Sum of the squares of the $p'$-character degrees

This paper investigates the sum of the squares of irreducible character degrees not divisible by a prime $p$ and its relation to the corresponding quantity in a $p$-Sylow normalizer, thereby proving a recent conjecture by E. Giannelli for the case $p=2$ and several other instances.

The Gist

Imagine you are a detective trying to solve a mystery about a massive, complex machine called a Finite Group. This machine is made of gears, levers, and springs (mathematical elements) that interact in specific ways.

The mathematicians in this paper are trying to understand a very specific property of this machine: The "Energy" of its parts.

In the world of these machines, every part has a "degree" (a measure of its complexity or size). Some parts are divisible by a specific prime number (let's call it $p$), and some are not. The authors are interested in the parts that are not divisible by $p$. They want to know: If you add up the squares of the sizes of all these "non-divisible" parts, how big is the total?

The Big Mystery: The "Normalizer" vs. The Whole Machine

The paper investigates a relationship between the whole machine ($G$) and a smaller, specialized workshop inside it called the Normalizer of a Sylow $p$-subgroup ($N_G(P)$).

Think of the Sylow $p$-subgroup ($P$) as a specific type of gear that only turns in multiples of $p$. The Normalizer is the "supervisor" of this gear; it's the set of all parts in the machine that can rearrange this gear without breaking it.

The Conjecture (The Guess):
The authors are testing a guess made by a mathematician named E. Giannelli. The guess says:

"The total 'energy' (sum of squares of degrees) of the non-$p$-divisible parts in the whole machine is always greater than or equal to the total 'energy' of the non-$p$-divisible parts in the supervisor's workshop."

Furthermore, they suspect that if these two totals are exactly equal, it means the supervisor's workshop has a special "exit door" (a normal complement) that allows the rest of the machine to function independently.

The Strategy: The "Matching Game"

To prove this, the authors use a clever trick. They try to set up a matching game (a bijection) between the parts in the whole machine and the parts in the workshop.

• The Old Rule: The famous "McKay Conjecture" (recently proven) says you can match every part in the machine to a part in the workshop.
• The New Rule (Giannelli's Conjecture): The authors want to prove you can match them in a way where the part in the workshop is never bigger than the part in the whole machine.

If you can prove this "Size Rule," then the "Energy Conjecture" (sum of squares) automatically follows. It's like saying: "If every student in the big school is taller than or equal to their partner in the small club, then the total height of the big school must be greater than the small club."

The Breakthroughs

The paper is a marathon of proving this "Size Rule" for different types of machines.

1. The "Even Number" Victory ($p=2$):
   The authors completely solve the mystery for the prime number 2. They prove that for any machine, if you look at the parts not divisible by 2, the "Energy" of the whole machine is indeed greater than or equal to the workshop. This is their Theorem B. It's a huge win because 2 is the most common prime number in these structures.

2. The "Equality" Clue (Theorem C):
   They also figure out exactly when the two energies are equal. They prove that this only happens if the machine has a very specific structure where the supervisor's workshop can be "peeled off" cleanly. This is like finding the exact condition where a puzzle fits together perfectly with no gaps.

3. The "Simple Groups" Detective Work:
   To prove the rule for all machines, they had to check the "atomic" machines (called Simple Groups). These are the Lego bricks that build all other machines.

   • They checked machines built from Lie Type (complex geometric shapes).
   • They checked Sporadic Groups (weird, one-off machines like the Monster).
   • They checked Alternating Groups (machines that just shuffle things around).

   They found that for most of these atomic machines, the "Size Rule" holds true. The only tricky cases left are for specific types of shuffling machines (Alternating groups) and some complex geometric ones where the prime $p$ is odd (not 2).

Why Should We Care?

You might ask, "Why do we care about adding up squares of numbers in a group?"

