len, weylgroup_element_t *current);
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/******** generate_left_and_ids and a pile of helper functions **************/
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static void generate_left_and_ids(semisimple_type_t type, weylgroup_element_t *group)
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{
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int rank = weyl_rank(type);
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int order = weyl_order(type);
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int positive = weyl_positive(type);
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queue_t queue;
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int current;
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int roots_known, elements, length_elements, nextids_count;
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int *cartan_matrix;
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int *root_vectors;
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int *vector;
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int *simple_roots;
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int *root_mapping;
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weylid_t *ids, *edges, *nextids;
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weylid_lookup_t *lookup;
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// allocate temporary stuff
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cartan_matrix = (int*)malloc(rank*rank *sizeof(int));
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root_vectors = (int*)malloc(2*positive*rank*sizeof(int));
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vector = (int*)malloc(rank *sizeof(int));
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root_mapping = (int*)malloc(positive*rank *sizeof(int));
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simple_roots = (int*)malloc(rank *sizeof(int));
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ids = (weylid_t*)malloc(order *sizeof(weylid_t));
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edges = (weylid_t*)malloc(rank*order *sizeof(weylid_t));
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nextids = (weylid_t*)malloc(rank*order *sizeof(weylid_t));
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lookup = (weylid_lookup_t*)malloc(order *sizeof(weylid_lookup_t));
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// get all information on the cartan type
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LOG("Get Cartan matrix.\n");
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weyl_cartan_matrix(type, cartan_matrix);
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// enumerate roots, first the simple ones, then all others by reflecting
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LOG("Enumerate roots.\n");
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memset(root_vectors, 0, 2*positive*rank*sizeof(int));
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roots_known = 0;
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queue_init(&queue);
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for(int i = 0; i < rank; i++) {
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root_vectors[rank*i + i] = 1; // (r_i)_j = delta_ij
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queue_put(&queue, i);
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roots_known++;
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}
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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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reflect_root_vector(cartan_matrix, rank, i, &root_vectors[rank*current], vector);
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int j;
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for(j = 0; j < roots_known; j++)
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if(compare_root_vectors(rank, &root_vectors[rank*j], vector) == 0)
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break;
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if(j == roots_known) {
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memcpy(&root_vectors[rank*roots_known], vector, rank*sizeof(int));
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queue_put(&queue, roots_known);
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roots_known++;
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}
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}
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}
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ERROR(roots_known != 2*positive, "Number of roots does not match!\n");
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// sort roots and restrict to positives
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LOG("Sort roots.\n");
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qsort_r(root_vectors, 2*positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
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memcpy(root_vectors, &root_vectors[positive*rank], positive*rank*sizeof(int)); // this just copies the second part of the list onto the first; source and destination are disjoint!
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// generate root_mapping, which gives the action of root reflections on positive roots (-1 if result is not a positive root)
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LOG("Compute root reflections.\n");
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for(int i = 0; i < positive; i++) {
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for(int j = 0; j < rank; j++) {
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reflect_root_vector(cartan_matrix, rank, j, &root_vectors[rank*i], vector);
