Projective Conjecture Research Articles

Character Triple Conjecture for p-solvable groups

In this paper, we prove the Character Triple Conjecture for p-solvable groups. This is a conjecture proposed by Späth during the reduction process of Dade's Projective Conjecture to quasisimple groups (see [29]). In addition, as suggested by Isaacs and Navarro in [13], we take into account the p-residue of characters.

A reduction theorem for Dade’s projective conjecture

In this paper, we propose a strengthening of Dade’s Conjecture. This version, called the Character Triple conjecture, once assumed for quasisimple groups, is shown to imply Dade’s Projective Conjecture for all finite groups. In particular Dade’s Projective Conjecture holds for a group whose nonabelian simple sections have only covering groups satisfying the Character Triple Conjecture. We verify the new conjecture for some classes of quasisimple groups.

Refinements of Dade's Projective Conjecture for p-solvable groups

Dade's Projective Conjecture is known to be true for finite p-solvable groups thanks to work of G.R. Robinson, but remains open in general. Work of Isaacs and Navarro suggested to Uno and Boltje refinements of this conjecture. These refinements were studied for finite p-solvable groups by Glesser. In the present paper, inspired by earlier work of Turull, we propose further refinements of the conjecture that take into account the Schur indices and the elements of the Brauer group. We prove that all these refinements of Dade's Projective Conjecture hold for all finite p-solvable groups. In particular, we obtain that the version of Dade's Projective Conjecture which involves character degree residues modulo p, fields of definition and Schur indices, as well as the full strength of Boltje's Conjecture both hold for all finite p-solvable groups. The proof develops a Clifford theory for normalizers of chains of p-subgroups which allows one to reduce the calculation of the relevant sums to simpler groups.

On the Gorenstein projective conjecture: IG-projective modules

Let [Formula: see text] be a commutative Noetherian local ring. A finitely generated Gorenstein projective [Formula: see text]-module [Formula: see text] is called IG-projective if there exists either an irreducible epimorphism [Formula: see text] or irreducible monomorphism [Formula: see text] with [Formula: see text] being a projective [Formula: see text]-module. In this paper, we give a note on Gorenstein Projective Conjecture: a finitely generated IG-projective module is projective if and only if it is self-orthogonal.

Derived equivalences and Cohen-Macaulay Auslander algebras

Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that Gorenstein projective conjecture is an invariant under standard derived equivalence between Artin R-algebras.

Dade's Invariant Conjecture for the Symplectic Group Sp4(2 n ) and the Special Unitary Group SU4(22n ) in Defining Characteristic

In this article, we verify Dade's projective invariant conjecture for the symplectic group Sp4(2 n ) and the special unitary group SU4(22n ) in the defining characteristic, that is, in characteristic 2. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for Sp4(2 n ) and SU4(22n ) in the defining characteristic, that is, Sp4(2 n ) and SU4(22n ) are good for the prime 2 in the sense of Isaacs, Malle, and Navarro.

Some refinements of Dade's projective conjecture

In this paper a further refinement of Dade's projective conjecture, due to Boltje, is presented. This new statement includes ideas first published by Isaacs and Navarro as well as the recent contractibility version of Alperin's conjecture introduced by Boltje. Leaning heavily on the work of Robinson, weaker forms of the conjecture are proved in the case of p-solvable groups.

Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

We show that Uno's refinement of the projective conjecture of Dade holds every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade's projective conjecture, Alperin-McKay conjecture and Puig's nilpotent block conjecture hold for all trivial intersection blocks.

The equivalence of some conjectures of Dade and Robinson

We demonstrate that Conjecture 4.1 of [G.R. Robinson, Proc. London Math. Soc. (3) 72 (1996) 312–330] and Dade's projective conjecture are equivalent in a way which is compatible with the p-local rank. Further we consider refinements of these conjectures similar to those of Isaacs, Navarro and Uno, show their equivalence and demonstrate that in order to verify them it suffices to consider only those groups with no non-central normal p-subgroup.

Dehn surgery on crosscap number two knots and projective planes

In the present paper, we will study the creation of projective planes by Dehn surgery on knots in the 3-sphere. It is shown that a projective plane cannot be created by Dehn surgery on a crosscap number two knot. As a corollary, we will prove that crosscap number two knots satisfy the projective space conjecture, which asserts that the projective 3-space cannot be obtained by Dehn surgery on a non-trivial knot in the 3-sphere.

On a Minimal Counterexample to Dade's Projective Conjecture

On a Minimal Counterexample to Dade's Projective Conjecture

On Finite Groups of p-Local Rank One and Conjectures of Dade and Robinson

We use the classification of finite simple groups to demonstrate that Conjecture 4.1 of G. R. Robinson (1996, Proc. London Math. Soc. (3)72, 312–330) and Dade's projective conjecture hold for finite groups of p-local rank one.

Dade's Projective Conjecture for p-Solvable Groups

Dade's Projective Conjecture for p-Solvable Groups

Verification of Dade's Conjecture for Janko GroupJ3

In [3] Dade made a conjecture expressing the numberk(B,d) of characters of a given defectdin a givenp-blockBof a finite groupGin terms of the corresponding numbersk(b,d) for blocksbof certainp-local subgroups ofG. Several different forms of this conjecture are given in [5].Dade claims that the most complicated form of this conjecture, called the “Inductive Conjecture 5.8” in [5], will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive conjecture for all covering groups of the third Janko groupJ3(in the notation of the Atlas [1]). This is one step in the inductive proof of the conjecture for all finite groups.Certain properties ofJ3simplify our task. The Schur Multiplier ofJ3is cyclic of order 3 (see [1, p. 82]). Hence, there are just two covering groups ofJ3, namelyJ3itself and a central extension 3·J3ofJ3by a cyclic groupZof order 3. We treat these two covering groups separately.The outer automorphism group Out(J3) ofJ3is cyclic of order 2 (see [1, p. 82]). In this case Dade affirms in [5, Section 6] that the inductive conjecture forJ3is equivalent to the much weaker “Invariant Conjecture 2.5” in [5]. Furthermore, Dade has proved in [6] that this Invariant Conjecture holds for all blocks with cyclic defect groups. The Sylowp-subgroups ofJ3are cyclic of orderpfor all primes dividing |J3| except 2 and 3. So we only need to verify the Invariant Conjecture for the two primesp=2 andp=3. We do that in Theorem 2.10.1 and Theorem 3.6.1 below.The group Out(3·J3|Z) of outer automorphisms of 3·J3centralizingZis trivial (see [1, p. 82]). In this case Dade affirms that the Inductive Conjecture is equivalent to the “Projective Conjecture 4.4” in [5]. Again, Dade has shown in [6] that this projective conjecture holds for all blocks with cyclic defect groups. So we only need to verify the projective conjecture forp=2 andp=3. We do that in Theorem 4.4.2 and Theorem 4.3.1.For the above reasons the four Theorems 2.10.1, 3.6.1, 4.4.2, and 4.3.1 below are sufficient to prove the inductive form of Dade's conjecture for all covering groups ofJ3.This paper is my Ph.D. thesis. It would not have been possible without the help and infinite patience of my advisor Everett Dade.

On a problem of Erdős and Larson

It is shown that a pairwise balanced design onn points in which each block is of size at leastn 1/2 −c can be embedded in a projective plane of ordern+i for somei≦c + 2 ifn is sufficiently large. Among other things this implies that if the projective plane conjecture is true, the conjecture of Erdős and Larson will not be true.