Slide 1: Introduction to Coxeter Groups
Coxeter groups are mathematical structures that describe symmetries in geometric spaces. They are named after H.S.M. Coxeter and have applications in various fields, including algebra, geometry, and combinatorics.
import networkx as nx
import matplotlib.pyplot as plt
def create_coxeter_diagram(matrix):
G = nx.Graph()
n = len(matrix)
for i in range(n):
for j in range(i+1, n):
if matrix[i][j] != 2:
G.add_edge(i, j, weight=matrix[i][j])
pos = nx.spring_layout(G)
nx.draw(G, pos, with_labels=True, node_color='lightblue', node_size=500)
edge_labels = nx.get_edge_attributes(G, 'weight')
nx.draw_networkx_edge_labels(G, pos, edge_labels=edge_labels)
plt.title("Coxeter Diagram")
plt.axis('off')
plt.show()
# Example: Coxeter matrix for A3 group
A3_matrix = [
[1, 3, 2],
[3, 1, 3],
[2, 3, 1]
]
create_coxeter_diagram(A3_matrix)Slide 2: Coxeter Matrix and Diagrams
A Coxeter group is defined by its Coxeter matrix, which encodes the relations between generators. The Coxeter diagram visually represents these relations.
def print_coxeter_matrix(matrix):
n = len(matrix)
print("Coxeter Matrix:")
for row in matrix:
print(" ".join(f"{x:2d}" for x in row))
# Example: Coxeter matrix for B3 group
B3_matrix = [
[1, 4, 2],
[4, 1, 3],
[2, 3, 1]
]
print_coxeter_matrix(B3_matrix)
create_coxeter_diagram(B3_matrix)Slide 3: Generating Words in Coxeter Groups
Words in Coxeter groups are sequences of generators. We can generate all words up to a certain length using recursive algorithms.
def generate_words(generators, max_length):
words = [[]]
for _ in range(max_length):
new_words = []
for word in words:
for gen in generators:
new_word = word + [gen]
if len(new_word) < 2 or new_word[-1] != new_word[-2]:
new_words.append(new_word)
words.extend(new_words)
return words
generators = ['s1', 's2', 's3']
max_length = 3
words = generate_words(generators, max_length)
print(f"Words of length up to {max_length}:")
for word in words:
print("".join(word))Slide 4: Word Reduction in Coxeter Groups
Word reduction is a fundamental operation in Coxeter groups. It involves applying relations to simplify words.
def reduce_word(word, relations):
reduced = True
while reduced:
reduced = False
for i in range(len(word) - 1):
for relation in relations:
if word[i:i+len(relation[0])] == relation[0]:
word = word[:i] + relation[1] + word[i+len(relation[0]):]
reduced = True
break
if reduced:
break
return word
# Example: A2 group relations
relations = [
(['s1', 's1'], []),
(['s2', 's2'], []),
(['s1', 's2', 's1'], ['s2', 's1', 's2'])
]
word = ['s1', 's2', 's1', 's2', 's1']
reduced_word = reduce_word(word, relations)
print("Original word:", "".join(word))
print("Reduced word:", "".join(reduced_word))Slide 5: Bruhat Order
The Bruhat order is a partial order on the elements of a Coxeter group, which plays a crucial role in the study of Schubert varieties and Kazhdan-Lusztig polynomials.
def subword(u, w):
i, j = 0, 0
while i < len(u) and j < len(w):
if u[i] == w[j]:
i += 1
j += 1
return i == len(u)
def bruhat_order(words):
order = {}
for w in words:
order[tuple(w)] = set()
for u in words:
if len(u) <= len(w) and subword(u, w):
order[tuple(w)].add(tuple(u))
return order
words = generate_words(['s1', 's2'], 2)
bruhat = bruhat_order(words)
for w, smaller in bruhat.items():
print(f"{w}: {smaller}")Slide 6: Length Function
The length function on a Coxeter group measures the minimal number of generators needed to express an element.
def length(word, relations):
reduced = reduce_word(word, relations)
return len(reduced)
# Example: A2 group relations
relations = [
(['s1', 's1'], []),
(['s2', 's2'], []),
(['s1', 's2', 's1'], ['s2', 's1', 's2'])
]
words = [
['s1'],
['s1', 's2'],
['s1', 's2', 's1'],
['s1', 's2', 's1', 's2']
]
for word in words:
print(f"Length of {''.join(word)}: {length(word, relations)}")Slide 7: Parabolic Subgroups
Parabolic subgroups are important substructures of Coxeter groups, obtained by considering only a subset of the generators.
def parabolic_subgroup(generators, subset):
return [w for w in generate_words(generators, len(generators)) if set(w).issubset(subset)]
generators = ['s1', 's2', 's3']
subset = {'s1', 's2'}
subgroup = parabolic_subgroup(generators, subset)
print(f"Parabolic subgroup generated by {subset}:")
for word in subgroup:
print("".join(word))Slide 8: Longest Element
The longest element is a unique element of maximal length in finite Coxeter groups, playing a special role in many contexts.
