florian/enumerate-balanced-ideals
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Major rewrite

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This commit is contained in:
Florian Stecker 2017-01-27 20:48:44 +01:00
parent ab546946c8
commit efd8e621ea
6 changed files with 641 additions and 893 deletions
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12
Makefile
View File

@ -6,20 +6,14 @@ SPECIAL_OPTIONS=-O3 -flto -funroll-loops -Winline
OPTIONS=-m64 -march=native -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS) OPTIONS=-m64 -march=native -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS)
all: generate process all: enumerate
generate: generate.o weyl.o thickenings.o enumerate: generate.o weyl.o thickenings.o
gcc $(OPTIONS) -o generate generate.o thickenings.o weyl.o gcc $(OPTIONS) -o generate generate.o thickenings.o weyl.o
process: process.o weyl.o thickenings.o
gcc $(OPTIONS) -o process process.o thickenings.o weyl.o
generate.o: generate.c $(HEADERS) generate.o: generate.c $(HEADERS)
gcc $(OPTIONS) -c generate.c gcc $(OPTIONS) -c generate.c
process.o: process.c $(HEADERS)
gcc $(OPTIONS) -c process.c
thickenings.o: thickenings.c $(HEADERS) thickenings.o: thickenings.c $(HEADERS)
gcc $(OPTIONS) -c thickenings.c gcc $(OPTIONS) -c thickenings.c
@ -27,4 +21,4 @@ weyl.o: weyl.c $(HEADERS)
gcc $(OPTIONS) -c weyl.c gcc $(OPTIONS) -c weyl.c
clean: clean:
rm -f generate process thickenings.o weyl.o generate.o process.o rm -f generate thickenings.o weyl.o generate.o

143
generate.c
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@ -9,20 +9,27 @@ char stringbuffer[100];
char stringbuffer2[100]; char stringbuffer2[100];
typedef struct { typedef struct {
node_t *graph; doublequotient_t *dq;
int cosets;
int rank; int rank;
int order; int order;
int hyperplanes; int positive;
semisimple_type_t type;
unsigned long left_invariance;
unsigned long right_invariance;
const char *alphabet;
int *buffer; int *buffer;
int level; int level;
} info_t; } info_t;
int shorten(int i, unsigned long left, unsigned long right, node_t *graph, int rank) static char* alphabetize(weylgroup_element_t *e, char *str)
{
if(e->wordlength == 0)
sprintf(str, "1");
else
for(int j = 0; j < e->wordlength; j++)
sprintf(str, "%c", e->word[j] + 'a');
return str;
}
/*
int shorten(int i, unsigned long left, unsigned long right, doublequotient_t *dq, int rank)
{ {
int other, shorter = i; int other, shorter = i;
do { do {
@ -41,6 +48,7 @@ int shorten(int i, unsigned long left, unsigned long right, node_t *graph, int r
return shorter; return shorter;
} }
*/
void balanced_thickening_callback(const bitvec_t *pos, int size, void *data) void balanced_thickening_callback(const bitvec_t *pos, int size, void *data)
{ {
@ -57,8 +65,8 @@ void balanced_thickening_callback(const bitvec_t *pos, int size, void *data)
for(int i = 0; i < size; i++) { for(int i = 0; i < size; i++) {
bit1 = i < size/2 ? bv_get_bit(pos, i) : !bv_get_bit(pos, size - 1 - i); bit1 = i < size/2 ? bv_get_bit(pos, i) : !bv_get_bit(pos, size - 1 - i);
for(int j = 0; j < info->rank; j++) { for(int j = 0; j < info->rank; j++) {
left = info->graph[i].left[j]; left = info->dq->cosets[i].min->left[j]->coset - info->dq->cosets;
right = info->graph[i].right[j]; right = info->dq->cosets[i].min->right[j]->coset - info->dq->cosets;
bit2left = left < size/2 ? bv_get_bit(pos, left) : !bv_get_bit(pos, size - 1 - left); bit2left = left < size/2 ? bv_get_bit(pos, left) : !bv_get_bit(pos, size - 1 - left);
bit2right = right < size/2 ? bv_get_bit(pos, right) : !bv_get_bit(pos, size - 1 - right); bit2right = right < size/2 ? bv_get_bit(pos, right) : !bv_get_bit(pos, size - 1 - right);
if(bit1 != bit2left) if(bit1 != bit2left)
@ -70,15 +78,16 @@ void balanced_thickening_callback(const bitvec_t *pos, int size, void *data)
printf("%4d left: ", totcount++); printf("%4d left: ", totcount++);
for(int j = 0; j < info->rank; j++) for(int j = 0; j < info->rank; j++)
printf("%c", left_invariance & (1 << j) ? info->alphabet[j] : ' '); printf("%c", left_invariance & (1 << j) ? j + 'a' : ' ');
printf(" right: "); printf(" right: ");
for(int j = 0; j < info->rank; j++) for(int j = 0; j < info->rank; j++)
printf("%c", right_invariance & (1 << j) ? info->alphabet[j] : ' '); printf("%c", right_invariance & (1 << j) ? j + 'a' : ' ');
/*
if(info->buffer) { if(info->buffer) {
printf(" generators:"); printf(" generators:");
queue_t queue; queue_t queue;
int current, left, right, shortest; int cur, left, right;
int *buffer = info->buffer; int *buffer = info->buffer;
for(int i = 0; i < size/2; i++) { for(int i = 0; i < size/2; i++) {
@ -87,32 +96,30 @@ void balanced_thickening_callback(const bitvec_t *pos, int size, void *data)
} }
for(int i = size-1; i >= 0; i--) { for(int i = size-1; i >= 0; i--) {
if(buffer[i]) { if(buffer[i]) {
int shortest = shorten(i, left_invariance, right_invariance, info->graph, info-> rank); weylgroup_element_t *shortest = shorten(i, left_invariance, right_invariance, info->dq);
printf(" %s", alphabetize(info->graph[shortest].word, printf(" %s", alphabetize(shortest, stringbuffer));
info->graph[shortest].wordlength,
info->alphabet,
stringbuffer));
buffer[i] = 0; buffer[i] = 0;
queue_init(&queue); queue_init(&queue);
queue_put(&queue, i); queue_put(&queue, i);
while((current = queue_get(&queue)) != -1) { while((cur = queue_get(&queue)) != -1) {
for(edgelist_t *edge = info->graph[current].bruhat_lower; edge != (edgelist_t*)0; edge = edge->next) { for(doublecoset_list_t *current = info->dq->coset[cur].bruhat_lower; current; current = current->next) {
if(buffer[edge->to]) { int idx = current->to - info->dq->cosets;
buffer[edge->to] = 0; if(buffer[idx]) {
queue_put(&queue, edge->to); buffer[idx] = 0;
queue_put(&queue, idx);
} }
} }
for(int j = 0; j < info->rank; j++) { for(int j = 0; j < info->rank; j++) {
left = info->graph[current].left[j]; left = info->dq->coset[cur].min.left[j] - info->dq->cosets;
if(left_invariance & (1 << j) && if(left_invariance & (1 << j) &&
info->graph[left].wordlength < info->graph[current].wordlength && info->graph[left].wordlength < info->graph[cur].wordlength &&
buffer[left]) { buffer[left]) {
buffer[left] = 0; buffer[left] = 0;
queue_put(&queue, left); queue_put(&queue, left);
} }
right = info->graph[current].left[j]; right = info->graph[cur].left[j];
if(right_invariance & (1 << j) && if(right_invariance & (1 << j) &&
info->graph[right].wordlength < info->graph[current].wordlength && info->graph[right].wordlength < info->graph[cur].wordlength &&
buffer[right]) { buffer[right]) {
buffer[right] = 0; buffer[right] = 0;