The authors give a great reason:

• The Dimension of Reality: This sum represents the "size" of a mathematical universe (an algebra) built from these specific parts.
• The Identity Card: There is a famous old question: "If two machines have the same 'identity card' (their group algebra), are they the same machine?"
  • If the sum of squares of the non-$p$ parts equals a specific number, it might tell us if the machine has a "normal $p$-complement" (a hidden, clean structure).
  • Solving this helps mathematicians understand the fundamental DNA of symmetry and structure in the universe.

The Bottom Line

This paper is a major step forward in the "McKay Detective Agency." They have:

1. Proven the "Energy Conjecture" is true for the prime number 2.
2. Proven that if the energies are equal, the machine has a very specific, clean structure.
3. Reduced the remaining mystery to a few specific, difficult cases involving "shuffling" machines and odd primes.

It's a story of taking a massive, intimidating mathematical wall and chipping away at it, proving that the "whole is indeed greater than the sum of its parts" (in a very specific, squared way), and figuring out exactly when the two sides become equal.

Technical Summary

Here is a detailed technical summary of the paper "Sum of the Squares of the $p'$-Character Degrees" by Nguyen N. Hung, J. Miquel Martínez, and Gabriel Navarro.

1. Problem Statement and Context

The paper investigates a specific inequality concerning the character degrees of finite groups, specifically focusing on $p'$-characters (irreducible complex characters whose degrees are not divisible by a prime $p$).

• The Core Conjecture (Conjecture A): Let $G$ be a finite group and $P \in \text{Syl}_p(G)$. The authors propose that the sum of the squares of the $p'$-character degrees of $G$ is at least the index of the derived subgroup of $P$ in the normalizer of $P$:
  $$\sum_{\chi \in \text{Irr}_{p'}(G)} \chi(1)^2 \geq |N_G(P) : P'|$$
  where $P' = [P, P]$ is the derived subgroup. Equality holds if and only if $N_G(P)$ has a normal complement in $G$.

• Motivation: This problem is motivated by the recently proven McKay Conjecture (which establishes a bijection between $p'$-characters of $G$ and $N_G(P)$). While the McKay conjecture guarantees the existence of a bijection, it does not immediately imply the inequality of the sums of squares. The authors note that characters in $G$ often have larger degrees than those in $N_G(P)$, but proving this systematically is non-trivial.

• Connection to Giannelli's Conjecture: The paper focuses on a strengthening of the McKay conjecture proposed by E. Giannelli (Conjecture 3.1), which posits the existence of a bijection $f: \text{Irr}_{p'}(G) \to \text{Irr}_{p'}(N_G(P))$ such that $f(\chi)(1) \leq \chi(1)$ for all $\chi$. If Giannelli's conjecture holds, Conjecture A follows immediately.

2. Methodology

The authors employ a combination of character theory, reduction to simple groups, and case-by-case analysis of finite simple groups of Lie type.

A. Reduction to Quasisimple Groups

The paper utilizes the local-global reduction strategy standard in the proof of the McKay conjecture.

• Theorem 3.5: The authors prove that if Giannelli's strengthening (Conjecture 3.1) holds for all quasisimple groups involved in $G$ (specifically satisfying a specific inductive condition, Conjecture 3.3), then it holds for $G$.
• Inductive Condition (Conjecture 3.3): This requires the existence of an $A$-equivariant bijection (where $A$ is the automorphism group) between $p'$-characters of a quasisimple group $S$ and a specific subgroup $M < S$, satisfying the degree inequality and a "central isomorphism" condition on character triples.

B. Analysis of Character Degrees

To verify the inductive condition, the authors analyze the relationship between:

1. $m_p(S)$: The smallest non-trivial $p'$-degree of $S$.
2. $b_p(N_S(P))$: The largest $p'$-degree of the normalizer of a Sylow $p$-subgroup in $S$.

Hypothesis 4.3: The authors introduce the condition $m_p(S) \geq b_p(N_S(P))$. If this holds, any bijection mapping the trivial character to the trivial character automatically satisfies the degree inequality $\chi(1) \geq f(\chi)(1)$.