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root_mapping[i*rank+j] =
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search(vector, root_vectors, positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
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}
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}
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// find simple roots in the list
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LOG("Find simple roots.\n");
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for(int i = 0; i < rank; i++) {
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memset(vector, 0, rank*sizeof(int));
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vector[i] = 1;
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simple_roots[i] = search(vector, root_vectors, positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
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}
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// enumerate weyl group elements using difference sets
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LOG("Enumerate Weyl group elements.\n");
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nextids[0] = 0;
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nextids_count = 1;
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elements = 0;
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for(int len = 0; len <= positive; len++) {
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length_elements = 0;
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// find unique ids in edges added in the last iteration
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qsort(nextids, nextids_count, sizeof(weylid_t), compare_weylid);
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for(int i = 0; i < nextids_count; i++)
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if(i == 0 || nextids[i] != nextids[i-1])
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ids[elements + length_elements++] = nextids[i];
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// add new edges
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nextids_count = 0;
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for(int i = elements; i < elements + length_elements; i++)
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for(int j = 0; j < rank; j++) {
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edges[i*rank+j] = multiply_generator(j, ids[i], simple_roots, root_mapping, rank, positive);
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if(!(ids[i] & BIT(simple_roots[j]))) // the new element is longer then the old one
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nextids[nextids_count++] = edges[i*rank+j];
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}
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elements += length_elements;
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}
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// translate the ids to list positions (i.e. local continuous ids)
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LOG("Reorder Weyl group elements.\n");
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for(int i = 0; i < order; i++) {
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lookup[i].id = ids[i];
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lookup[i].position = i;
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}
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qsort(lookup, order, sizeof(weylid_lookup_t), compare_weylid_lookup);
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// fill in results
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LOG("Compute left multiplication.\n");
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for(int i = 0; i < order; i++) {
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group[i].id = ids[i];
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for(int j = 0; j < rank; j++)
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group[i].left[j] = group + lookup_id(edges[i*rank+j], lookup, order);
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}
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// free temporary stuff
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free(cartan_matrix);
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free(root_vectors);
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free(vector);
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free(root_mapping);
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free(simple_roots);
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free(ids);
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free(edges);
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free(nextids);
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free(lookup);
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}
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// glibc search function, but with user pointer and returning index (or -1 if not found)
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static int search(const void *key, const void *base, size_t nmemb, size_t size, int (*compar) (const void *, const void *, void *), void *arg)
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{
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size_t l, u, idx;
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const void *p;
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int comparison;
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l = 0;
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u = nmemb;
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while (l < u) {
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idx = (l + u) / 2;
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p = (void *) (((const char *) base) + (idx * size));