queue_put(&queue, right); queue_put(&queue, right);
@ -132,6 +139,7 @@ void balanced_thickening_callback(const bitvec_t *pos, int size, void *data)
printf("]"); printf("]");
} }
} }
*/
printf("\n"); printf("\n");
} }
@ -151,12 +159,11 @@ int main(int argc, const char *argv[])
{ {
semisimple_type_t type; semisimple_type_t type;
unsigned long right_invariance, left_invariance; unsigned long right_invariance, left_invariance;
int rank, order, hyperplanes, cosets; int rank, order, positive;
int fixpoints; int fixpoints;
node_t *graph; doublequotient_t *dq;
char string_buffer1[1000];
const char *alphabet = "abcdefghijklmnopqrstuvwxyz"; const char *alphabet = "abcdefghijklmnopqrstuvwxyz";
// read arguments // read arguments
@ -190,10 +197,7 @@ int main(int argc, const char *argv[])
// generate graph // generate graph
graph = graph_alloc(type); dq = weyl_generate_bruhat(type, left_invariance, right_invariance);
cosets = prepare_simplified_graph(type, left_invariance, right_invariance, graph);
ERROR(cosets < 0, "The left invariance is not preserved by the opposition involution!\n");
// print stuff // print stuff
@ -203,7 +207,7 @@ int main(int argc, const char *argv[])
rank = weyl_rank(type); // number of simple roots rank = weyl_rank(type); // number of simple roots
order = weyl_order(type); // number of Weyl group elements order = weyl_order(type); // number of Weyl group elements
hyperplanes = weyl_hyperplanes(type); // number of positive roots positive = weyl_positive(type); // number of positive roots
if(output_level >= 1) { if(output_level >= 1) {
if(left_invariance) { if(left_invariance) {
@ -226,59 +230,48 @@ int main(int argc, const char *argv[])
} }
fprintf(stdout, "\n"); fprintf(stdout, "\n");
fprintf(stdout, "Rank: %d\tOrder: %d\tPositive Roots: %d\tCosets: %d\n\n", rank, order, hyperplanes, cosets); fprintf(stdout, "Rank: %d\tOrder: %d\tPositive Roots: %d\tCosets: %d\n\n", rank, order, positive, dq->count);
} }
if(output_level >= 3) { if(output_level >= 3) {
fprintf(stdout, "Shortest coset representatives: \n"); fprintf(stdout, "Shortest coset representatives: \n");
for(int i = 0, wl = 0; i < cosets; i++) { for(int i = 0, wl = 0; i < dq->count; i++) {
if(i == 0) { if(dq->cosets[i].min->wordlength > wl) {
fprintf(stdout, "1"); printf("\n");
} else if(graph[i].wordlength > wl) { wl = dq->cosets[i].min->wordlength;
fprintf(stdout, "\n%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1)); }
wl = graph[i].wordlength; fprintf(stdout, "\n%s ", alphabetize(dq->cosets[i].min, stringbuffer));
} else
fprintf(stdout, "%s ", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
} }
fprintf(stdout, "\n\n"); fprintf(stdout, "\n\n");
} }
if(output_level >= 4) { if(output_level >= 4) {
edgelist_t *edge;
fprintf(stdout, "Bruhat order in graphviz format:\n"); fprintf(stdout, "Bruhat order in graphviz format:\n");
fprintf(stdout, "digraph test123 {\n"); fprintf(stdout, "digraph test123 {\n");
for(int i = 0; i < cosets; i++) { for(int i = 0; i < dq->count; i++)
edge = graph[i].bruhat_lower; for(doublecoset_list_t *current = dq->cosets[i].bruhat_lower; current; current = current->next)
while(edge) {
fprintf(stdout, "%s -> %s;\n", fprintf(stdout, "%s -> %s;\n",
alphabetize(graph[i].word, graph[i].wordlength, alphabet, stringbuffer), alphabetize(dq->cosets[i].min, stringbuffer),
alphabetize(graph[edge->to].word, graph[edge->to].wordlength, alphabet, stringbuffer2)); alphabetize(current->to->min, stringbuffer2));
edge = edge->next;
}
}
fprintf(stdout, "}\n\n"); fprintf(stdout, "}\n\n");
} }
if(output_level >= 4) { if(output_level >= 4) {
fprintf(stdout, "Opposites:\n"); fprintf(stdout, "Opposites:\n");
for(int i = 0; i < cosets; i++) { for(int i = 0; i < dq->count; i++)
fprintf(stdout, "%s <-> %s\n", fprintf(stdout, "%s <-> %s\n",
alphabetize(graph[i].word, graph[i].wordlength, alphabet, stringbuffer), alphabetize(dq->cosets[i].min, stringbuffer),
alphabetize(graph[graph[i].opposite].word, graph[graph[i].opposite].wordlength, alphabet, stringbuffer2)); alphabetize(dq->cosets[i].opposite->min, stringbuffer2));
}
fprintf(stdout, "\n"); fprintf(stdout, "\n");
} }
fixpoints = 0; fixpoints = 0;
for(int i = 0; i < cosets; i++) for(int i = 0; i < dq->count; i++)
if(graph[i].opposite == i) { if(dq->cosets[i].opposite == &dq->cosets[i]) {
if(output_level >= 1) { if(output_level >= 1) {
if(fixpoints == 0) if(fixpoints == 0)
fprintf(stdout, "No thickenings since the longest element fixes the following cosets: %s", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1)); fprintf(stdout, "No thickenings since the longest element fixes the following cosets:");
else fprintf(stdout, " %s", alphabetize(dq->cosets[i].min, stringbuffer));
fprintf(stdout, " %s", alphabetize(graph[i].word, graph[i].wordlength, alphabet, string_buffer1));
} }
fixpoints++; fixpoints++;
} }
@ -287,37 +280,31 @@ int main(int argc, const char *argv[])
if(!fixpoints) { if(!fixpoints) {
int *buffer = (int*)malloc(cosets*sizeof(int)); int *buffer = (int*)malloc(dq->count*sizeof(int));
info_t info; info_t info;
info.graph = graph; info.dq = dq;
info.cosets = cosets; info.rank = weyl_rank(type);
info.rank = rank; info.order = weyl_order(type);
info.order = order; info.positive = weyl_positive(type);
info.hyperplanes = hyperplanes;
info.type = type;
info.left_invariance = left_invariance;
info.right_invariance = right_invariance;
info.alphabet = alphabet;
info.buffer = buffer; info.buffer = buffer;
info.level = output_level; info.level = output_level;
long count; long count;
if(output_level >= 2) { if(output_level >= 2) {
fprintf(stdout, "Balanced ideals:\n", count); fprintf(stdout, "Balanced ideals:\n", count);
count = enumerate_balanced_thickenings(graph, cosets, balanced_thickening_callback, &info); count = enumerate_balanced_thickenings(dq, balanced_thickening_callback, &info);
fprintf(stdout, "\n", count); fprintf(stdout, "\n", count);
} else { } else {
long outputcount = 0; long outputcount = 0;
count = enumerate_balanced_thickenings(graph, cosets, balanced_thickening_simple_callback, &outputcount); count = enumerate_balanced_thickenings(dq, balanced_thickening_simple_callback, &outputcount);
} }
if(output_level >= 1) if(output_level >= 1)
fprintf(stdout, "Found %ld balanced ideal%s\n", count, count == 1 ? "" : "s"); fprintf(stdout, "Found %ld balanced ideal%s\n", count, count == 1 ? "" : "s");
} }
graph_free(type, graph); weyl_destroy_bruhat(dq);
free(type.factors); free(type.factors);
return 0; return 0;

629
thickenings.c
View File

@ -8,606 +8,6 @@
#include "weyl.h" #include "weyl.h"
#include "queue.h" #include "queue.h"
