Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by the second author. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower bound for the maximal degree of an irreducible Brauer character belonging to B in terms of the dimension of B and well-known invariants like the defect and the number of irreducible Brauer characters. We also propose a weaker version of this conjecture for which a slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks.
We then show that the strong conjecture is true for blocks with one simple module, blocks of p-solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. We also show that the weak conjecture is true for blocks of tame representation type.
We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Späth's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups.
We provide a self-contained proof of the main properties of Brauer quotients of Young modules. We then use these results to give a new inductive proof of Nakayama's Conjecture on the blocks of the symmetric group.
In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.
We study Brauer's long-standing $k(B)$-conjecture on the number of characters in $p$-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for $p\ge5$ nor in the case of abelian defect. For $p=3$ we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes.
Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p'$-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p'$-degree of $G$ and $N_G (P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Broué abeli…
The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Spaeth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive condition for all finite simple groups. The first is to establish an equivariant bijection between irreducible Brauer characters and weights for the universal covering groups of simple groups. Assume q is a power of some odd prime p. We first prove the blockwise Alperin weight conjecture for Sp2n(q) and odd non-defining characteristics. If the decomposition matri…
This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpoten…
In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$.
A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite $p$-solvable group and $φ$ is an irreducible $p$-Brauer character of $G$ with vertex $Q$, then the number of lifts of $φ$ is at most $|Q:Q'|$. This conjecture is now known to be true in several situations for $p$ odd, but there has been little progress for $p$ even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift are neither linear nor conjugate. In this paper we show that if $χ$ is a lift of an irreducible $2$-Brauer character in a solvable group, then $χ$ has a linear Navarro vertex …
We study the cohomology with modular coefficients of Deligne-Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal l-block of a finite reductive group G(F_q) when the order of q modulo l is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, Lübeck and Malle) and the derived equivalence predicted by the geometric version of Broué's conjecture.
Blocks with a single automorphism orbit of irreducible Brauer characters
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）；Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\rm Ann}(J)=\{α\in F[G]|Jα=0\}$. We have another algebra $F[G]/I$. We study the decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$. This decomposition establishs some connections between Cartan invariants and chief composition factors of $G$. We find that existing zero-defect $p$-block in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we consider the Cartan invariants for a block algebra $B$ of $G$, the decomposition is…
In this paper, we study some variations of the well-known Itô-Michler theorem for $p$-Brauer characters using various inequalities involving the $p$-Brauer character degrees of finite groups. Several new criteria for the existence of a normal Sylow $p$-subgroup of finite groups are obtained using the $p$-parts and $p'$-parts of the $p$-Brauer character degrees.
We describe two methods for computing $p$-modular Brauer character tables for groups of Lie type $G(p^f)$ in defining characteristic $p$, assuming that the ordinary character table of $G(p^f)$ is known, and the weight multiplicities of the corresponding algebraic group $G$ are known for $p$-restricted highest weights.
We prove new results in generalized Harish-Chandra theory providing a description of the so-called Brauer--Lusztig blocks in terms of the information encoded in the $\ell$-adic cohomology of Deligne--Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the $\ell$-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between $\ell$-structures and their geometric counte…
We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for (some) quasi-simple groups.
We show that the decomposition matrix of a given group $G$ is unitriangular, whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of $N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to their stabilizer in $G$. Using the recent result by Brunat--Dudas--Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix, whenever they are related via Bonnafé--Dat--Rouquier's equivalence to a unipotent block. Th…
Brauer Theory for a finite group can be viewed as a method for comparing the representations of the group in characteristic 0 with those in prime characteristic. Here we generalize much of the machinery of Brauer theory to the setting of profinite groups. By regarding Grothendieck groups as functors we describe corresponding Grothendieck groups for profinite groups, and generalize the decomposition map, regarded as a natural transformation. We discuss characters and Brauer characters for profinite groups. We give a functorial description of the block theory of a profinite group. We finish with…
It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of $g_1^2\ldots g_n^2=1$ with $g_1,\ldots,g_n\in G$.