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comparison = (*compar) (key, p, arg);
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if (comparison < 0)
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u = idx;
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else if (comparison > 0)
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l = idx + 1;
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else
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return idx;
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}
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return -1;
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}
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// maybe we want a different ordering here?
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static int compare_root_vectors(int rank, const int *x, const int *y)
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{
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for(int i = 0; i < rank; i++)
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if(x[i] != y[i])
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return x[i] - y[i];
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return 0;
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}
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static int compare_root_vectors_qsort(const void *x, const void *y, void *arg)
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{
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return compare_root_vectors(*((int*)arg), x, y);
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}
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static int compare_weylid(const void *x, const void *y)
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{
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weylid_t u = *((weylid_t*)x);
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weylid_t v = *((weylid_t*)y);
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return u > v ? 1 : u < v ? -1 : 0;
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}
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static int compare_weylid_lookup(const void *x, const void *y)
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{
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weylid_t u = ((weylid_lookup_t*)x)->id;
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weylid_t v = ((weylid_lookup_t*)y)->id;
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return u > v ? 1 : u < v ? -1 : 0;
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}
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static int lookup_id(weylid_t id, weylid_lookup_t *list, int len)
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{
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weylid_lookup_t key;
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key.id = id;
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weylid_lookup_t *p = (weylid_lookup_t*)bsearch(&key, list, len, sizeof(weylid_lookup_t), compare_weylid_lookup);
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return p->position;
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}
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static weylid_t multiply_generator(int s, weylid_t w, const int* simple, const int* mapping, int rank, int positive)
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{
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weylid_t sw = 0;
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for(int i = 0; i < positive; i++) {
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if(w & BIT(i))
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if(mapping[i*rank+s] != -1)
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sw |= BIT(mapping[i*rank+s]);
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}
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if(w & BIT(simple[s]))
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return sw;
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else
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return sw | BIT(simple[s]);
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}
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static void reflect_root_vector(const int *cartan, int rank, int i, int *old, int *new)
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{
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memcpy(new, old, rank*sizeof(int));
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for(int j = 0; j < rank; j++)
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new[i] -= cartan[i*rank + j]*old[j];
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}
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/************* Weyl group infos ************************/
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static int weyl_exists(simple_type_t type)
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{
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if(type.series < 'A' || type.series > 'G' || type.rank < 1 ||
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type.series == 'B' && type.rank < 2 ||
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type.series == 'C' && type.rank < 2 ||
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type.series == 'D' && type.rank < 3 ||
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type.series == 'E' && type.rank != 6 && type.rank != 7 && type.rank != 8 ||
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type.series == 'F' && type.rank != 4 ||
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type.series == 'G' && type.rank != 2)
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return 0;
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else
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return 1;
