Irreducible Brauer Characters Research Articles

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

AbstractLet k(B0) and l(B0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B0 of a finite group G. We prove that, if k(B0)−l(B0) = 1, then l(B0) ≥ p − 1 or else p = 11 and l(B0) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B0) ≥ p − 1 or else p = 11 and $$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ G / O p ′ ( G ) ≅ C 11 2 ⋊ SL ( 2 , 5 ) . These results are useful in the study of principal blocks with few characters.We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least $$2\sqrt {p - 1} + 1 - {k_p}(G)$$ 2 p − 1 + 1 − k p ( G ) , where kp(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

Irreducible Characters with Cyclic Anchor Group

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group.

Navarro vertices and lifts in solvable groups

Let Q be a p-subgroup of a finite p-solvable group G, where p is a prime, and suppose that δ is a linear character of Q with the property that whenever are conjugate in G. In this situation, we show that restriction to p-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of G with Navarro vertex onto the set of irreducible Brauer characters of G with Green vertex Q. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups. Communicated by Mandi Schaeffer Fry

On e-cuspidal pairs of finite groups of exceptional Lie type

Let G be a simple, simply connected algebraic group of exceptional type defined over F q with Frobenius endomorphism F : G → G . Let ℓ ∤ q be a good prime for G . We determine the number of irreducible Brauer characters in the quasi-isolated ℓ -blocks of G F . This is done by proving that generalized e -Harish-Chandra theory holds for the Lusztig series associated to quasi-isolated elements of G ⁎ F .

Real Brauer characters

In this paper, we determine the structure of finite groups with a small number of real-valued irreducible Brauer characters.

Bounds on the number of irreducible Brauer characters in blocks of finite groups of exceptional Lie type

Let G be a simple, simply connected linear algebraic group of exceptional type defined over Fq with Frobenius endomorphism F:G→G. In this work we give upper bounds for the number of irreducible Brauer characters in the quasi-isolated ℓ-blocks of GF and GF/Z(GF) when the prime ℓ is bad for G.

Low-dimensional representations of finite orthogonal groups

AbstractWe determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

Self-dual modules in characteristic two and normal subgroups

We prove Clifford theoretic results which only hold in characteristic 2.Let G be a finite group, let N be a normal subgroup of G and let φ be an irreducible 2-Brauer character of N. We show that φ occurs with odd multiplicity in the restriction of some self-dual irreducible Brauer character θ of G if and only if φ is G-conjugate to its dual. Moreover, if φ is self-dual then θ is unique and the multiplicity is 1.Next suppose that θ is a self-dual irreducible 2-Brauer character of G which is not of quadratic type. We prove that the restriction of θ to N is a sum of distinct self-dual irreducible Brauer character of N, none of which have quadratic type. Moreover, G has no self-dual irreducible 2-Brauer character of non-quadratic type if and only if N and G/N satisfy the same property.Finally, suppose that b is a real 2-block of N. We show that there is a unique real 2-block of G covering b which is weakly regular with respect to N.

Basic Sets for Unipotent Blocks of Finite Reductive Groups in Bad Characteristic

Abstract We show the existence of a unitriangular basic set for unipotent blocks of simple reductive groups of classical type in bad characteristic with some exceptions. Then, we introduce an algorithm to count unipotent irreducible Brauer characters and we verify that this algorithm is effective for exceptional adjoint groups.

An equivariant bijection between irreducible Brauer characters and weights for Sp(2n,q)

The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Späth 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 matrix of Sp2n(q) is unitriangular with respect to an Aut(Sp2n(q))-stable basic set (this assumption holds for linear primes), we can establish an equivariant bijection between the irreducible Brauer characters and weights.

