Height Zero Research Articles

Brauer's Height Zero Conjecture

Brauer's Height Zero Conjecture

A 12/7-approximation algorithm for the discrete Bamboo Garden Trimming problem

We study the discrete Bamboo Garden Trimming problem (BGT), where we are given n bamboos with different growth rates. At the end of each day, one can cut down one bamboo to height zero. The goal in BGT is to make a perpetual schedule of cuts such that the height of the tallest bamboo ever is minimized. Here, we improve the current best approximation guarantee by designing a 12/7-approximation algorithm.

Brauer’s Height Zero Conjecture for principal blocks

Abstract We prove the other half of Brauer’s Height Zero Conjecture in the case of principal blocks.

Brauer’s height zero conjecture for two primes

Let p and q be two primes. We propose that Brauer’s Height Zero Conjecture for the principal p-blocks of finite groups can naturally be extended from the perspective of q. We prove one direction of this new conjecture, and show the reverse direction assuming that the Inductive Alperin–McKay condition holds for the finite simple groups.

P-Blocks relative to a character of a normal subgroup

Let G be a finite group, let N◃G, and let θ∈Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|θ) relative to p. We call each member Bθ of this partition a θ-block, and to each θ-block Bθ we naturally associate a conjugacy class of p-subgroups of G/N, which we call the θ-defect groups of Bθ. If N is trivial, then the θ-blocks are the Brauer p-blocks. Using θ-blocks, we can unify the Gluck–Wolf–Navarro–Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every θ-block Bθ has size less than or equal the size of any of its θ-defect groups, thus bringing normal subgroups to the k(B)-conjecture.

Alperin\u2013McKay natural correspondences in solvable and symmetric groups for the prime p=2

Let G be a finite solvable or symmetric group, and let B be a 2-block of G. We construct a canonical correspondence between the irreducible characters of height zero in B and those in its Brauer first main correspondent. For symmetric groups our bijection is compatible with restriction of characters.

On Brauer’s Height Zero Conjecture

In this paper, the unproven half of Richard Brauer’s Height Zero Conjecture is reduced to a question on simple groups.

Zero character heights and Abelian defect groups of blocks in finite groups

Let G be a group, Irr(G) be the set of all the irreducible ordinary characters of G, B ∊ Bl(G) be a block of G and Irr(B) be the set of all the irreducible ordinary characters of G which are in the block B. In this paper the object is to study the ordinary irreducible characters of height zero in relation to the Abelian defect groups of the blocks in which such characters sit. We prove that if B ∊ Bl(G) is a block of G with a defect group D such that every χ ∊ Irr(B) has height zero, then D is Abelian.

Characters of relative p'-degree over normal subgroups

Let Z be a normal subgroup of a finite group G , let ??Irr(Z) be an irreducible complex character of Z , and let p be a prime number. If p does not divide the integers ?(1)/?(1) for all ??Irr(G) lying over ? , then we prove that the Sylow p -subgroups of G/Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture

ON RELATIVE HEIGHT ZERO BRAUER CHARACTERS

Let $N \triangleleft G$ where $G$ is a finite group and let $B$ be a $p$-block of $G$, where $p$ is a prime. A Brauer character $\psi \in \mathop{\mathrm{IBr}}_{p}(B)$ is said to be of relative height zero with respect to $N$ provided that the height of $\psi$ is equal to that of an irreducible constituent of $\psi_{N}$. Now assume $G$ is $p$-solvable. In this paper, we count the number of relative height zero irreducible Brauer characters of $B$ with respect to $N$ that lie over any given $\varphi \in \mathop{\mathrm{IBr}}_{p}(N)$. As a consequence, we show that if $D$ is a defect group of $B$ and $\hat{B}$ is the unique $p$-block of $NN_{G}(D)$ with defect group $D$ such that $\hat{B}^{G} = B$, then $B$ and $\hat{B}$ have equal numbers of relative height zero irreducible Brauer characters with respect to $N$.

Canonical heights for Hénon maps

We consider the arithmetic of Henon maps f(x, y)=(ay, x+f(y)) defined over number fields and function fields, usually with the restriction that a=1. We prove a result on the variation of Kawaguchi's canonical height in families of Henon maps, and derive from this a specialization theorem, showing that the set of parameters above which a given non-periodic point becomes periodic is a set of bounded height. Proving this involves showing that the only points of canonical height zero for a Henon map over a function field are those which are periodic (in the non-isotrivial case). In the case of quadratic Henon maps f(x, y)=(y, x+y^2+b), we obtain a stronger result, bounding the canonical height below by a quantity which grows linearly in the height of b, once the number of places of bad reduction is fixed. Finally, we propose a conjecture regarding rational periodic points for quadratic Henon maps defined over the rational numbers, namely that they can only have period 1, 2, 3, 4, 6, or 8. We check this conjecture for the first million values of the parameter b, ordered by height.

Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups

We propose a generalization of Brauer?s Height Zero Conjecture that considers positive heights. We give strong evidence supporting one half of the generalization and obtain some partial results regarding the other half.

Brauer’s height zero conjecture for metacyclic defect groups

We prove that Brauer’s Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and p-rational irreducible characters. Then the Alperin-McKay Conjecture follows provided p = 3. Along the way we verify a few other conjectures. Finally we consider the extraspecial defect group of order p 3 and exponent p 2 for an odd prime more closely. Here for blocks with inertial index 2 we prove the Galois-Alperin-McKay Conjecture by computing k0(B). Then for p 11 also Alperin’s Weight Conjecture follows. This improves some results of [Gao, 2012], [Holloway-Koshitani-Kunugi, 2010] and [Hendren, 2005].

The heights of irreducible Brauer characters in 2-blocks of the symmetric groups

We prove that there exists a unique irreducible Brauer character of height zero in any 2-blocks of the symmetric group. This generalizes the theorem of P. Fong and G.D. James that the dimension of every non-trivial 2-modular simple module of the symmetric group is even.

Local Schur indices, blocks, and character values

It is shown that ordinary irreducible characters of height zero in 2-blocks of finite groups have trivial 2-local Schur index. In view of Brauer's height zero conjecture, it is therefore extremely plausible that the irreducible characters in 2-blocks with abelian defect groups have trivial 2-local Schur index. If q is an odd prime, then there is no relationship expected between the height of an irreducible character in a q-block and the q-local Schur index, but a condition for a prime p≠q not to divide this index is given in terms of a rationality property of a defect class of the character's p-block.

On the focal defect group of a block, characters of height zero, and lower defect group multiplicities

We discuss the focal subgroup of the defect group D of a p-block B, which we refer to as the focal defect group, and denote by D 0 . We note that (the character group) of D / D 0 acts (in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducible characters of height zero is semi-regular. We also prove that all orbits under this action have length divisible by [ Z ( D ) : D 0 ∩ Z ( D ) ] . As applications, we prove that all Cartan invariants for B are divisible by [ Z ( D ) : D 0 ∩ Z ( D ) ] , that if Out ( D ) is a p-group (and D ≠ 1 ), then the number of irreducible characters of height zero in B is divisible by p and that if Z ( D ) ⩽̸ D 0 , then the block B is of Lefschetz type (see [R. Knörr, G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1) (1989) 48–60]).

A class of blocks behaving like blocks of p-solvable groups

We define a class of blocks having similar properties to blocks of p-solvable groups, and show that a version of the Fong–Swan theorem holds for irreducible Brauer characters in such blocks. We also show that the height of an irreducible character in such block is bounded by the exponent of the central quotient of a defect group, which in particular implies that if further the defect groups are abelian, then every irreducible character in the block has height zero.

Heights and preperiodic points of polynomials over function fields

Let K be a function field in one variable over an arbitrary field F. Given a rational function f(z) in K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of f all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial f, such points exist only if f is isotrivial. In fact, such K-rational points exist only if f is defined over the constant field of K after a K-rational change of coordinates.

Brauer's Height Zero Conjecture for Central Quotients of Special Linear and Special Unitary Groups

Brauer's Height Zero Conjecture for Central Quotients of Special Linear and Special Unitary Groups

On conjectures of Olsson, Brauer, and Alperin

Let G be a finite group, p a prime number, B a p-block of the group G, k(B) the number of irreducible complex characters of R belonging to B, k0(B) the number of irreducible characters of height zero in B, and let D be the defect group of B. This article considers the relationship between Brauer's conjecture (k(B) ≤ ¦D¦), Olsson's conjecture (k0(B) ≤¦D/D'¦), and Alperin's conjecture (k0(B) = k0(~B), where ~B is a p-block NG(D) such that ~BG = B). In particular, Olsson's conjecture is proved for p-blocks for those p-solvable groups G for which a Hall p′-subgroup of the group NG(D) is either supersolvable or has odd order.