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💡 Pro tip: This is one of those techniques that will make you look like a data science wizard! Introduction to Coxeter Groups - Made Simple!

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.

Here’s where it gets exciting! Here’s how we can tackle this:

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)

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🎉 You’re doing great! This concept might seem tricky at first, but you’ve got this! Coxeter Matrix and Diagrams - Made Simple!

A Coxeter group is defined by its Coxeter matrix, which encodes the relations between generators. The Coxeter diagram visually represents these relations.

Let’s break this down together! Here’s how we can tackle this:

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)

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✨ Cool fact: Many professional data scientists use this exact approach in their daily work! Generating Words in Coxeter Groups - Made Simple!

Words in Coxeter groups are sequences of generators. We can generate all words up to a certain length using recursive algorithms.

Here’s a handy trick you’ll love! Here’s how we can tackle this:

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))

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🔥 Level up: Once you master this, you’ll be solving problems like a pro! Word Reduction in Coxeter Groups - Made Simple!

Word reduction is a fundamental operation in Coxeter groups. It involves applying relations to simplify words.

Here’s where it gets exciting! Here’s how we can tackle this:

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))

🚀 Bruhat Order - Made Simple!

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.

This next part is really neat! Here’s how we can tackle this:

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}")

🚀 Length Function - Made Simple!

The length function on a Coxeter group measures the minimal number of generators needed to express an element.

Let’s make this super clear! Here’s how we can tackle this:

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)}")

🚀 Parabolic Subgroups - Made Simple!

Parabolic subgroups are important substructures of Coxeter groups, obtained by considering only a subset of the generators.

Ready for some cool stuff? Here’s how we can tackle this:

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))

🚀 Longest Element - Made Simple!

The longest element is a unique element of maximal length in finite Coxeter groups, playing a special role in many contexts.

Don’t worry, this is easier than it looks! Here’s how we can tackle this:

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))

🚀 Weak Order - Made Simple!

The weak order is another partial order on Coxeter groups, based on subwords of reduced expressions.

Don’t worry, this is easier than it looks! Here’s how we can tackle this:

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}")

🚀 Counting Elements - Made Simple!

Counting elements in Coxeter groups involves generating functions and can be done smartly using recurrence relations.

Let me walk you through this step by step! Here’s how we can tackle this:

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}")

🚀 Kazhdan-Lusztig Polynomials - Made Simple!

Kazhdan-Lusztig polynomials are important invariants associated with pairs of elements in the Bruhat order of a Coxeter group.

Let’s make this super clear! Here’s how we can tackle this:

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)}")

🚀 Real-Life Example: Crystallography - Made Simple!

Coxeter groups are used in crystallography to describe symmetries of crystal structures. Here’s an example of generating symmetry operations for a cubic crystal.

Let’s make this super clear! Here’s how we can tackle this:

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])

🚀 Real-Life Example: Error-Correcting Codes - Made Simple!

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.

Here’s a handy trick you’ll love! Here’s how we can tackle this:

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}")

🚀 Additional Resources - Made Simple!

For further exploration of Combinatorics of Coxeter Groups, consider the following resources:

1. Björner, A., & Brenti, F. (2005). Combinatorics of Coxeter Groups. Springer. ArXiv: https://arxiv.org/abs/math/0503457
2. Humphreys, J. E. (1990). Reflection Groups and Coxeter Groups. Cambridge University Press.
3. 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.

🎊 Awesome Work!

You’ve just learned some really powerful techniques! Don’t worry if everything doesn’t click immediately - that’s totally normal. The best way to master these concepts is to practice with your own data.

What’s next? Try implementing these examples with your own datasets. Start small, experiment, and most importantly, have fun with it! Remember, every data science expert started exactly where you are right now.

Keep coding, keep learning, and keep being awesome! 🚀