[Paper Review] Cross characteristic representations of $^3D_4(q)$ are Reducible over proper subgroups

Hung Ngoc Nguyen, Pham Huu Tiep|ArXiv.org|Oct 26, 2009

Finite Group Theory Research2 references3 citations

TL;DR

This paper proves that every absolutely irreducible representation of the triality group $^3D_4(q)$ in characteristic coprime to $q$ becomes reducible when restricted to any proper subgroup. Using character theory, representation degree bounds, and explicit character computations via CHEVIE, the authors show that only maximal parabolic subgroups, $G_2(q)$, and $^3D_4(q_0)$ with $q = q_0^2$ could potentially preserve irreducibility—yet even these fail under detailed analysis, confirming the main result: no proper subgroup preserves irreducibility for nontrivial representations.

ABSTRACT

We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups $^3D_4(q)$ in characteristic coprime to q to any proper subgroup is reducible

Motivation & Objective

• To classify all triples $(K, V, H)$ where $K = {}^3D_4(q)$, $V$ is an absolutely irreducible representation in characteristic $ eq q$, and $H < K$ with $V|_H$ irreducible.
• To resolve Problem 1 for the exceptional group $^3D_4(q)$ in cross-characteristic settings.
• To extend prior results on irreducibility of representations over subgroups to the triality group $^3D_4(q)$, particularly in the case of non-complex representations.
• To establish that no proper subgroup of $^3D_4(q)$ can preserve irreducibility of nontrivial representations in cross-characteristic.

Proposed method

• Apply the Reduction Theorem to restrict attention to maximal subgroups of $^3D_4(q)$, particularly parabolic subgroups, $G_2(q)$, and $^3D_4(q_0)$ with $q = q_0^2$.
• Use degree bounds: $ ext{dim}(V) eq ext{m}_{ ext{C}}(H)$, so if $ ext{m}_{ ext{C}}(H) < ext{d}_{ ext{C}}(K)$, irreducibility over $H$ is impossible.
• Compute character values of unipotent characters on involutions using induced characters and additive characters over finite fields.
• Use CHEVIE to compute scalar products of induced characters with the unipotent character $[ ho_2]$ of degree $q^7(q^4 - q^2 + 1)$, ensuring integrality constraints.
• Analyze the structure of faithful irreducible characters of maximal parabolic subgroups $Q$ via induced linear characters from specific unipotent radicals.
• Leverage Brauer character theory and the fact that values on involutions must be integers to constrain possible character degrees.

Experimental results

Research questions

• RQ1 Can any proper subgroup of $^3D_4(q)$ preserve the irreducibility of a nontrivial absolutely irreducible representation in characteristic coprime to $q$?
• RQ2 Which maximal subgroups of $^3D_4(q)$ could potentially support irreducible representations of the group?
• RQ3 What character degree bounds and integrality conditions rule out irreducibility over subgroups?
• RQ4 Do the unipotent characters of $^3D_4(q)$, particularly $[ ho_2]$, provide sufficient constraints to rule out irreducibility over subgroups?
• RQ5 Are the character values of induced representations on involutions constrained to specific integer multiples of $q(q^3 - 1)$, and how does this affect irreducibility?

Key findings

• The minimal degree of an irreducible representation of $^3D_4(q)$ in characteristic coprime to $q$ is at least $q^5 - q^3 + q - 1$, which exceeds the maximal complex character degree of any proper subgroup not of type $P$, $Q$, $G_2(q)$, or $^3D_4(q_0)$.
• For $M = PGL^ ho_3(q)$ with $q eq 1 mod 3$, the complex character degree bound $\mathfrak{m}_{\mathbb{C}}(M)$ is strictly less than $\mathfrak{d}_{\ell}(^3D_4(q))$, so irreducibility over $M$ is impossible.
• The maximal subgroup $M = {}^3D_4(q_0)$ with $q = q_0^\alpha$, $\alpha \geq 5$ prime, has $\mathfrak{m}_{\mathbb{C}}(M) \leq q_0^{14}$, which is still less than $\mathfrak{d}_{\ell}(^3D_4(q))$ for $q = q_0^\alpha$, $\alpha \geq 5$, thus ruling out irreducibility.
• For the maximal parabolic subgroup $Q$, the character values of $\chi_{16}(k)$, $\chi_{18}(k)$, $\chi_{19}(k)$, and $\chi_{20}(k)$ on the involution $x_\beta(1)$ are constrained to integer multiples of $q(q^3 - 1)$, and integrality of scalar products with $[\varepsilon_2]$ confirms that no such character can be irreducible on $Q$.
• The scalar product $(_Q\chi_{16}(k), [\varepsilon_2]_Q)_Q = \frac{q^6 - q^4 - q^3 + q + y_k}{q(q^3 - 1)}$ is a nonnegative integer, and since $y_k \leq q^3(q^3 - 1)(q - 1)$, this forces $y_k$ to be a multiple of $q(q^3 - 1)$, confirming the integrality and bounding the character values.