Problem 883: Remarkable Representations

Mathematical Analysis

Theorem 1 (Sum of Squares)

For any finite group $G$ with irreducible representations $\rho_1, \ldots, \rho_h$:

Proof. Consider the regular representation $R: G \to GL(\mathbb{C}[G])$. By Maschke’s theorem, $R$ decomposes into irreducibles: $R \cong \bigoplus_{i=1}^{h} (\dim \rho_i) \cdot \rho_i$. Taking dimensions: $|G| = \dim R = \sum (\dim \rho_i)^2$. $\square$

Theorem 2 (Number of Irreducibles)

$h$ = number of conjugacy classes of $G$.

Proof. The center of the group algebra $Z(\mathbb{C}[G])$ has dimension equal to the number of conjugacy classes, and also equal to $h$ since the Wedderburn decomposition gives $h$ simple summands. $\square$

Theorem 3 (Orthogonality Relations)

For irreducible characters $\chi_i, \chi_j$:

Theorem 4 (Divisibility)

$\dim \rho_i \mid |G|$ for each irreducible representation.

Concrete Examples

Cyclic Group $\mathbb{Z}/n\mathbb{Z}$

All irreducible representations are 1-dimensional: $\rho_k(g) = e^{2\pi i k g / n}$ for $k = 0, \ldots, n-1$.

Check: $\sum_{k=0}^{n-1} 1^2 = n = |\mathbb{Z}/n\mathbb{Z}|$. $\checkmark$

Symmetric Group $S_3$

$|S_3| = 6$, conjugacy classes: $\{e\}, \{(12),(13),(23)\}, \{(123),(132)\}$. So $h = 3$.

Dimensions: $d_1 = 1, d_2 = 1, d_3 = 2$.

Check: $1^2 + 1^2 + 2^2 = 1 + 1 + 4 = 6 = |S_3|$. $\checkmark$

The character table:

Symmetric Group $S_4$

$|S_4| = 24$, $h = 5$ conjugacy classes. Dimensions: $1, 1, 2, 3, 3$.

Check: $1 + 1 + 4 + 9 + 9 = 24$. $\checkmark$

Dihedral Group $D_n$ ($n$ odd)

$|D_n| = 2n$, $(n+3)/2$ conjugacy classes. Dimensions: two 1-dimensional, $(n-1)/2$ of dimension 2.

Check: $2 \cdot 1 + \frac{n-1}{2} \cdot 4 = 2 + 2(n-1) = 2n$. $\checkmark$

Verification Table

| Group | $|G|$ | Dimensions | $\sum d_i^2$ | Match | |:—|:-:|:—|:-:|:-:| | $\mathbb{Z}/4$ | 4 | 1,1,1,1 | 4 | Yes | | $S_3$ | 6 | 1,1,2 | 6 | Yes | | $D_4$ | 8 | 1,1,1,1,2 | 8 | Yes | | $Q_8$ | 8 | 1,1,1,1,2 | 8 | Yes | | $A_4$ | 12 | 1,1,1,3 | 12 | Yes | | $S_4$ | 24 | 1,1,2,3,3 | 24 | Yes |

Finding All Solutions to $\sum d_i^2 = n$

This is equivalent to representing $n$ as a sum of perfect squares where each $d_i \mid n$.

Burnside’s Lemma Connection

The number of orbits of a group $G$ acting on a set $X$:

This uses the character of the permutation representation, connecting representation theory to counting problems.

Schur Orthogonality (Column Version)

where $C_g$ is the conjugacy class of $g$ and $[g \sim h]$ means $g$ and $h$ are conjugate.

Complexity Analysis

Answer

/*
 * Problem 883: Remarkable Representations
 * sum(dim rho_i)^2 = |G| for irreducible representations.
 */
#include <bits/stdc++.h>
using namespace std;

bool verify(int order, vector<int> dims) {
    int s = 0;
    for (int d : dims) { s += d * d; if (order % d != 0) return false; }
    return s == order;
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);

    // Known groups
    cout << "=== Sum of Squares ===" << endl;
    struct Group { string name; int order; vector<int> dims; };
    vector<Group> groups = {
        {"Z/4", 4, {1,1,1,1}},
        {"S_3", 6, {1,1,2}},
        {"D_4", 8, {1,1,1,1,2}},
        {"Q_8", 8, {1,1,1,1,2}},
        {"A_4", 12, {1,1,1,3}},
        {"S_4", 24, {1,1,2,3,3}},
        {"S_5", 120, {1,1,4,4,5,5,6}},
    };
    for (auto& g : groups) {
        int s = 0;
        for (int d : g.dims) s += d * d;
        bool ok = verify(g.order, g.dims);
        cout << g.name << ": |G|=" << g.order << ", sum=" << s
             << (ok ? " OK" : " FAIL") << endl;
        assert(ok);
    }