char *alphabetize(int *word, int len, const char *alphabet, char *buffer)
{
if(len == 0) {
buffer[0] = '1';
buffer[1] = 0;
return buffer;
}
int i = 0;
for(i = 0; i < len; i++)
buffer[i] = alphabet[word[i]];
buffer[i] = 0;
return buffer;
}
void print_thickening(int rank, int order, const signed char *thickening, int upto_level, const char *alphabet, FILE *f)
{
for(int i = 0; i < order; i++) {
if(thickening[i] == HEAD_MARKER)
fprintf(f, "\e[41;37mx\e[0m");
else if(thickening[i] < - upto_level || thickening[i] > upto_level)
fprintf(f, " ");
else if(thickening[i] < 0 && thickening[i] > -10)
fprintf(f, "\e[47;30m%d\e[0m", -thickening[i]);
else if(thickening[i] <= -10)
fprintf(f, "\e[47;30m+\e[0m");
else if(thickening[i] > 0 && thickening[i] < 10)
fprintf(f, "\e[40;37m%d\e[0m", thickening[i]);
else if(thickening[i] >= 10)
fprintf(f, "\e[40;37m+\e[0m");
else
fprintf(f, " ");
}
fprintf(f, "\e[K");
}
static int compare_wordlength(const void *a, const void *b, void *gr)
{
int i = *((int*)a);
int j = *((int*)b);
node_t *graph = (node_t*)gr;
return graph[i].wordlength - graph[j].wordlength;
}
void prepare_graph(semisimple_type_t type, node_t *graph)
{
queue_t queue;
edgelist_t *edgelists_lower, *edgelists_higher;
int rank, order, hyperplanes;
edgelist_t *edge, *previous;
int edgelist_count, hyperplane_count;
int current;
weylgroup_element_t *graph_data;
node_t *graph_unsorted;
int *ordering, *reverse_ordering, *seen;
// initialize
rank = weyl_rank(type);
order = weyl_order(type);
hyperplanes = weyl_hyperplanes(type);
edgelists_higher = graph[0].bruhat_higher;
edgelists_lower = &graph[0].bruhat_higher[order*hyperplanes/2];
graph_data = weyl_alloc(type);
graph_unsorted = (node_t*)malloc(order*sizeof(node_t));
ordering = (int*)malloc(order*sizeof(int));
reverse_ordering = (int*)malloc(order*sizeof(int));
seen = (int*)malloc(order*sizeof(int));
for(int i = 0; i < order; i++) {
graph_unsorted[i].wordlength = INT_MAX;
graph[i].bruhat_lower = 0;
graph[i].bruhat_higher = 0;
graph[i].is_hyperplane_reflection = 0;
}
LOG("Generate Weyl group.\n");
weyl_generate(type, graph_data);
for(int i = 0; i < order; i++)
for(int j = 0; j < rank; j++) {
graph_unsorted[i].left = graph_data[i].left;
graph_unsorted[i].id = graph_data[i].id;
}
// find wordlengths
LOG("Determine word lengths.\n");
graph_unsorted[0].wordlength = 0;
queue_init(&queue);
queue_put(&queue, 0);
while((current = queue_get(&queue)) != -1) {
for(int i = 0; i < rank; i++) {
int neighbor = graph_unsorted[current].left[i];
if(graph_unsorted[neighbor].wordlength > graph_unsorted[current].wordlength + 1) {
graph_unsorted[neighbor].wordlength = graph_unsorted[current].wordlength + 1;
queue_put(&queue, neighbor);
}
}
}
LOG("Sort by wordlength.\n");
for(int i = 0; i < order; i++)
ordering[i] = i;
qsort_r(ordering, order, sizeof(int), compare_wordlength, graph_unsorted); // so ordering is a map new index -> old index
for(int i = 0; i < order; i++)
reverse_ordering[ordering[i]] = i; // reverse_ordering is a map old index -> new index
for(int i = 0; i < order; i++) {
// we have only set left, wordlength and id so far, so just copy these
graph[i].wordlength = graph_unsorted[ordering[i]].wordlength;
graph[i].id = graph_unsorted[ordering[i]].id;
for(int j = 0; j < rank; j++)
graph[i].left[j] = reverse_ordering[graph_unsorted[ordering[i]].left[j]]; // rewrite references
}
LOG("Find shortest words.\n");
for(int i = 0; i < order; i++)
memset(graph[i].word, 0, hyperplanes*sizeof(int));
queue_init(&queue);
queue_put(&queue, 0);
while((current = queue_get(&queue)) != -1) {
for(int i = 0; i < rank; i++) {
int neighbor = graph[current].left[i];
if(graph[neighbor].wordlength == graph[current].wordlength + 1 && graph[neighbor].word[0] == 0) {
memcpy(&graph[neighbor].word[1], &graph[current].word[0], graph[current].wordlength*sizeof(int));
graph[neighbor].word[0] = i;
queue_put(&queue, neighbor);
}
}
}
LOG("Generate right edges.\n");
for(int i = 0; i < order; i++) {
for(int j = 0; j < rank; j++) {
current = graph[0].left[j];
for(int k = graph[i].wordlength - 1; k >= 0; k--) { // apply group element from right to left
current = graph[current].left[graph[i].word[k]];
}
graph[i].right[j] = current;
}
}
LOG("Find opposites.\n");
node_t *longest = &graph[order-1];
for(int i = 0; i < order; i++) {
current = i;
for(int k = longest->wordlength - 1; k >= 0; k--)
current = graph[current].left[longest->word[k]];
graph[i].opposite = current;
}
LOG("Enumerate hyperplanes.\n");
hyperplane_count = 0;
for(int i = 0; i < order; i++) {
for(int j = 0; j < rank; j++) {
current = 0;
int *word1 = graph[i].word;
int word1len = graph[i].wordlength;
int *word2 = graph[graph[i].right[j]].word; // want to calculate word2 * word1^{-1}
int word2len = graph[graph[i].right[j]].wordlength;
for(int k = 0; k < word1len; k++) // apply inverse, i.e. go from left to right
current = graph[current].left[word1[k]];
for(int k = word2len - 1; k >= 0; k--) // now from right to left
current = graph[current].left[word2[k]];
if(graph[current].is_hyperplane_reflection == 0) {
graph[current].is_hyperplane_reflection = 1;
hyperplane_count++;
}
}
}
LOG("Determine Bruhat order.\n");
edgelist_count = 0;
for(int i = 0; i < order; i++) {
if(graph[i].is_hyperplane_reflection) {
for(int j = 0; j < order; j++) {
current = j;
for(int k = graph[i].wordlength - 1; k >= 0; k--) // apply hyperplane reflection
current = graph[current].left[graph[i].word[k]];
if(graph[j].wordlength < graph[current].wordlength) { // current has higher bruhat order than j
edgelists_lower[edgelist_count].to = j;
edgelists_lower[edgelist_count].next = graph[current].bruhat_lower;
graph[current].bruhat_lower = &edgelists_lower[edgelist_count];
edgelist_count++;
} else if(graph[j].wordlength > graph[current].wordlength) { // j has higher bruhat order than current; these are already included from the other side
} else {
ERROR(1, "Chambers of equal word lengths should not be folded on each other!\n");
}
}
}
}
LOG("Perform transitive reduction.\n");
for(int i = 0; i < order; i++) {
memset(seen, 0, order*sizeof(int));
queue_init(&queue);
for(int len = 1; len <= graph[i].wordlength; len++) {
// remove all edges originating from i of length len which connect to something already seen using shorter edges
edge = graph[i].bruhat_lower;
previous = (edgelist_t*)0;
while(edge) {
if(graph[i].wordlength - graph[edge->to].wordlength != len) {
previous = edge;
} else if(seen[edge->to]) {
if(previous)
previous->next = edge->next;
else
graph[i].bruhat_lower = edge->next;
} else {
previous = edge;
seen[edge->to] = 1;
queue_put(&queue, edge->to);
}
edge = edge->next;