C. Case-by-Case Verification

The authors verify Conjecture 3.3 (and Hypothesis 4.3) for various classes of groups:

• Groups of Lie Type in Characteristic $p$: They analyze unitary groups ($SU_n(q)$) and other classical groups. They use Lusztig series, semisimple characters, and the structure of Sylow normalizers (often involving tori and Weyl groups) to bound the degrees.
• Groups of Lie Type in Characteristic $\neq p$: They utilize $d$-Harish-Chandra theory and Sylow $d$-tori (where $d$ is the order of $q$ modulo $p$) to bound the degrees of characters in the normalizer.
• Exceptional Covers: They handle exceptional covering groups (e.g., covers of sporadic groups and groups with non-generic Schur multipliers) using Dade's theory of blocks with cyclic defect groups and explicit computations (often referencing GAP).

D. Theorem C (Equality Condition)

The paper also proves Theorem C, which characterizes when equality holds in Conjecture A. It establishes that all $p'$-characters of $N_G(P)$ extend to $G$ if and only if there exists a normal subgroup $K \trianglelefteq G$ such that $K N_G(P) = G$ and $N_K(P) = 1$. This proof relies on a relative version of the McKay conjecture and induction on the group order.

3. Key Contributions and Results

Main Theorems

1. Theorem B (Main Result): Conjecture A holds for $p=2$.

   • The authors prove that for $p=2$, the sum of squares of $2'$-character degrees in$G$is at least$|N_G(P) : P'|$.
   • This is achieved by proving Giannelli's conjecture (Conjecture 3.1) for all quasisimple groups when $p=2$ (Theorem 5.1).

2. Theorem C: Provides a structural characterization for the equality case in Conjecture A, extending a previous result by Ito. It links the extension of characters to the existence of a normal complement.

3. Verification for $p=2$:

   • For $p=2$, the authors show that Hypothesis 4.3 ($m_2(S) \geq b_2(N_S(P))$) holds for almost all quasisimple groups.
   • For the few exceptions (specifically certain unitary groups and sporadic groups like $Co_2, Co_3, McL$), they construct explicit bijections or use the fact that the normalizer has a unique large-degree character to satisfy the inequality.

Specific Findings on Groups of Lie Type

• Unitary Groups ($SU_n(q)$): The paper provides a detailed analysis of Weil characters and semisimple characters. They prove that for $p=2$, even in cases where Hypothesis 4.3 fails (e.g., $SU_n(q)$ with $n$ odd), the specific structure of the automorphism group allows for the construction of the required degree-decreasing bijection.
• Sporadic Groups: The authors verify the conjecture for exceptional covers of sporadic groups, noting that for groups like $Co_2$ and $Co_3$, the normalizer is not perfect, allowing for linear characters that facilitate the degree inequality.

4. Significance

• Advancement of the McKay Program: This paper moves beyond the existence of the McKay bijection to address quantitative properties (sums of squares and degree inequalities). It demonstrates that the "local" structure (normalizer) controls the "global" sum of squares in a specific direction.
• Proof for $p=2$: The resolution of Conjecture A for $p=2$ is a significant milestone. The case $p=2$ is often the most difficult in finite group theory due to the complexity of Sylow 2-subgroups and their automorphisms.
• Methodological Innovation: The paper introduces and utilizes a "relative" version of the McKay conjecture and Theorem C to handle equality conditions. This approach offers a new tool for studying character extensions and normal complements.
• Open Problems: The authors explicitly state that the conjecture remains open for $p > 2$ in general. The difficulty lies in the fact that for $p > 2$, the normalizer $N_S(P)$ often possesses $p'$-characters with degrees larger than the minimal $p'$-degree of $S$, making the simple degree inequality approach (Hypothesis 4.3) fail more frequently.

Conclusion

The paper successfully proves that the sum of the squares of the $p'$-character degrees of a finite group $G$ is bounded below by the index $|N_G(P) : P'|$ when $p=2$. This result is derived by proving a strengthened version of the McKay conjecture (Giannelli's conjecture) for $p=2$, utilizing deep structural analysis of finite simple groups of Lie type and sporadic groups. The work also provides a complete characterization of the equality case, linking it to the existence of normal complements.