    // Dihedral groups
    cout << "\n=== Dihedral Groups ===" << endl;
    for (int n = 3; n <= 10; n++) {
        int order = 2 * n;
        vector<int> dims;
        if (n % 2 == 1) {
            dims = {1, 1};
            for (int i = 0; i < (n - 1) / 2; i++) dims.push_back(2);
        } else {
            dims = {1, 1, 1, 1};
            for (int i = 0; i < (n - 2) / 2; i++) dims.push_back(2);
        }
        int s = 0;
        for (int d : dims) s += d * d;
        cout << "D_" << n << ": |D|=" << order << ", sum=" << s
             << (s == order ? " OK" : " FAIL") << endl;
        assert(s == order);
    }

    cout << "\nAnswer: sum d_i^2 = |G| (verified for all test groups)" << endl;
    return 0;
}

"""
Problem 883: Remarkable Representations
Sum of squares of irreducible representation dimensions equals group order.
"""

from math import gcd
from itertools import product

def verify_sum_of_squares(group_order, dims):
    """Verify sum(d^2) = |G| and each d divides |G|."""
    s = sum(d * d for d in dims)
    divides = all(group_order % d == 0 for d in dims)
    return s == group_order and divides

def cyclic_dims(n):
    """Cyclic group Z/nZ: n representations, all 1-dimensional."""
    return [1] * n

def symmetric_dims(n):
    """Known irreducible dimensions for S_n."""
    known = {
        1: [1], 2: [1, 1], 3: [1, 1, 2],
        4: [1, 1, 2, 3, 3], 5: [1, 1, 4, 4, 5, 5, 6]
    }
    return known.get(n, None)

def dihedral_dims(n):
    """Dihedral group D_n of order 2n."""
    if n % 2 == 1:  # odd n
        return [1, 1] + [2] * ((n - 1) // 2)
    else:  # even n
        return [1, 1, 1, 1] + [2] * ((n - 2) // 2)

def factorials():
    return {1: 1, 2: 2, 3: 6, 4: 24, 5: 120, 6: 720}

def find_dim_partitions(n, max_d=None):
    """Find all ways to write n as sum of squares d_i^2 where d_i | n."""
    if max_d is None:
        max_d = n
    divisors = [d for d in range(1, n + 1) if n % d == 0 and d <= max_d]
    results = []
    def backtrack(remaining, min_d, current):
        if remaining == 0:
            results.append(tuple(current))
            return
        for d in divisors:
            if d >= min_d and d * d <= remaining:
                backtrack(remaining - d * d, d, current + [d])
    backtrack(n, 1, [])
    return results

# --- Verification ---
print("=== Sum of Squares Verification ===")
groups = [
    ("Z/4", 4, cyclic_dims(4)),
    ("Z/6", 6, cyclic_dims(6)),
    ("S_3", 6, [1, 1, 2]),
    ("D_4", 8, dihedral_dims(4)),
    ("Q_8", 8, [1, 1, 1, 1, 2]),
    ("A_4", 12, [1, 1, 1, 3]),
    ("S_4", 24, [1, 1, 2, 3, 3]),
]
for name, order, dims in groups:
    ok = verify_sum_of_squares(order, dims)
    s = sum(d*d for d in dims)
    print(f"  {name:>4}: |G|={order:>3}, dims={dims}, sum={s}, {'OK' if ok else 'FAIL'}")
    assert ok

print("\n=== Dihedral Groups ===")
for n in range(3, 11):
    dims = dihedral_dims(n)
    order = 2 * n
    ok = verify_sum_of_squares(order, dims)
    print(f"  D_{n}: |D|={order}, dims={dims}, sum={sum(d*d for d in dims)}, {'OK' if ok else 'FAIL'}")
    assert ok

print("\n=== Dimension Partitions of 12 ===")
parts = find_dim_partitions(12)
for p in parts:
    print(f"  {p}: sum of squares = {sum(d*d for d in p)}")

answer = sum(d*d for d in [1, 1, 2, 3, 3])
print(f"\nAnswer: S_4 dimension sum = {answer} = |S_4| = 24")

# --- 4-Panel Visualization ---