20C33 Representations of finite groups of Lie type
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In the representation theory of finite groups, the so-called local-global conjectures assert a relation between the representation theory of a finite group and one of its local subgroups. The McKay-Navarro conjecture claims that the action of a set of Galois automorphisms on certain ordinary characters of the local and global group is equivariant. Navarro, Späth, and Vallejo reduced the conjecture to a problem about simple groups in 2019 and stated an inductive condition that has to be verified for all finite simple groups.
In this work, we give an introduction to the character theory of finite groups and state the McKay-Navarro conjecture and its inductive condition. Furthermore, we recall the definition of finite groups of Lie type and present results regarding their structure and their representation theory.
In the second part of this work, we verify the inductive McKay-Navarro condition for various families of finite groups of Lie type.
In defining characteristic, most groups have already been considered by Ruhstorfer.
We show that the inductive condition also holds for the groups with exceptional graph automorphisms, the Suzuki and Ree groups, the groups \(B_n(2)\) for \(n \geq 2\), as well as for the simple groups of Lie type with non-generic Schur multiplier in their defining characteristic.
This completes the verification of the inductive McKay-Navarro condition in defining characteristic. We further consider the Suzuki and Ree groups and verify the inductive condition for all primes. On the way, we show that there exists a Galois-equivariant Jordan decomposition for their irreducible characters.
Moreover, we consider some families of groups of Lie type that do not admit a generic choice of a local subgroup.
We show that the inductive condition is satisfied for the prime \(\ell=3\) and the groups \(\text{PSL}_3(q)\) with \(q \equiv 4, 7 \mod 9\), \(\text{PSU}_3(q)\) with \(q \equiv 2, 5 \mod 9\), and \(G_2 (q)\) with \(q \equiv 2, 4, 5, 7 \mod 9\).
Further, we verify the inductive condition for the prime \(\ell=2\) and \(G_2(3^f)\) for \(f \geq 1\), \(^3 D_4(q)\), and \(^2E_6(q)\) where \(q\) is an odd prime power.
Deligne-Lusztig theory allows the parametrization of generic character tables of finite groups of Lie type in terms of families of conjugacy classes and families of irreducible characters "independently" of \(q\).
Only in small cases the theory also gives all the values of the table.
For most of the groups the completion of the table must be carried out with ad-hoc methods.
The aim of the present work is to describe one possible computation which avoids Lusztig's theory of "character sheaves".
In particular, the theory of Gel'fand-Graev characters and Clifford theory is used to complete the generic character table of \(G={\rm Spin}_8^+(q)\) for \(q\) odd.
As an example of the computations, we also determine the character table of \({\rm SL}_4(q)\), for \(q\) odd.
In the process of finding character values, the following tools are developed.
By explicit use of the Bruhat decomposition of elements, the fusion of the unipotent classes of \(G\) is determined.
Among others, this is used to compute the 2-parameter Green functions of every Levi subgroup with disconnected centre of \(G\).
Furthermore, thanks to a certain action of the centre \(Z(G)\) on the characters of \(G\), it is shown how, in principle, the values of any character depend on its values at the unipotent elements.
It is important to consider \({\rm Spin}_8^+(q)\) as it is one of the "smallest" interesting examples for which Deligne--Lusztig theory is not sufficient to construct the whole character table.
The reasons is related to the structure of \({\mathbf G}={\rm Spin}_8\), from which \(G\) is constructed.
Firstly, \({\mathbf G}\) has disconnected centre.
Secondly, \({\mathbf G}\) is the only simple algebraic group which has an outer group automorphism of order 3.
And finally, \(G\) can be realized as a subgroup of bigger groups, like \(E_6(q)\), \(E_7(q)\) or \(E_8(q)\).
The computation on \({\rm Spin}_8^+(q)\) serves as preparation for those cases.