}
// see which nodes we can reach using only edges up to length len, mark them as seen
while((current = queue_get(&queue)) != -1) {
edge = graph[current].bruhat_lower;
while(edge) {
if(!seen[edge->to]) {
seen[edge->to] = 1;
queue_put(&queue, edge->to);
}
edge = edge->next;
}
}
}
}
LOG("Revert Bruhat order.\n");
edgelist_count = 0;
for(int i = 0; i < order; i++) {
edge = graph[i].bruhat_lower;
while(edge) {
edgelists_higher[edgelist_count].to = i;
edgelists_higher[edgelist_count].next = graph[edge->to].bruhat_higher;
graph[edge->to].bruhat_higher = &edgelists_higher[edgelist_count];
edgelist_count++;
edge = edge->next;
}
}
LOG("Sort opposites.\n");
// additional sorting step to force opposite property (opposite of j is at n - j - 1)
for(int i = 0; i < order; i++)
reverse_ordering[i] = -1;
for(int i = 0, j = 0; i < order; i++) { // i = old index, j = new index
if(reverse_ordering[i] == -1) {
reverse_ordering[i] = j;
ordering[j] = i;
reverse_ordering[graph[i].opposite] = order - j - 1;
ordering[order - j - 1] = graph[i].opposite;
j++;
}
}
memcpy(graph_unsorted, graph, order*sizeof(node_t));
for(int i = 0; i < order; i++) {
graph[i] = graph_unsorted[ordering[i]];
graph[i].opposite = reverse_ordering[graph[i].opposite];
for(int j = 0; j < rank; j++) {
graph[i].left[j] = reverse_ordering[graph[i].left[j]];
graph[i].right[j] = reverse_ordering[graph[i].right[j]];
}
for(edge = graph[i].bruhat_lower; edge; edge = edge->next)
edge->to = reverse_ordering[edge->to];
for(edge = graph[i].bruhat_higher; edge; edge = edge->next)
edge->to = reverse_ordering[edge->to];
}
weyl_free(graph_data);
free(graph_unsorted);
free(ordering);
free(reverse_ordering);
free(seen);
}
static int edgelist_contains(edgelist_t *list, int x) {
while(list) {
if(list->to == x)
return 1;
list = list->next;
}
return 0;
}
static edgelist_t *edgelist_add(edgelist_t *list, int new, edgelist_t *storage, int *storage_index)
{
edgelist_t *new_link = &storage[*storage_index];
new_link->next = list;
new_link->to = new;
(*storage_index)++;
return new_link;
}
int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigned long right, node_t *simplified_graph)
{
node_t *full_graph;
int edgelists_used;
int rank, order, hyperplanes;
int *reduced, *group, *simplified;
int *seen;
int current;
edgelist_t *edge, *previous;
queue_t queue;
int ncosets;
rank = weyl_rank(type);
order = weyl_order(type);
hyperplanes = weyl_hyperplanes(type);
for(int i = 0; i < rank; i++) {
int oppi = weyl_opposition(type, i);
if(left & BIT(i) && !(left & BIT(oppi)) ||
left & BIT(oppi) && !(left & BIT(i)))
return -1;
}
edgelist_t *edgelists_higher = &simplified_graph[0].bruhat_higher[0];
edgelist_t *edgelists_lower = &simplified_graph[0].bruhat_higher[order*hyperplanes/2];
// get full graph
full_graph = graph_alloc(type);
prepare_graph(type, full_graph);
LOG("Full graph generated.\n");
// initialize stuff
reduced = (int*)malloc(order*sizeof(int));
group = (int*)malloc(order*sizeof(int));
simplified = (int*)malloc(order*sizeof(int));
for(int i = 0; i < order; i++) {
group[i] = -1;
reduced[i] = i;
}
LOG("Group by double coset.\n");
// step 1: group
for(int i = 0; i < order; i++) {
if(group[i] != -1)
continue;
queue_init(&queue);
queue_put(&queue, i);
while((current = queue_get(&queue)) != -1) {
if(group[current] != -1)
continue;
group[current] = i;
for(int j = 0; j < rank; j++) {
if(left & (1 << j))
queue_put(&queue, full_graph[current].left[j]);
if(right & (1 << j))
queue_put(&queue, full_graph[current].right[j]);
}
}
}
LOG("Find minimal length elements.\n");
// step 2: find minimum
for(int i = 0; i < order; i++)
if(full_graph[i].wordlength < full_graph[reduced[group[i]]].wordlength)
reduced[group[i]] = i;
// step 3: assign minimum to all
for(int i = 0; i < order; i++)
reduced[i] = reduced[group[i]];
// step 4: assign indices to cosets
ncosets = 0;
for(int i = 0; i < order; i++)
if(reduced[i] == i)
simplified[i] = ncosets++;
for(int i = 0; i < order; i++)
simplified[i] = simplified[reduced[i]];
seen = (int*) malloc(ncosets*sizeof(int));
edgelists_used = 0;
LOG("Copy minimal elements.\n");
// step 5: set up nodes from minima
current = 0;
for(int i = 0; i < order; i++)
if(reduced[i] == i) { // is minimum
memcpy(simplified_graph[simplified[i]].word, full_graph[i].word, full_graph[i].wordlength*sizeof(int));
simplified_graph[simplified[i]].wordlength = full_graph[i].wordlength;
simplified_graph[simplified[i]].opposite = simplified[full_graph[i].opposite];
simplified_graph[simplified[i]].id = full_graph[i].id;
simplified_graph[simplified[i]].bruhat_lower = (edgelist_t*)0;
simplified_graph[simplified[i]].bruhat_higher = (edgelist_t*)0;
for(int j = 0; j < rank; j++) {
simplified_graph[simplified[i]].left[j] = simplified[full_graph[i].left[j]];
simplified_graph[simplified[i]].right[j] = simplified[full_graph[i].right[j]];
}
}
LOG("Find induced order.\n");
// step 6: find order relations
for(int i = 0; i < order; i++) {
edge = full_graph[i].bruhat_lower;
while(edge) {
int this = simplified[i];
int that = simplified[edge->to];
if(this != that) {
// found something
if(!edgelist_contains(simplified_graph[this].bruhat_lower, that))
simplified_graph[this].bruhat_lower = edgelist_add(simplified_graph[this].bruhat_lower, that, edgelists_lower, &edgelists_used);
ERROR(simplified_graph[this].wordlength <= simplified_graph[that].wordlength, "The order assumption is being violated!\n");
}
edge = edge->next;
}
}
LOG("Perform transitive reduction.\n");
// step 7: remove redundant edges
for(int i = 0; i < ncosets; i++) {
memset(seen, 0, ncosets*sizeof(int));
queue_init(&queue);
for(int len = 1; len <= simplified_graph[i].wordlength; len++) {
edge = simplified_graph[i].bruhat_lower;
previous = (edgelist_t*)0;
while(edge) {
// only look at edges of this length now
if(simplified_graph[i].wordlength - simplified_graph[edge->to].wordlength != len) {
// we only consider edges of length len in this pass
previous = edge;
} else if(seen[edge->to]) {
// this edge is redundant, remove it
if(previous)
previous->next = edge->next;
else
simplified_graph[i].bruhat_lower = edge->next;
} else {
// this edge was not redundant, add to seen
previous = edge;
seen[edge->to] = 1;
queue_put(&queue, edge->to);
}
edge = edge->next;
}
// calculate transitive closure of seen nodes
while((current = queue_get(&queue)) != -1) {
edge = simplified_graph[current].bruhat_lower;
while(edge) {
if(!seen[edge->to]) {
seen[edge->to] = 1;
queue_put(&queue, edge->to);
}
edge = edge->next;
}
}
}
}
LOG("Revert order.\n");
// step 8: revert order
edgelists_used = 0;
for(int i = 0; i < ncosets; i++) {