This paper provides some properties of Brauer symmetry classes of tensors. We derive a dimension formula for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups having cyclic groups support on non-linear Brauer characters. Using the derived formula, we investigate the necessary and sufficient condition for the existence of the o-basis of Dicyclic groups, Semi-dihedral groups and also reinvestigate those things on Dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors as…
Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of $G$ in characteristic other than p to a maximal subgroup of $G_2(q)$ is still irreducible. Similar results are obtained for $^2B_2(q)$ and $^2G_2(q)$.
The Fong-Swan theorem shows that for a $p$-solvable group $G$ and Brauer character $ϕ\in \ibrg$, there is an ordinary character $χ\in \irrg$ such that $χ^0 = ϕ$, where $^0$ denotes restriction to the $p$-regular elements of $G$. This still holds in the generality of $π$-separable groups \cite{bpi}, where $\ibrg$ is replaced by $\ipig$. For $ϕ\in \ipig$, let $L_ϕ = \{χ\in \irrg \mid χ^0 = ϕ\}$. In this paper we give a lower bound for the size of $L_ϕ$ in terms of the structure of the normal nucleus of $ϕ$ and, if $G$ is assumed to be odd and $π= \{p' \}$, we give an upper bound for $L_ϕ$ in ter…
A super-Brauer character theory of a group $G$ and a prime $p$ is a pair consisting of a partition of the irreducible $p$-Brauer characters and a partition of the $p$-regular elements of $G$ that satisfy certain properties. We classify the groups and primes that have exactly one super-Brauer character theory. We discuss the groups with exactly two super-Brauer character theories.
We introduce a generalization of Brauer character to allow arbitrary finite length modules over discrete valuation rings. We show that the generalized super Brauer character of Tate cohomology is a linear combination of trace functions. Using this result, we find a counterexample to a conjecture of Borcherds about vanishing of Tate cohomology for Fricke elements of the Monster.
We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the Itô--Michler theorem for the prime~2, again with Galois automorphisms. We close, this time for odd primes $p$, with a new characterisation of $p$-closed groups via the decomposition numbers of certain characters.
If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.
Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer charact…
The computation of the Brauer group BM of modified supergroup algebras is perfomed, yielding, in particular, the computation of the Brauer group of all finite-dimensional triangular Hopf algebras when the base field is algebraically closed and of characteristic zero. The results are compared with the computation of lazy cohomology and with Yinhuo Zhang's exact sequence. As an example, we compute explicitely the Brauer group and lazy cohomology for modified supergroup algebras with (extensions of) Weyl groups of irreducible root systems as a group datum and their standard representation as a re…
Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $φ(1)$ for every irreducible monomial $p$-Brauer character $φ$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.
We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.
In this paper we examine the behavior of lifts of Brauer characters in p-solvable groups where p is an odd prime. In the main result, we show that if ϕ\in IBrp(G) is a Brauer character of a solvable group such that ϕhas an abelian vertex subgroup Q, then the number of lifts of ϕin Irr(G) is at most |Q|. In order to accomplish this, we develop several results about lifts of Brauer characters in p-solvable groups that were previously only known to be true in the case of groups of odd order.
A well-known theorem of Burnside says that if $ρ$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $ρ$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of $ρ$ takes on $r$ distinct values, then the first $r$ tensor powers of $ρ$ already contain amongst them a…
Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such that $p^{\bar e_p(G)}$ divides $χ(1)$ for some $χ\in IBr_p(G)$. We show that $|G:O_p(G)|_p \leq p^{k \bar e_p(G)}$ for an explicitly given constant $k$. We also study the analogous problem for the $p$-parts of the conjugacy class sizes of $p$-regular elements of finite groups.
In this paper, we discuss O-basis of symmetry classes of polynomials associated with the Brauer character of the Dicyclic group. Also, necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the Dicyclic group.
In this paper, we discuss O-basis of symmetry classes of polynomials associated with the Brauer character of the Semi-Dihedral groups and Dihedral groups. Also, necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the Semi-Dihedral groups.