def longest_element(generators, relations):
words = generate_words(generators, len(generators) ** 2)
return max(words, key=lambda w: length(w, relations))
# Example: A3 group relations
relations = [
(['s1', 's1'], []),
(['s2', 's2'], []),
(['s3', 's3'], []),
(['s1', 's2', 's1'], ['s2', 's1', 's2']),
(['s2', 's3', 's2'], ['s3', 's2', 's3']),
(['s1', 's3'], ['s3', 's1'])
]
generators = ['s1', 's2', 's3']
longest = longest_element(generators, relations)
print("Longest element:", "".join(longest))
print("Length:", length(longest, relations))Slide 9: Weak Order
The weak order is another partial order on Coxeter groups, based on subwords of reduced expressions.
def weak_order(words, relations):
order = {}
for w in words:
order[tuple(w)] = set()
for i in range(len(w)):
u = w[:i] + w[i+1:]
if length(u, relations) < length(w, relations):
order[tuple(w)].add(tuple(u))
return order
generators = ['s1', 's2']
words = generate_words(generators, 3)
weak = weak_order(words, relations)
for w, smaller in weak.items():
print(f"{w}: {smaller}")Slide 10: Counting Elements
Counting elements in Coxeter groups involves generating functions and can be done efficiently using recurrence relations.
def count_elements(generators, max_length):
counts = [1] # Empty word
for n in range(1, max_length + 1):
count = len(generators) * counts[-1]
for i in range(n - 1):
count -= len(generators) * counts[i]
counts.append(count)
return counts
generators = ['s1', 's2', 's3']
max_length = 10
counts = count_elements(generators, max_length)
for n, count in enumerate(counts):
print(f"Elements of length {n}: {count}")Slide 11: Kazhdan-Lusztig Polynomials
Kazhdan-Lusztig polynomials are important invariants associated with pairs of elements in the Bruhat order of a Coxeter group.
from sympy import symbols, expand
def kl_polynomial(u, w, bruhat_order):
if u == w:
return 1
if u not in bruhat_order[w]:
return 0
q = symbols('q')
P = 1
for v in bruhat_order[w]:
if u in bruhat_order[v]:
P -= q ** ((len(w) - len(v)) // 2) * kl_polynomial(u, v, bruhat_order)
return expand(P)
generators = ['s1', 's2']
words = generate_words(generators, 3)
bruhat = bruhat_order(words)
u = tuple(['s1'])
w = tuple(['s1', 's2', 's1'])
print(f"KL polynomial P_{u},{w} = {kl_polynomial(u, w, bruhat)}")Slide 12: Real-Life Example: Crystallography
Coxeter groups are used in crystallography to describe symmetries of crystal structures. Here's an example of generating symmetry operations for a cubic crystal.
import numpy as np
def generate_cubic_symmetries():
# Generators for cubic symmetry group (Oh)
generators = [
np.array([[0, 1, 0], [-1, 0, 0], [0, 0, 1]]), # 90° rotation around z
np.array([[1, 0, 0], [0, 0, 1], [0, -1, 0]]), # 90° rotation around x
np.array([[-1, 0, 0], [0, -1, 0], [0, 0, 1]]) # Mirror reflection in xy-plane
]
symmetries = [np.eye(3)]
for _ in range(3): # Iterate to generate all 48 symmetry operations
new_symmetries = []
for sym in symmetries:
for gen in generators:
new_sym = sym @ gen
if not any(np.allclose(new_sym, s) for s in symmetries):
new_symmetries.append(new_sym)
symmetries.extend(new_symmetries)
return symmetries
cubic_symmetries = generate_cubic_symmetries()
print(f"Number of symmetry operations: {len(cubic_symmetries)}")
print("Example symmetry operation:")
print(cubic_symmetries[1])Slide 13: Real-Life Example: Error-Correcting Codes
Coxeter groups, particularly the Weyl groups, are used in the construction of certain error-correcting codes. Here's an example of generating a simple code based on the A2 Coxeter group.
def generate_a2_code():
# Generators for A2 group
s1 = lambda x, y: (y, x)
s2 = lambda x, y: (x, y-x)
codewords = set([(0, 0)])
for _ in range(2): # Apply generators twice
new_codewords = set()
for x, y in codewords:
new_codewords.add(s1(x, y))
new_codewords.add(s2(x, y))
codewords.update(new_codewords)
return sorted(codewords)
code = generate_a2_code()
print("A2-based code:")
for codeword in code:
print(codeword)
# Simple encoding function
def encode(message, code):
return code[message % len(code)]
# Example usage
message = 4
encoded = encode(message, code)
print(f"Encoded message {message}: {encoded}")Slide 14: Additional Resources
For further exploration of Combinatorics of Coxeter Groups, consider the following resources:
- Björner, A., & Brenti, F. (2005). Combinatorics of Coxeter Groups. Springer. ArXiv: https://arxiv.org/abs/math/0503457
- Humphreys, J. E. (1990). Reflection Groups and Coxeter Groups. Cambridge University Press.
- Reading, N. (2007). Clusters, Coxeter-sortable elements and noncrossing partitions. ArXiv: https://arxiv.org/abs/math/0507186
These resources provide in-depth coverage of the topic and can help deepen your understanding of Coxeter groups and their combinatorial aspects.