edge = simplified_graph[i].bruhat_lower;
while(edge) {
simplified_graph[edge->to].bruhat_higher =
edgelist_add(simplified_graph[edge->to].bruhat_higher,
i, edgelists_higher, &edgelists_used);
edge = edge->next;
}
}
LOG("Sort opposites.\n");
int *ordering = (int*)malloc(ncosets*sizeof(int));
int *reverse_ordering = (int*)malloc(ncosets*sizeof(int));
node_t *unsorted = (node_t*)malloc(ncosets*sizeof(node_t));
int opp, pos;
pos = 0;
for(int i = 0; i < ncosets; i++) { // first all the pairs
opp = simplified_graph[i].opposite;
if(opp > i) { // first occurrence of this pair
ordering[pos] = i;
ordering[ncosets-1-pos] = opp;
reverse_ordering[i] = pos;
reverse_ordering[opp] = ncosets-1-pos;
pos++;
}
}
for(int i = 0; i < ncosets; i++) // and finally the self-opposites
if(simplified_graph[i].opposite == i) {
ordering[pos] = i;
reverse_ordering[i] = pos;
pos++;
}
// now really do it
memcpy(unsorted, simplified_graph, ncosets*sizeof(node_t));
for(int i = 0; i < ncosets; i++) {
simplified_graph[i] = unsorted[ordering[i]];
simplified_graph[i].opposite = reverse_ordering[simplified_graph[i].opposite];
for(edgelist_t *edge = simplified_graph[i].bruhat_lower; edge != (edgelist_t*)0; edge = edge->next)
edge->to = reverse_ordering[edge->to];
for(edgelist_t *edge = simplified_graph[i].bruhat_higher; edge != (edgelist_t*)0; edge = edge->next)
edge->to = reverse_ordering[edge->to];
for(int j = 0; j < rank; j++) {
simplified_graph[i].left[j] = reverse_ordering[simplified_graph[i].left[j]];
simplified_graph[i].right[j] = reverse_ordering[simplified_graph[i].right[j]];
}
}
free(ordering);
free(reverse_ordering);
free(unsorted);
free(seen);
free(reduced);
free(group);
free(simplified);
graph_free(type, full_graph);
LOG("Simplified graph generated.\n");
return ncosets;
}
node_t *graph_alloc(semisimple_type_t type)
{
int rank = weyl_rank(type);
int order = weyl_order(type);
int hyperplanes = weyl_hyperplanes(type);
node_t *graph = (node_t*)malloc(order*sizeof(node_t));
int *left = (int*)malloc(order*rank*sizeof(int));
int *right = (int*)malloc(order*rank*sizeof(int));
edgelist_t *edgelists = (edgelist_t*)malloc(order*hyperplanes*sizeof(edgelist_t));
int *words = (int*)malloc(order*hyperplanes*sizeof(int));
for(int i = 0; i < order; i++) {
graph[i].left = &left[rank*i];
graph[i].right = &right[rank*i];
graph[i].word = &words[hyperplanes*i];
}
graph[0].bruhat_higher = edgelists;
return graph;
}
void graph_free(semisimple_type_t type, node_t *graph)
{
free(graph[0].left);
free(graph[0].right);
free(graph[0].word);
int order = weyl_order(type);
// find the head of all edgelists by just taking the one having the lowest address
edgelist_t *edgelists = graph[0].bruhat_lower;
for(int i = 0; i < order; i++) {
if(graph[i].bruhat_lower < edgelists && graph[i].bruhat_lower != 0)
edgelists = graph[i].bruhat_lower;
if(graph[i].bruhat_higher < edgelists && graph[i].bruhat_higher != 0)
edgelists = graph[i].bruhat_higher;
}
free(edgelists);
}
/*********************************** THE ACTUAL ENUMERATION ****************************************/
typedef struct { typedef struct {
int size; // the size of the weyl group. We store however only the first size/2 elements int size; // the size of the weyl group. We store however only the first size/2 elements
bitvec_t *principal_pos; bitvec_t *principal_pos;
@ -680,18 +80,15 @@ static long enumerate_tree(const enumeration_info_t *info, const bitvec_t *pos,
next_next_neg = bv_next_zero(&known, next_next_neg + 1); next_next_neg = bv_next_zero(&known, next_next_neg + 1);
} while(next_next_neg <= info->size/2); } while(next_next_neg <= info->size/2);
// multiprocessing stuff
// if(level == 3)
// fprintf(stderr, "%d\n", count);
return count; return count;
} }
void generate_principal_ideals(node_t *graph, int size, bitvec_t *pos, bitvec_t *neg, int *is_slim) static void generate_principal_ideals(doublequotient_t *dq, bitvec_t *pos, bitvec_t *neg, int *is_slim)
{ {
queue_t queue; queue_t queue;
int current; int current;
edgelist_t *edge; doublecoset_list_t *edge;
int size = dq->count;
// generate principal ideals // generate principal ideals
int *principal = (int*)malloc(size*sizeof(int)); int *principal = (int*)malloc(size*sizeof(int));
@ -701,10 +98,10 @@ void generate_principal_ideals(node_t *graph, int size, bitvec_t *pos, bitvec_t
queue_init(&queue); queue_init(&queue);
queue_put(&queue, i); queue_put(&queue, i);
while((current = queue_get(&queue)) != -1) while((current = queue_get(&queue)) != -1)
for(edge = graph[current].bruhat_lower; edge; edge = edge->next) for(edge = dq->cosets[current].bruhat_lower; edge; edge = edge->next)
if(!principal[edge->to]) { if(!principal[edge->to - dq->cosets]) {
principal[edge->to] = 1; principal[edge->to - dq->cosets] = 1;
queue_put(&queue, edge->to); queue_put(&queue, edge->to - dq->cosets);
} }
// copy the first half into bitvectors // copy the first half into bitvectors
@ -726,7 +123,7 @@ void generate_principal_ideals(node_t *graph, int size, bitvec_t *pos, bitvec_t
fprintf(stderr, " ids: [0"); fprintf(stderr, " ids: [0");
for(int j = 1; j < size; j++) for(int j = 1; j < size; j++)
if(principal[j]) if(principal[j])
fprintf(stderr, ", %d", graph[j].id); fprintf(stderr, ", %d", dq->cosets[j].min->id);
fprintf(stderr, "]\n"); fprintf(stderr, "]\n");
} }
#endif #endif
@ -757,12 +154,12 @@ void generate_principal_ideals(node_t *graph, int size, bitvec_t *pos, bitvec_t
returns the number of balanced ideals returns the number of balanced ideals
*/ */
long enumerate_balanced_thickenings(node_t *graph, int size, void (*callback) (const bitvec_t *, int, void*), void *callback_data) long enumerate_balanced_thickenings(doublequotient_t *dq, void (*callback) (const bitvec_t *, int, void*), void *callback_data)
{ {
long count = 0; long count = 0;
enumeration_info_t info; enumeration_info_t info;
info.size = size; info.size = dq->count;
info.callback = callback; info.callback = callback;
info.callback_data = callback_data; info.callback_data = callback_data;
info.principal_pos = (bitvec_t*)malloc(info.size*sizeof(bitvec_t)); info.principal_pos = (bitvec_t*)malloc(info.size*sizeof(bitvec_t));
@ -771,15 +168,15 @@ long enumerate_balanced_thickenings(node_t *graph, int size, void (*callback) (c
// the algorithm only works if the opposition pairing does not stabilize any element // the algorithm only works if the opposition pairing does not stabilize any element
// if this happens, there can be no balanced thickenings // if this happens, there can be no balanced thickenings
for(int i = 0; i < info.size; i++) for(int i = 0; i < dq->count; i++)