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}
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int weyl_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 weyl_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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ERROR(!weyl_exists(type.factors[i]), "A Weyl group of type %c%d does not exist!\n", type.factors[i].series, type.factors[i].rank);
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switch(type.factors[i].series) {
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case 'A':
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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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break;
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case 'B': case 'C':
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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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break;
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case 'D':
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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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break;
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case '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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break;
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case 'F':
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order *= 1152;
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break;
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case 'G':
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order *= 12;
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break;
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}
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}
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return order;
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}
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int weyl_positive(semisimple_type_t type)
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{
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int positive = 0;
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for(int i = 0; i < type.n; i++) {
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ERROR(!weyl_exists(type.factors[i]), "A Weyl group of type %c%d does not exist!\n", type.factors[i].series, type.factors[i].rank);
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switch(type.factors[i].series) {
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case 'A':
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positive += (type.factors[i].rank * (type.factors[i].rank + 1)) / 2;
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break;
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case 'B': case 'C':
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positive += type.factors[i].rank * type.factors[i].rank;
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break;
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case 'D':
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positive += type.factors[i].rank * (type.factors[i].rank - 1);
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break;
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case 'E':
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if(type.factors[i].rank == 6)
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positive += 36;
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else if(type.factors[i].rank == 7)
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positive += 63;
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else if(type.factors[i].rank == 8)
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positive += 120;
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break;
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case 'F':
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positive += 24;
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break;
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case 'G':
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positive += 6;
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break;
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}
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}
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return positive;
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}
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int weyl_opposition(semisimple_type_t type, int simple_root)
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{
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int offset = 0;
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int factor = 0;
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int r, iota_r;
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for(factor = 0; factor < type.n; factor++)
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if(simple_root < offset + type.factors[factor].rank)
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break;
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else
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offset += type.factors[factor].rank;
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r = simple_root - offset;
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ERROR(!weyl_exists(type.factors[factor]), "A Weyl group of type %c%d does not exist!\n", type.factors[factor].series, type.factors[factor].rank);
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switch(type.factors[factor].series) {
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case 'B': case 'C': case 'F': case 'G':
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iota_r = r;
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break;
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case 'A':