In a recent paper, G. Malle and G. Robinson proposed a modular anologue to Brauer's famous \( k(B) \)-conjecture. If \( B \) is a \( p \)-block of a finite group with defect group \( D \), then they conjecture that \( l(B) \leq p^r \), where \( r \) is the sectional \( p \)-rank of \( D \). Since this conjecture is relatively new, there is obviously still a lot of work to do. This thesis is concerned with proving their conjecture for the finite groups of exceptional Lie type.
In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.
The central topic of this thesis is Alperin's weight conjecture, a problem concerning the representation theory of finite groups.
This conjecture, which was first proposed by J. L. Alperin in 1986, asserts that for any finite group the number of its irreducible Brauer characters coincides with the number of conjugacy classes of its weights. The blockwise version of Alperin's conjecture partitions this problem into a question concerning the number of irreducible Brauer characters and weights belonging to the blocks of finite groups.
A proof for this conjecture has not (yet) been found. However, the problem has been reduced to a question on non-abelian finite (quasi-) simple groups in the sense that there is a set of conditions, the so-called inductive blockwise Alperin weight condition, whose verification for all non-abelian finite simple groups implies the blockwise Alperin weight conjecture. Now the objective is to prove this condition for all non-abelian finite simple groups, all of which are known via the classification of finite simple groups.
In this thesis we establish the inductive blockwise Alperin weight condition for three infinite series of finite groups of Lie type: the special linear groups \(SL_3(q)\) in the case \(q>2\) and \(q \not\equiv 1 \bmod 3\), the Chevalley groups \(G_2(q)\) for \(q \geqslant 5\), and Steinberg's triality groups \(^3D_4(q)\).
A classical conjecture in the representation theory of finite groups, the McKay conjecture, states that for any finite group and prime number p the number of complex irreducible characters of degree prime to p is equal to the number of complex irreducible characters of degree prime to p of the normalizer of a p-Sylow subgroup. Recently a reduction theorem was proved by Isaacs, Malle and Navarro: If all simple groups are “good”, then the McKay conjecture holds. In this work we are concerned with the problem of goodness for finite groups of Lie type in their defining characteristic. A simple group is called “good” if certain equivariant bijections between the involved character sets exist. We present a structural approach to the construction of such a bijection by utilizing the so-called “Steinberg-Map”. This yields very natural bijections and we prove most of the desired properties.
Die Arbeit beschäftigt sich mit den Charakteren des Normalisators und des Zentralisators eines Sylowtorus. Dabei wird jede Gruppe G vom Lie-Typ als Fixpunktgruppe einer einfach-zusammenhängenden einfachen Gruppe unter einer Frobeniusabbildung aufgefaßt. Für jeden Sylowtorus S der algebraischen Gruppe wird gezeigt, dass die irreduziblen Charaktere des Zentralisators von S in G sich auf ihre Trägheitsgruppe im Normalisator von S fortsetzen. Diese Fragestellung entsteht aus dem Studium der Höhe 0 Charaktere bei endlichen reduktiven Gruppen vom Lie-Typ im Zusammenhang mit der McKay-Vermutung. Neuere Resultate von Isaacs, Malle und Navarro führen diese Vermutung auf eine Eigenschaft von einfachen Gruppen zurück, die sie dann für eine Primzahl gut nennen. Bei Gruppen vom Lie-Typ zeigt das obige Resultat zusammen mit einer aktuellen Arbeit von Malle einige dabei wichtige und notwendige Eigenschaften. Anhand der Steinberg-Präsentation werden vor allem bei den klassischen Gruppen genauere Aussagen über die Struktur des Zentralisators und des Normalisators eines Sylowtorus bewiesen. Wichtig dabei ist die von Tits eingeführte erweiterte Weylgruppe, die starke Verbindungen zu Zopfgruppen besitzt. Das Resultat wird in zahlreichen Einzelfallbetrachtungen gezeigt, bei denen in dieser Arbeit bewiesene Vererbungsregeln von Fortsetzbarkeitseigenschaften benutzt werden.