if(graph[i].opposite == i) if(dq->cosets[i].opposite->min->id == dq->cosets[i].min->id)
return 0; return 0;
// we can only handle bitvectors up to 64*BV_QWORD_RANK bits, but we only store half of the weyl group // we can only handle bitvectors up to 64*BV_QWORD_RANK bits, but we only store half of the weyl group
if(info.size > 128*BV_QWORD_RANK) if(info.size > 128*BV_QWORD_RANK)
return -1; return -1;
generate_principal_ideals(graph, size, info.principal_pos, info.principal_neg, info.principal_is_slim); generate_principal_ideals(dq, info.principal_pos, info.principal_neg, info.principal_is_slim);
// enumerate balanced ideals // enumerate balanced ideals
bitvec_t pos, neg; bitvec_t pos, neg;

39
thickenings.h
View File

@ -3,48 +3,11 @@
#define BV_QWORD_RANK 10 #define BV_QWORD_RANK 10
#include "bitvec.h" #include "bitvec.h"
#include "weyl.h" #include "weyl.h"
#define DEBUG(msg, ...) do{fprintf(stderr, msg, ##__VA_ARGS__); }while(0) #define DEBUG(msg, ...) do{fprintf(stderr, msg, ##__VA_ARGS__); }while(0)
#define MAX_THICKENINGS 0 // 0 means infinite
#define HEAD_MARKER 127
typedef struct _edgelist {
int to;
struct _edgelist *next;
} edgelist_t;
// describes an element of the Weyl group; only "opposite" and "bruhat_lower" are being used for enumerating thickenings; everything else is just needed for initialization or output
typedef struct {
int *word;
int wordlength;
int *left;
int *right;
int opposite;
edgelist_t *bruhat_lower;
edgelist_t *bruhat_higher;
int is_hyperplane_reflection; // boolean value
weylid_t id;
} node_t;
// printing functions
char *alphabetize(int *word, int len, const char *alphabet, char *buffer);
void print_thickening(int rank, int order, const signed char *thickening, int level, const char *alphabet, FILE *f);
// generating the graph of the bruhat order
node_t *graph_alloc(semisimple_type_t type);
void graph_free(semisimple_type_t type, node_t *graph);
void prepare_graph(semisimple_type_t type, node_t *graph);
int prepare_simplified_graph(semisimple_type_t type, unsigned long left, unsigned long right, node_t *simplified_graph);
// enumerating balanced thickenings // enumerating balanced thickenings
long enumerate_balanced_thickenings(node_t *graph, int size, void (*callback) (const bitvec_t *, int, void*), void *callback_data); long enumerate_balanced_thickenings(doublequotient_t *dq, void (*callback) (const bitvec_t *, int, void*), void *callback_data);
// various helper functions
static int compare_wordlength(const void *a, const void *b, void *gr);
static int edgelist_contains(edgelist_t *list, int x);
static edgelist_t *edgelist_add(edgelist_t *list, int new, edgelist_t *storage, int *storage_index);
#endif #endif

653
weyl.c
View File

@ -12,15 +12,167 @@ typedef struct {
int position; int position;
} weylid_lookup_t; } weylid_lookup_t;
static void generate_left_and_ids(semisimple_type_t type, weylgroup_element_t *group);
static int search(const void *key, const void *base, size_t nmem, size_t size, int (*compar) (const void *, const void *, void *), void *arg); static int search(const void *key, const void *base, size_t nmem, size_t size, int (*compar) (const void *, const void *, void *), void *arg);
static int compare_root_vectors(int rank, const int *x, const int *y); static int compare_root_vectors(int rank, const int *x, const int *y);
static int compare_root_vectors_qsort(const void *x, const void *y, void *arg); static int compare_root_vectors_qsort(const void *x, const void *y, void *arg);
static int compare_weylid(const void *x, const void *y);
static int compare_weylid_lookup(const void *x, const void *y); static int compare_weylid_lookup(const void *x, const void *y);
static int lookup_id(weylid_t id, weylid_lookup_t *list, int len); static int lookup_id(weylid_t id, weylid_lookup_t *list, int len);
static weylid_t multiply_generator(int s, weylid_t w, const int *simple, const int *mapping, int rank, int positive); static weylid_t multiply_generator(int s, weylid_t w, const int *simple, const int *mapping, int rank, int positive);
static void reflect_root_vector(const int *cartan, int rank, int i, int *old, int *new); static void reflect_root_vector(const int *cartan, int rank, int i, int *old, int *new);
static weylgroup_element_t* apply_word(int *word, int len, weylgroup_element_t *current);
static weylgroup_element_t* apply_word_reverse(int *word, int len, weylgroup_element_t *current);
/***************** simple helper functions **********************************/ /******** generate_left_and_ids and a pile of helper functions **************/
static void generate_left_and_ids(semisimple_type_t type, weylgroup_element_t *group)
{
int rank = weyl_rank(type);
int order = weyl_order(type);
int positive = weyl_positive(type);
queue_t queue;
int current;
int roots_known, elements, length_elements, nextids_count;
int *cartan_matrix;
int *root_vectors;
int *vector;
int *simple_roots;
int *root_mapping;
weylid_t *ids, *edges, *nextids;
weylid_lookup_t *lookup;
// allocate temporary stuff
cartan_matrix = (int*)malloc(rank*rank *sizeof(int));
root_vectors = (int*)malloc(2*positive*rank*sizeof(int));
vector = (int*)malloc(rank *sizeof(int));
root_mapping = (int*)malloc(positive*rank *sizeof(int));
simple_roots = (int*)malloc(rank *sizeof(int));
ids = (weylid_t*)malloc(order *sizeof(weylid_t));
edges = (weylid_t*)malloc(rank*order *sizeof(weylid_t));
nextids = (weylid_t*)malloc(rank*order *sizeof(weylid_t));
lookup = (weylid_lookup_t*)malloc(order *sizeof(weylid_lookup_t));
// get all information on the cartan type
LOG("Get Cartan matrix.\n");
weyl_cartan_matrix(type, cartan_matrix);
// enumerate roots, first the simple ones, then all others by reflecting
LOG("Enumerate roots.\n");
memset(root_vectors, 0, 2*positive*rank*sizeof(int));
roots_known = 0;
queue_init(&queue);
for(int i = 0; i < rank; i++) {
root_vectors[rank*i + i] = 1; // (r_i)_j = delta_ij
queue_put(&queue, i);
roots_known++;
}
while((current = queue_get(&queue)) != -1) {
for(int i = 0; i < rank; i++) {
reflect_root_vector(cartan_matrix, rank, i, &root_vectors[rank*current], vector);
int j;
for(j = 0; j < roots_known; j++)
if(compare_root_vectors(rank, &root_vectors[rank*j], vector) == 0)
break;
if(j == roots_known) {
memcpy(&root_vectors[rank*roots_known], vector, rank*sizeof(int));
queue_put(&queue, roots_known);
roots_known++;
}
}
}
ERROR(roots_known != 2*positive, "Number of roots does not match!\n");
// sort roots and restrict to positives
LOG("Sort roots.\n");
qsort_r(root_vectors, 2*positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
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!