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iota_r = type.factors[factor].rank - 1 - r;
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break;
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case 'D':
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if(type.factors[factor].rank % 2 == 0)
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iota_r = r;
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else
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iota_r = r == 0 ? 1 : r == 1 ? 0 : r;
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break;
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case 'E':
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if(type.factors[factor].rank != 6)
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iota_r = r;
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else
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iota_r = r == 2 || r == 3 ? r : 5 - r;
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break;
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}
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return iota_r + offset;
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}
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void weyl_cartan_matrix(semisimple_type_t type, int *m)
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{
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int offset = 0;
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int rank = weyl_rank(type);
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int **A = (int**)malloc(rank*sizeof(int*));
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memset(m, 0, rank*rank*sizeof(int));
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for(int i = 0; i < rank; i++)
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m[i*rank+i] = 2;
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for(int k = 0; k < type.n; k++) {
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ERROR(!weyl_exists(type.factors[k]), "A Weyl group of type %c%d does not exist!\n", type.factors[k].series, type.factors[k].rank);
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for(int i = 0; i < type.factors[k].rank; i++) // A is the submatrix corresponding to the current simple factor
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A[i] = &m[(i+offset)*rank + offset];
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for(int i = 1; i < type.factors[k].rank; i++) {
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A[i][i-1] = -1;
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A[i-1][i] = -1;
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}
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switch(type.factors[k].series) {
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case 'A':
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break;
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case 'B':
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A[0][1] = -2;
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break;
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case 'C':
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A[1][0] = -2;
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break;
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case 'D':
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A[0][1] = A[1][0] = 0;
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A[0][2] = A[2][0] = -1;
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break;
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case 'E':
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A[1][2] = A[2][1] = 0;
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A[1][3] = A[3][1] = -1;
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break;
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case 'F':
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A[2][1] = -2;
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break;
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case 'G':
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A[1][0] = -3;
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break;
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}
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offset += type.factors[k].rank;
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}
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free(A);
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}
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/************ weyl_generate etc. ********************/
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static weylgroup_element_t* apply_word(int *word, int len, weylgroup_element_t *current)
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{
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for(int k = len - 1; k >= 0; k--) // apply group element from right to left
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current = current->left[word[k]];
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return current;
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}
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static weylgroup_element_t* apply_word_reverse(int *word, int len, weylgroup_element_t *current)
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{
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for(int k = 0; k < len; k++) // apply group element from left to right (i.e. apply inverse)
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current = current->left[word[k]];
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return current;
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}
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weylgroup_t *weyl_generate(semisimple_type_t type)
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{