The Inductive Blockwise Alperin Weight Condition for PSp4(q)

Let G be a finite group and ℓ be any prime dividing [Formula: see text]. The blockwise Alperin weight conjecture states that the number of the irreducible Brauer characters in an ℓ-block B of G equals the number of the G-conjugacy classes of ℓ-weights belonging to B. Recently, this conjecture has been reduced to the simple groups, which means that to prove the blockwise Alperin weight conjecture, it suffices to prove that all non-abelian simple groups satisfy the inductive blockwise Alperin weight condition. In this paper, we verify this inductive condition for the finite simple groups [Formula: see text] and non-defining characteristic, where q is a power of an odd prime.

The Alperin weight conjecture for symmetric and general linear groups revisited

Explicit bijections are given between the weights associated to a block of a symmetric group and the irreducible Brauer characters in the block. Similar bijections are given between the weights associated to a block of a general linear group and a basic set of the block.

2-blocks with abelian hyperfocal subgroup of rank 3

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G, whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has an abelian hyperfocal subgroup of rank 3.

Blocks with the hyperfocal subgroup [formula omitted

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G, whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has hyperfocal subgroup Z2n×Z2n for a positive integer number n, when the block is controlled by the normalizer NG(P) and the hyperfocal subgroup is contained in the center of P, or when the block is not controlled by NG(P) and the hyperfocal subgroup is contained in the center of the unique essential subgroup in the fusion system and has order at most 16. In particular, Alperin's weight conjecture holds in the considered cases.

Principal 2‐blocks and Sylow 2‐subgroups

Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal $2$-block of $G$ are fixed by a certain Galois automorphism $\sigma \in \mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite groups.

Quasi-projective Brauer characters

We study p-Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p-singular elements for a fixed prime p dividing the order of G. Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character φ there exists a minimal p-power, say pa(φ), such that pa(φ)φ is quasi-projective. The exponent a(φ) only depends on the Cartan matrix of the block to which φ belongs. Moreover pa(φ) is bounded by the vertex of the module affording φ, and equality holds in case that G is p-solvable. We give some evidence for the conjecture that a(φ)=0 occurs if and only if φ belongs to a block of defect 0. Finally, we study indecomposable quasi-projective Brauer characters of a block B. This set is finite and corresponds to a minimal Hilbert basis of the rational cone defined by the Cartan matrix of B.

Blocks of small defect in alternating groups and squares of Brauer character degrees

Abstract Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G / P {G/P} is nilpotent if and only if φ ⁢ ( 1 ) 2 {\varphi(1)^{2}} divides | G : ker ( φ ) | {|G:{\rm ker}(\varphi)|} for every irreducible Brauer character φ of G.

On the number of irreducible characters in a 3-block with a minimal nonabelian defect group

Let B be a 3-block of a finite group G with a defect group D. In this paper, we are mainly concerned with the number of characters in a particular block, so we shall use Isaacs’ approach to block structure. We consider the block B of a group G as a union of two sets, namely a set of irreducible ordinary characters of G having cardinality k(B) and a set of irreducible Brauer characters of G having cardinality l(B). We calculate k(B) and l(B) provided that D is normal in G and $$D \cong \left\langle {x,y,z|x^{3^n } = y^{3^m } = z^3 = \left[ {x,z} \right] = \left[ {y,z} \right] = 1,\left[ {x,y} \right] = z} \right\rangle \left( {n > m \geqslant 2} \right)$$ .

Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters

This paper provides some properties of Brauer symmetry classes of tensors. A dimension formula is derived for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups whose non-linear Brauer characters have support being a cyclic group. Using the derived formula, necessary and sufficient condition are investigated for the existence of an o-basis of dicyclic groups, semi-dihedral groups, and also those things are reinvestigated on dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.

On 2-blocks with $${k(B) - l(B) = 1}$$ k ( B ) - l ( B ) = 1

We investigate some properties of Cartan matrices of symmetric algebras. In particular, we study the Cartan matrices of p-blocks B of finite groups for the cases that \({k(B) - l(B) = 1}\) and that \({k(B) = 3}\) where k(B) and l(B) are the numbers of irreducible ordinary and Brauer characters associated to B, respectively.