// generate root_mapping, which gives the action of root reflections on positive roots (-1 if result is not a positive root)
LOG("Compute root reflections.\n");
for(int i = 0; i < positive; i++) {
for(int j = 0; j < rank; j++) {
reflect_root_vector(cartan_matrix, rank, j, &root_vectors[rank*i], vector);
root_mapping[i*rank+j] =
search(vector, root_vectors, positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
}
}
// find simple roots in the list
LOG("Find simple roots.\n");
for(int i = 0; i < rank; i++) {
memset(vector, 0, rank*sizeof(int));
vector[i] = 1;
simple_roots[i] = search(vector, root_vectors, positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
}
// enumerate weyl group elements using difference sets
LOG("Enumerate Weyl group elements.\n");
nextids[0] = 0;
nextids_count = 1;
elements = 0;
for(int len = 0; len <= positive; len++) {
length_elements = 0;
// find unique ids in edges added in the last iteration
qsort(nextids, nextids_count, sizeof(weylid_t), compare_weylid);
for(int i = 0; i < nextids_count; i++)
if(i == 0 || nextids[i] != nextids[i-1])
ids[elements + length_elements++] = nextids[i];
// add new edges
nextids_count = 0;
for(int i = elements; i < elements + length_elements; i++)
for(int j = 0; j < rank; j++) {
edges[i*rank+j] = multiply_generator(j, ids[i], simple_roots, root_mapping, rank, positive);
if(!(ids[i] & BIT(simple_roots[j]))) // the new element is longer then the old one
nextids[nextids_count++] = edges[i*rank+j];
}
elements += length_elements;
}
// translate the ids to list positions (i.e. local continuous ids)
LOG("Reorder Weyl group elements.\n");
for(int i = 0; i < order; i++) {
lookup[i].id = ids[i];
lookup[i].position = i;
}
qsort(lookup, order, sizeof(weylid_lookup_t), compare_weylid_lookup);
// fill in results
LOG("Compute left multiplication.\n");
for(int i = 0; i < order; i++) {
group[i].id = ids[i];
for(int j = 0; j < rank; j++)
group[i].left[j] = group + lookup_id(edges[i*rank+j], lookup, order);
}
// free temporary stuff
free(cartan_matrix);
free(root_vectors);
free(vector);
free(root_mapping);
free(simple_roots);
free(ids);
free(edges);
free(nextids);
free(lookup);
}
// glibc search function, but with user pointer and returning index (or -1 if not found) // glibc search function, but with user pointer and returning index (or -1 if not found)
static int search (const void *key, const void *base, size_t nmemb, size_t size, int (*compar) (const void *, const void *, void *), void *arg) static int search (const void *key, const void *base, size_t nmemb, size_t size, int (*compar) (const void *, const void *, void *), void *arg)
@ -176,46 +328,46 @@ int weyl_order(semisimple_type_t type)
return order; return order;
} }
int weyl_hyperplanes(semisimple_type_t type) int weyl_positive(semisimple_type_t type)
{ {
int hyperplanes = 0; int positive = 0;
for(int i = 0; i < type.n; i++) { for(int i = 0; i < type.n; i++) {
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); 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);
switch(type.factors[i].series) { switch(type.factors[i].series) {
case 'A': case 'A':
hyperplanes += (type.factors[i].rank * (type.factors[i].rank + 1)) / 2; positive += (type.factors[i].rank * (type.factors[i].rank + 1)) / 2;
break; break;
case 'B': case 'C': case 'B': case 'C':
hyperplanes += type.factors[i].rank * type.factors[i].rank; positive += type.factors[i].rank * type.factors[i].rank;
break; break;
case 'D': case 'D':
hyperplanes += type.factors[i].rank * (type.factors[i].rank - 1); positive += type.factors[i].rank * (type.factors[i].rank - 1);
break; break;
case 'E': case 'E':
if(type.factors[i].rank == 6) if(type.factors[i].rank == 6)
hyperplanes += 36; positive += 36;
else if(type.factors[i].rank == 7) else if(type.factors[i].rank == 7)
hyperplanes += 63; positive += 63;
else if(type.factors[i].rank == 8) else if(type.factors[i].rank == 8)
hyperplanes += 120; positive += 120;
break; break;
case 'F': case 'F':
hyperplanes += 24; positive += 24;
break; break;
case 'G': case 'G':
hyperplanes += 6; positive += 6;
break; break;
} }
} }
return hyperplanes; return positive;
} }
int weyl_opposition(semisimple_type_t type, int simple_root) int weyl_opposition(semisimple_type_t type, int simple_root)
@ -313,157 +465,354 @@ void weyl_cartan_matrix(semisimple_type_t type, int *m)
free(A); free(A);
} }
/************ memory allocation ********************/ /************ weyl_generate etc. ********************/
weylgroup_element_t *weyl_alloc(semisimple_type_t type) static weylgroup_element_t* apply_word(int *word, int len, weylgroup_element_t *current)
{
for(int k = len - 1; k >= 0; k--) // apply group element from right to left
current = current->left[word[k]];
return current;
}
static weylgroup_element_t* apply_word_reverse(int *word, int len, weylgroup_element_t *current)
{
for(int k = 0; k < len; k++) // apply group element from left to right (i.e. apply inverse)
current = current->left[word[k]];
return current;
}
weylgroup_t *weyl_generate(semisimple_type_t type)
{ {
int rank = weyl_rank(type); int rank = weyl_rank(type);
int order = weyl_order(type); int order = weyl_order(type);
int positive = weyl_positive(type);
int *left = (int*)malloc(rank*order*sizeof(int));
int *right = (int*)malloc(rank*order*sizeof(int));
weylgroup_element_t *group = (weylgroup_element_t*)malloc(order*sizeof(weylgroup_element_t));
for(int i = 0; i < order; i++) {
group[i].left = &left[i*rank];
group[i].right = &right[i*rank];
}
return group;
}
void weyl_free(weylgroup_element_t *x)
{
free(x[0].left);
free(x[0].right);
free(x);
}
void weyl_generate(semisimple_type_t type, weylgroup_element_t *group)
{
int rank, order, positive;
queue_t queue;
int current;
int roots_known, elements, length_elements, nextids_count;
int *cartan_matrix;
int *root_vectors;
int *vector;
int *simple_roots;
int *root_mapping;
weylid_t *ids, *edges, *nextids;
weylid_lookup_t *lookup;
rank = weyl_rank(type);
order = weyl_order(type);
positive = weyl_hyperplanes(type);
ERROR(positive > 64, "We can't handle root systems with more than 64 positive roots!\n"); ERROR(positive > 64, "We can't handle root systems with more than 64 positive roots!\n");
cartan_matrix = (int*)malloc(rank*rank *sizeof(int)); // allocate result
root_vectors = (int*)malloc(2*positive*rank*sizeof(int));
vector = (int*)malloc(rank *sizeof(int));
root_mapping = (int*)malloc(positive*rank *sizeof(int));
simple_roots = (int*)malloc(rank *sizeof(int));
ids = (weylid_t*)malloc(order *sizeof(weylid_t));
edges = (weylid_t*)malloc(rank*order *sizeof(weylid_t));
nextids = (weylid_t*)malloc(rank*order *sizeof(weylid_t));
lookup = (weylid_lookup_t*)malloc(order *sizeof(weylid_lookup_t));
weyl_cartan_matrix(type, cartan_matrix); weylgroup_element_t *group = (weylgroup_element_t*)malloc(order*sizeof(weylgroup_element_t));
weylgroup_t *result = malloc(sizeof(weylgroup_t));
// enumerate roots result->type = type;
memset(root_vectors, 0, 2*positive*rank*sizeof(int)); result->elements = group;
result->lists = (weylgroup_element_t**)malloc(2*order*rank*sizeof(weylgroup_element_t*));
// first the simple roots
queue_init(&queue);
for(int i = 0; i < rank; i++) {
root_vectors[rank*i + i] = 1;
queue_put(&queue, i);
}
// and then we get all others by reflecting
roots_known = rank;
while((current = queue_get(&queue)) != -1) {
for(int i = 0; i < rank; i++) {
reflect_root_vector(cartan_matrix, rank, i, &root_vectors[rank*current], vector);
int j;
for(j = 0; j < roots_known; j++)
if(compare_root_vectors(rank, &root_vectors[rank*j], vector) == 0)
break;
if(j == roots_known) {
memcpy(&root_vectors[rank*roots_known], vector, rank*sizeof(int));
queue_put(&queue, roots_known);
roots_known++;
}
}
}
ERROR(roots_known != 2*positive, "Number of roots does not match!\n")
// sort roots and restrict to positives
qsort_r(root_vectors, 2*positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
memcpy(root_vectors, &root_vectors[positive*rank], positive*rank*sizeof(int));
for(int i = 0; i < positive; i++) {
for(int j = 0; j < rank; j++) {
reflect_root_vector(cartan_matrix, rank, j, &root_vectors[rank*i], vector);
root_mapping[i*rank+j] =
search(vector, root_vectors, positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
}
}
// where in the list are the simple roots?
