A counterexample to a conjecture of Kiyota, Murai and Wada

TL;DR

This paper presents a counterexample disproving a 2002 conjecture that the largest eigenvalue of a block's Cartan matrix is rational if and only if all eigenvalues are rational, challenging previous assumptions in group theory.

Contribution

The authors provide the first known counterexample to the conjecture, showing that the largest eigenvalue can be irrational even when all eigenvalues are rational.

Findings

Counterexample with irrational largest eigenvalue

Disproves the if-and-only-if condition of the conjecture

Discusses implications for the theory of Cartan matrices

Abstract

Kiyota, Murai and Wada conjectured in 2002 that the largest eigenvalue of the Cartan matrix C of a block of a finite group is rational if and only if all eigenvalues of C are rational. We provide a counterexample to this conjecture and discuss related questions.

Equations

$$C=\left(\begin{array}{ccc}10& 10& 5\\ 10& 13& 6\\ 5& 6& 7\end{array}\right).$$

Full text

**Keywords: Cartan matrices, eigenvalues, rationality

AMS classification: 20C20**

Let $B$ be a block of a finite group $G$ with respect to an algebraically closed field of characteristic $p>0$. It is well-known that the Cartan matrix $C\in\mathbb{Z}^{l\times l}$ of $B$ is symmetric, positive definite, non-negative and indecomposable (here $l=l(B)$ is the number of simple modules of $B$). Let $ED$ (respectively $EV$) be the multiset of elementary divisors (respectively eigenvalues) of $C$. Note that these multisets do not depend on the order of the simple modules of $B$. Let $D$ be a defect group of $B$. Then the elementary divisors of $C$ divide $|D|$ and $|D|$ occurs just once in $ED$. On the other hand, the eigenvalues of $C$ are real, positive algebraic integers. By Perron-Frobenius theory, the largest eigenvalue $\rho(C)$ (i. e. the spectral radius) of $C$ occurs with multiplicity $1$ in $EV$. Moreover, $\prod_{\lambda\in ED}\lambda=\det(C)=\prod_{\lambda\in EV}\lambda$. Apart from these facts, there seems little correlation between $EV$ and $ED$.

According to (the weak) Donovan’s Conjecture, there should be an upper bound on $\rho(C)$ in terms of $|D|$. However, it can happen that $\rho(C)>l(B)|D|$. For example, if $B$ is the principal $2$-block of $G=\operatorname{PSp}(4,4).4$, a computation with GAP [1] shows that $\rho(C)>7201>5\cdot 2^{10}=l(B)|D|$. This is even more striking than the observation $\operatorname{tr}(C)>l(B)|D|$ made in [7] for the same block. Conversely, $|D|$ cannot be bounded in terms of $\rho(C)$: for $p\geq 5$ the principal $p$-block of $\operatorname{SL}(2,p)$ satisfies $\rho(C)<4<p=|D|$ (see [4, Example on p. 3843]).

If $\lambda\in EV\cap\mathbb{Z}$, then $|D|/\lambda$ is an eigenvalue of $|D|C^{-1}\in\mathbb{Z}^{l\times l}$ and therefore it is an algebraic integer. This shows that $\lambda$ divides $|D|$. By a similar argument, $\lambda$ is divisible by the smallest elementary divisor of $C$. In [3, Questions 1 and 2], Kiyota, Murai and Wada proposed the following conjecture on the rationality of eigenvalues (see also [10, Conjecture]).

1. (1)

$EV=ED$. 2. (2)

$\rho(C)=|D|$. 3. (3)

$\rho(C)\in\mathbb{Z}$. 4. (4)

$EV\subseteq\mathbb{Z}$.

Clearly, (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Leftarrow$ (4) $\Leftarrow$ (1) holds and it remains to prove (3) $\Rightarrow$ (1). This has been done for blocks of finite or tame representation type (see [3, Propositions 3 and 4]). For $p$-solvable $G$ we have (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (4) and $\rho(C)\leq|D|$ (see [3, Theorem 1], [8, Corollary 3.6] and [4, Corollary 3.6]). Other special cases were considered in [5, 6, 9, 12]. If $D\unlhd G$, then (1)–(4) are satisfied (see [3, Proposition 2]). This holds in particular for the Brauer correspondent $b$ of $B$ in the normalizer $\operatorname{N}_{G}(D)$. In view of Broué’s Abelian Defect Group Conjecture, Kiyota, Murai and Wada [3, Question 3] raised the following question.

It was proved in [6, 5] that the answer to A counterexample to a conjecture of Kiyota, Murai and Wada is yes for principal $p$-blocks whenever $p\in\{2,3\}$. However, the following counterexample shows not only that A counterexample to a conjecture of Kiyota, Murai and Wada is false, but also that A counterexample to a conjecture of Kiyota, Murai and Wada has a negative answer (for principal blocks) in general:

Let $B$ be the principal $5$-block of $G=\operatorname{PSU}(3,4)$. The Atlas of Brauer characters [2] (or [11]) gives

[TABLE]

It follows that EV=\bigl{\{}\frac{1}{2}(5+\sqrt{5}),\,\frac{1}{2}(5-\sqrt{5}),\,25\bigr{\}} and $ED=\{1,\,5,\,25\}$. Therefore, $\rho(C)=25=|D|$, but $EV\neq ED$. Moreover, $D$ is abelian since $|D|=25$, but $B$ cannot be Morita equivalent to $b$, since the eigenvalues of the Cartan matrix of $b$ are rational integers as explained above.

We do not know whether the implications (3) $\Rightarrow$ (2), (4) $\Rightarrow$ (1) or (4) $\Rightarrow$ (2) in A counterexample to a conjecture of Kiyota, Murai and Wada might hold in general. Wada [9, Decomposition Conjecture] strengthened all three implications as follows.

• •

$|E_{i}|=|F_{i}|$* for $i=1,\ldots,n$.*

• •

$\prod_{\lambda\in E_{i}}\lambda=\prod_{\lambda\in F_{i}}\lambda$* for $i=1,\ldots,n$.*

• •

$\prod_{\lambda\in E_{i}}(X-\lambda)\in\mathbb{Z}[X]$* is irreducible for $i=1,\ldots,n$.*

• •

$\rho(C)\in E_{1}$, $|D|\in F_{1}$.

Again we found a counterexample: The group $\operatorname{PSU}(3,3)$ has a faithful $7$-dimensional representation over $\mathbb{F}_{3}$. Let $G=\mathbb{F}_{3}^{7}\rtimes\operatorname{PSU}(3,3)$ be the corresponding semidirect product, and let $B$ be the principal $3$-block of $G$. This group and its character table can be accessed as $\texttt{PrimitiveGroup}(3^{7},35)$ and CharacterTable("P49/G1/L1/V1/ext3") in GAP. In this way we obtain $9\in EV$, but $9\notin ED$. Obviously, this contradicts A counterexample to a conjecture of Kiyota, Murai and Wada.

Acknowledgment

I thank Thomas Breuer for some explanations about character tables in GAP. Moreover, I am grateful to Gabriel Navarro for getting me interested in counterexamples. This work is supported by the German Research Foundation (project SA 2864/1-1).