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int rank = weyl_rank(type);
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int order = weyl_order(type);
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int positive = weyl_positive(type);
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ERROR(positive > 64, "We can't handle root systems with more than 64 positive roots!\n");
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// allocate result
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weylgroup_element_t *group = (weylgroup_element_t*)malloc(order*sizeof(weylgroup_element_t));
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weylgroup_t *result = malloc(sizeof(weylgroup_t));
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result->type = type;
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result->elements = group;
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result->lists = (weylgroup_element_t**)malloc(2*order*rank*sizeof(weylgroup_element_t*));
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for(int i = 0; i < order; i++) {
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group[i].left = result->lists + 2*i*rank;
|
|
group[i].right = result->lists + (2*i+1)*rank;
|
|
group[i].coset = (doublecoset_t*)0;
|
|
group[i].index = i;
|
|
}
|
|
|
|
// the main part
|
|
LOG("Start generating Weyl group.\n");
|
|
|
|
generate_left_and_ids(type, group);
|
|
|
|
// word length is just the number of 1s in the binary id
|
|
LOG("Find word lengths.\n");
|
|
|
|
for(int i = 0; i < order; i++) {
|
|
group[i].wordlength = 0;
|
|
for(int j = 0; j < positive; j++)
|
|
if(group[i].id & BIT(j))
|
|
group[i].wordlength++;
|
|
}
|
|
|
|
// allocate letters
|
|
|
|
int total_wordlength = 0;
|
|
for(int i = 0; i < order; i++)
|
|
total_wordlength += group[i].wordlength;
|
|
result->letters = (int*)malloc(total_wordlength*sizeof(int));
|
|
total_wordlength = 0;
|
|
for(int i = 0; i < order; i++) {
|
|
group[i].word = result->letters + total_wordlength;
|
|
total_wordlength += group[i].wordlength;
|
|
}
|
|
|
|
// find shortest words (using that the elements are already ordered by word length)
|
|
LOG("Find shortest words.\n");
|
|
|
|
memset(result->letters, -1, total_wordlength*sizeof(int));
|
|
for(int i = 0; i < order - 1; i++) {
|
|
weylgroup_element_t *this = &group[i];
|
|
for(int j = 0; j < rank; j++) {
|
|
weylgroup_element_t *that = group[i].left[j];
|
|
if(that->wordlength > this->wordlength && that->word[0] == -1) {
|
|
memcpy(that->word + 1, this->word, this->wordlength*sizeof(int));
|
|
that->word[0] = j;
|
|
}
|
|
}
|
|
}
|
|
|
|
// generate right edges
|
|
LOG("Compute right multiplication.\n");
|
|
|
|
for(int i = 0; i < order; i++)
|
|
for(int j = 0; j < rank; j++)
|
|
group[i].right[j] = apply_word(group[i].word, group[i].wordlength, group[0].left[j]);
|
|
|
|
// find opposites
|
|
LOG("Find opposites.\n");
|
|
|
|
weylgroup_element_t *longest = &group[order-1];
|
|
for(int i = 0; i < order; i++)
|
|
group[i].opposite = apply_word(longest->word, longest->wordlength, &group[i]);
|
|
|
|
// check for root reflections
|
|
LOG("Find root reflections.\n");
|
|
|
|
for(int i = 0; i < order; i++)
|
|
group[i].is_root_reflection = 0;
|
|
for(int i = 0; i < order; i++)
|
|
for(int j = 0; j < rank; j++) // we want to calculate word^{-1} * j * word; this is a root reflection
|
|
apply_word_reverse(group[i].word, group[i].wordlength, group[i].left[j]) -> is_root_reflection = 1;
|
|
|
|
return result;
|
|
}
|
|
|
|
void weyl_destroy(weylgroup_t *group)
|
|
{
|
|
free(group->elements);
|
|
free(group->lists);
|
|
free(group->letters);
|
|
free(group);
|
|
}
|
|
|
|
doublequotient_t *weyl_generate_bruhat(semisimple_type_t type, int left_invariance, int right_invariance)
|
|
{
|
|
int rank = weyl_rank(type);
|
|
int order = weyl_order(type);
|
|
int positive = weyl_positive(type);
|
|
int count;
|
|
|
|
int is_minimum, is_maximum;
|
|
|
|
weylgroup_t *wgroup = weyl_generate(type);
|
|
weylgroup_element_t *group = wgroup->elements;
|
|
doublecoset_t *cosets;
|
|
|
|
for(int i = 0; i < rank; i++) {
|
|
int oppi = weyl_opposition(type, i);
|
|
if(left_invariance & BIT(i) && !(left_invariance & BIT(oppi)) ||
|
|
left_invariance & BIT(oppi) && !(left_invariance & BIT(i)))
|
|
ERROR(1, "The specified left invariance is not invariant under the opposition involution!\n");
|
|
}
|
|
|
|
doublequotient_t *result = (doublequotient_t*)malloc(sizeof(doublequotient_t));
|
|
result->type = type;
|
|
result->left_invariance = left_invariance;
|
|
result->right_invariance = right_invariance;
|
|
result->group = wgroup->elements;
|
|
result->grouplists = wgroup->lists;
|
|
result->groupletters = wgroup->letters;
|
|
|
|
free(wgroup); // dissolved in result and not needed anymore
|
|
|
|
LOG("Count cosets.\n"); // count cosets by finding the minimum length element in every coset
|
|
|
|
count = 0;
|
|
for(int i = 0; i < order; i++) {
|
|
is_minimum = 1;
|
|
for(int j = 0; j < rank; j++)
|
|
if(left_invariance & BIT(j) && group[i].left[j]->wordlength < group[i].wordlength ||
|
|
right_invariance & BIT(j) && group[i].right[j]->wordlength < group[i].wordlength)
|
|
is_minimum = 0;
|
|
if(is_minimum)
|
|
count++;
|
|
}
|
|
result->count = count;
|
|
|
|
// alloc more stuff
|
|
|
|
cosets = result->cosets = (doublecoset_t*)malloc(count*sizeof(doublecoset_t));
|
|
for(int i = 0; i < count; i++) {
|
|
cosets[i].bruhat_lower = cosets[i].bruhat_higher = (doublecoset_list_t*)0;
|
|
}
|
|
result->lists = (doublecoset_list_t*)malloc(2*count*positive*sizeof(doublecoset_list_t)); // 2 times, for bruhat lower and higher
|
|
|
|
LOG("Find minimal length elements in cosets.\n"); // basically same code as above
|
|
|
|
count = 0;
|
|
for(int i = 0; i < order; i++) {
|
|
is_minimum = 1;
|
|
for(int j = 0; j < rank; j++)
|
|
if(left_invariance & BIT(j) && group[i].left[j]->wordlength < group[i].wordlength ||