for(int i = 0; i < rank; i++) {
memset(vector, 0, rank*sizeof(int));
vector[i] = 1;
simple_roots[i] = search(vector, root_vectors, positive, rank*sizeof(int), compare_root_vectors_qsort, &rank);
}
// enumerate weyl group elements using difference sets
nextids[0] = 0;
nextids_count = 1;
elements = 0;
for(int len = 0; len <= positive; len++) {
length_elements = 0;
// find unique ids in edges added in the last iteration
qsort(nextids, nextids_count, sizeof(weylid_t), compare_weylid);
for(int i = 0; i < nextids_count; i++)
if(i == 0 || nextids[i] != nextids[i-1])
ids[elements + length_elements++] = nextids[i];
// add new edges
nextids_count = 0;
for(int i = elements; i < elements + length_elements; i++)
for(int j = 0; j < rank; j++) {
edges[i*rank+j] = multiply_generator(j, ids[i], simple_roots, root_mapping, rank, positive);
if(!(ids[i] & BIT(simple_roots[j]))) // the new element is longer then the old one
nextids[nextids_count++] = edges[i*rank+j];
}
elements += length_elements;
}
// translate the ids to list positions (i.e. local continuous ids)
for(int i = 0; i < order; i++) {
lookup[i].id = ids[i];
lookup[i].position = i;
}
qsort(lookup, order, sizeof(weylid_lookup_t), compare_weylid_lookup);
for(int i = 0; i < order; i++) { for(int i = 0; i < order; i++) {
group[i].id = ids[i]; group[i].left = result->lists + 2*i*rank;
group[i].right = result->lists + (2*i+1)*rank;
group[i].coset = (doublecoset_t*)0;
}
// 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++) for(int j = 0; j < rank; j++)
group[i].left[j] = lookup_id(edges[i*rank+j], lookup, order); 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");
} }
free(cartan_matrix); doublequotient_t *result = (doublequotient_t*)malloc(sizeof(doublequotient_t));
free(root_vectors); result->type = type;
free(vector); result->left_invariance = left_invariance;
free(root_mapping); result->right_invariance = right_invariance;
free(simple_roots); result->group = wgroup->elements;
free(ids); result->grouplists = wgroup->lists;
free(edges); result->groupletters = wgroup->letters;
free(nextids);
free(lookup); free(wgroup); // dissolved in result and not needed anymore
// count cosets by finding the minimum length element in every coset
LOG("Count cosets.\n");
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
// find minima (basically same code as above)
LOG("Find minimal length elements in cosets.\n");
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++;
}
}
// generate quotient map
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;
}
}
// find maxima
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];
}
}
// opposites
LOG("Find opposites.\n");
for(int i = 0; i < count; i++)
cosets[i].opposite = cosets[i].min->opposite->coset;
// bruhat order
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;
}
}
}
}
// transitive reduction
LOG("Perform transitive reduction.\n");
doublecoset_t *offset = &cosets[0];
doublecoset_t *origin;
doublecoset_list_t *current;
doublecoset_list_t *prev;
queue_t queue;
int cur;
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 - offset]) {
if(prev)
prev->next = current->next;
else
origin->bruhat_lower = current->next;
} else {
prev = current;
seen[current->to - offset] = 1;
queue_put(&queue, current->to - offset);
}
}
// see which nodes we can reach using only edges up to length len, mark them as seen
while((cur = queue_get(&queue)) != -1) {
current = (cur + offset)->bruhat_lower;
for(current = (cur+offset)->bruhat_lower; current; current = current->next) {
if(!seen[current->to - offset]) {
seen[current->to - offset] = 1;
queue_put(&queue, current->to - offset);
}
}
}
}
}
free(seen);
// reverse bruhat order
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;
}
}
// sort opposites and rewrite everything
LOG("Sort opposites.\n");
int *old2newindices = (int*)malloc(count*sizeof(int));
int *new2oldindices = (int*)malloc(count*sizeof(int));
doublecoset_t *oldcosets = (doublecoset_t*)malloc(count*sizeof(doublecoset_t));
memcpy(oldcosets, cosets, count*sizeof(doublecoset_t));
int j = 0;
for(int i = 0; i < count; i++)
if(&cosets[i] < cosets[i].opposite) {
old2newindices[i] = j;
old2newindices[cosets[i].opposite - cosets] = count-1-j;
j++;
}
for(int i = 0; i < count; i++)
if(i == cosets[i].opposite - cosets)
old2newindices[i] = j++;
for(int i = 0; i < count; i++)
new2oldindices[old2newindices[i]] = i;
for(int i = 0; i < count; i++) {
cosets[i].min = oldcosets[new2oldindices[i]].min;
cosets[i].max = oldcosets[new2oldindices[i]].max;
cosets[i].opposite = old2newindices[oldcosets[new2oldindices[i]].opposite - cosets] + cosets;
cosets[i].bruhat_lower = oldcosets[new2oldindices[i]].bruhat_lower;
cosets[i].bruhat_higher = oldcosets[new2oldindices[i]].bruhat_higher;
for(current = cosets[i].bruhat_lower; current; current = current -> next)
current->to = old2newindices[current->to - cosets] + cosets;
for(current = cosets[i].bruhat_higher; current; current = current -> next)
current->to = old2newindices[current->to - cosets] + cosets;
}
for(int i = 0; i < order; i++)
group[i].coset = old2newindices[group[i].coset - cosets] + cosets;
free(old2newindices);
free(new2oldindices);
free(oldcosets);
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);
} }

94
weyl.h
View File

@ -3,34 +3,92 @@
#include <inttypes.h> #include <inttypes.h>
typedef struct { struct _simple_type;
char series; struct _semisimple_type;
int rank; struct _weylgroup_element;
} simple_type_t; struct _weylgroup;
struct _doublecoset;
typedef struct { struct _doublecoset_list;
int n; struct _doublequotient;
simple_type_t *factors;
} semisimple_type_t;
typedef uint64_t weylid_t; typedef uint64_t weylid_t;
typedef struct _simple_type simple_type_t;
typedef struct _semisimple_type semisimple_type_t;
typedef struct _weylgroup_element weylgroup_element_t;
typedef struct _weylgroup weylgroup_t;
typedef struct _doublecoset doublecoset_t;
typedef struct _doublecoset_list doublecoset_list_t;
typedef struct _doublequotient doublequotient_t;
typedef struct { /***************************** structures *******************************/
int *left;
int *right; struct _simple_type {
int opposite; char series;
int rank;
};
struct _semisimple_type {
int n;
simple_type_t *factors;
};
struct _weylgroup_element {
int *word;
int wordlength;
weylgroup_element_t **left;
weylgroup_element_t **right;
weylgroup_element_t *opposite;
int is_root_reflection; // boolean value
weylid_t id; weylid_t id;
} weylgroup_element_t;
// only set if quotient is generated
doublecoset_t *coset;
};
struct _weylgroup {
semisimple_type_t type;
weylgroup_element_t *elements;
weylgroup_element_t **lists;
int *letters;
};
struct _doublecoset {
doublecoset_list_t *bruhat_lower;
doublecoset_list_t *bruhat_higher;
doublecoset_t *opposite;
weylgroup_element_t *max;
weylgroup_element_t *min;
};
struct _doublecoset_list {
doublecoset_t *to;
doublecoset_list_t *next;
};
struct _doublequotient {
semisimple_type_t type;
int left_invariance; // bitmask with rank bits
int right_invariance;
int count; // number of cosets
doublecoset_t *cosets;
weylgroup_element_t *group;
doublecoset_list_t *lists; // only for memory allocation / freeing
weylgroup_element_t **grouplists; // only for memory allocation / freeing
int *groupletters; // only for memory allocation / freeing
};
/***************************** functions **************************************/
int weyl_rank(semisimple_type_t type); int weyl_rank(semisimple_type_t type);
int weyl_order(semisimple_type_t type); int weyl_order(semisimple_type_t type);
int weyl_hyperplanes(semisimple_type_t type); int weyl_positive(semisimple_type_t type);
void weyl_cartan_matrix(semisimple_type_t type, int *m); void weyl_cartan_matrix(semisimple_type_t type, int *m);
int weyl_opposition(semisimple_type_t type, int simple_root); int weyl_opposition(semisimple_type_t type, int simple_root);
weylgroup_element_t *weyl_alloc(semisimple_type_t type); weylgroup_t *weyl_generate(semisimple_type_t type);
void weyl_free(weylgroup_element_t *x); void weyl_destroy(weylgroup_t *group);
void weyl_generate(semisimple_type_t type, weylgroup_element_t *group); doublequotient_t *weyl_generate_bruhat(semisimple_type_t type, int left_invariance, int right_invariance);
void weyl_destroy_bruhat(doublequotient_t *dq);
#endif #endif