|
|
right_invariance & BIT(j) && group[i].right[j]->wordlength < group[i].wordlength)
|
|
is_minimum = 0;
|
|
if(is_minimum) {
|
|
cosets[count].min = &group[i];
|
|
group[i].coset = &cosets[count];
|
|
count++;
|
|
}
|
|
}
|
|
|
|
LOG("Generate quotient map.\n");
|
|
|
|
for(int i = 0; i < order; i++) {
|
|
for(int j = 0; j < rank; j++) {
|
|
if(left_invariance & BIT(j) && group[i].left[j]->wordlength > group[i].wordlength)
|
|
group[i].left[j]->coset = group[i].coset;
|
|
if(right_invariance & BIT(j) && group[i].right[j]->wordlength > group[i].wordlength)
|
|
group[i].right[j]->coset = group[i].coset;
|
|
}
|
|
}
|
|
|
|
LOG("Find maximal length elements.\n");
|
|
|
|
for(int i = 0; i < order; i++) {
|
|
is_maximum = 1;
|
|
for(int j = 0; j < rank; j++)
|
|
if(left_invariance & BIT(j) && group[i].left[j]->wordlength > group[i].wordlength ||
|
|
right_invariance & BIT(j) && group[i].right[j]->wordlength > group[i].wordlength)
|
|
is_maximum = 0;
|
|
if(is_maximum) {
|
|
group[i].coset->max = &group[i];
|
|
}
|
|
}
|
|
|
|
LOG("Find opposites.\n");
|
|
|
|
for(int i = 0; i < count; i++)
|
|
cosets[i].opposite = cosets[i].min->opposite->coset;
|
|
|
|
LOG("Sort opposites.\n");
|
|
|
|
int *old2newindices = (int*)malloc(count*sizeof(int));
|
|
int *new2oldindices = (int*)malloc(count*sizeof(int));
|
|
|
|
// give the cosets some temporary indices
|
|
for(int i = 0; i < count; i++)
|
|
cosets[i].index = i;
|
|
|
|
// generate a nice ordering, where element j is opposite to n-j, except the self-opposite ones, which are in the middle
|
|
int j = 0;
|
|
for(int i = 0; i < count; i++)
|
|
if(i < cosets[i].opposite->index) {
|
|
old2newindices[i] = j;
|
|
old2newindices[cosets[i].opposite->index] = count-1-j;
|
|
j++;
|
|
}
|
|
for(int i = 0; i < count; i++)
|
|
if(i == cosets[i].opposite->index)
|
|
old2newindices[i] = j++;
|
|
|
|
for(int i = 0; i < count; i++)
|
|
new2oldindices[old2newindices[i]] = i;
|
|
|
|
// rewrite everything in the new ordering
|
|
doublecoset_t *oldcosets = (doublecoset_t*)malloc(count*sizeof(doublecoset_t));
|
|
memcpy(oldcosets, cosets, count*sizeof(doublecoset_t));
|
|
for(int i = 0; i < count; i++) {
|
|
cosets[i].min = oldcosets[new2oldindices[i]].min;
|
|
cosets[i].max = oldcosets[new2oldindices[i]].max;
|
|
cosets[i].opposite = cosets + old2newindices[oldcosets[new2oldindices[i]].opposite->index];
|
|
// cosets[i].bruhat_lower = oldcosets[new2oldindices[i]].bruhat_lower;
|
|
// cosets[i].bruhat_higher = oldcosets[new2oldindices[i]].bruhat_higher;
|
|
// for(doublecoset_list_t *current = cosets[i].bruhat_lower; current; current = current -> next)
|
|
// current->to = &cosets[old2newindices[current->to->index]];
|
|
// for(doublecoset_list_t *current = cosets[i].bruhat_higher; current; current = current -> next)
|
|
// current->to = &cosets[old2newindices[current->to->index]];
|
|
}
|
|
for(int i = 0; i < order; i++)
|
|
group[i].coset = old2newindices[group[i].coset->index] + cosets;
|
|
for(int i = 0; i < count; i++) // do this in the end, so we can use the "index" attribute before to translate pointers to indices
|
|
cosets[i].index = i;
|
|
|
|
free(old2newindices);
|
|
free(new2oldindices);
|
|
free(oldcosets);
|
|
|
|
LOG("Find bruhat order.\n");
|
|
|
|
int edgecount = 0;
|
|
for(int i = 0; i < order; i++) {
|
|
if(group[i].is_root_reflection) {
|
|
for(int j = 0; j < count; j++) {
|
|
weylgroup_element_t *this = cosets[j].min;
|
|
weylgroup_element_t *that = apply_word(group[i].word, group[i].wordlength, cosets[j].min);
|
|
if(this->wordlength > that->wordlength) { // this is higher in bruhat order than that
|
|
doublecoset_list_t *new = &result->lists[edgecount++];
|
|
new->next = this->coset->bruhat_lower;
|
|
this->coset->bruhat_lower = new;
|
|
new->to = that->coset;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
LOG("Perform transitive reduction.\n"); // eliminating redudant order relations
|
|
|
|
doublecoset_t *origin;
|
|
doublecoset_list_t *current;
|
|
doublecoset_list_t *prev;
|
|
queue_t queue;
|
|
int idx;
|
|
int *seen = malloc(count*sizeof(int));
|
|
for(int i = 0; i < count; i++) {
|
|
memset(seen, 0, count*sizeof(int));
|
|
queue_init(&queue);
|
|
|
|
for(int len = 1; len <= cosets[i].min->wordlength; len++) {
|
|
|
|
// remove all edges originating from i of length len which connect to something already seen using shorter edges
|
|
origin = &cosets[i];
|
|
prev = (doublecoset_list_t*)0;
|
|
|
|
for(current = origin->bruhat_lower; current; current = current->next) {
|
|
if(origin->min->wordlength - current->to->min->wordlength != len) {
|
|
prev = current;
|
|
} else if(seen[current->to->index]) {
|
|
if(prev)
|
|
prev->next = current->next;
|
|
else
|
|
origin->bruhat_lower = current->next;
|
|
} else {
|
|
prev = current;
|
|
seen[current->to->index] = 1;
|
|
queue_put(&queue, current->to->index);
|
|
}
|
|
}
|
|
|
|
// see which nodes we can reach using only edges up to length len, mark them as seen
|
|
while((idx = queue_get(&queue)) != -1) {
|
|
current = cosets[idx].bruhat_lower;
|
|
for(current = cosets[idx].bruhat_lower; current; current = current->next) {
|
|
if(!seen[current->to->index]) {
|
|
seen[current->to->index] = 1;
|
|
queue_put(&queue, current->to->index);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
free(seen);
|
|
|
|
LOG("Revert bruhat order.\n");
|
|
|
|
for(int i = 0; i < count; i++) {
|
|
for(current = cosets[i].bruhat_lower; current; current = current->next) {
|
|
doublecoset_list_t *new = &result->lists[edgecount++];
|
|
new->to = &cosets[i];
|
|
new->next = current->to->bruhat_higher;
|
|
current->to->bruhat_higher = new;
|
|
}
|
|
}
|
|
|
|
return result;
|
|
}
|
|
|
|
void weyl_destroy_bruhat(doublequotient_t *dq)
|
|
{
|
|
free(dq->group);
|
|
free(dq->grouplists);
|
|
free(dq->groupletters);
|
|
free(dq->cosets);
|
|
free(dq->lists);
|
|
free(dq);
|
|
}
|