Brauer's Height Zero Conjecture

The Brauer's Height Zero Conjecture is a conjecture in modular representation theory of finite groups relating the degrees of the complex irreducible characters in a Brauer block and the structure of its defect groups. It was formulated by Richard Brauer in 1955.

Statement

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Let   be a finite group and   a prime. The set   of irreducible complex characters can be partitioned into Brauer  -blocks. Each  -block   is canonically associated to a conjugacy class of  -subgroups, called the defect groups of B. The set of irreducible characters belonging to   is denoted by  .

Let   be the discrete valuation defined on the integers by   where   is coprime to  . Brauer proved that if   is a block with defect group   then   for each  . Brauer's Height Zero Conjecture asserts that   for all   if and only if   is abelian.

History

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Brauer's Height Zero Conjecture was formulated by Richard Brauer in 1955.[1] It also appeared as Problem 23 in Brauer's list of problems.[2] Brauer's Problem 12 of the same list asks whether the character table of a finite group   determines if its Sylow  -subgroups are abelian. Solving Brauer's height zero conjecture for blocks whose defect groups are Sylow  -subgroups (or equivalently, that contain a character of degree coprime to  ) also gives a solution to Brauer's Problem 12.

Proof

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The proof of the if direction of the conjecture was completed by Radha Kessar and Gunter Malle[3] in 2013 after a reduction to finite simple groups by Thomas R. Berger and Reinhard Knörr.[4]

The only if direction was proved for  -solvable groups by David Gluck and Thomas R. Wolf.[5] The so called generalized Gluck—Wolf theorem, which was a main obstacle towards a proof of the Height Zero Conjecture was proven by Gabriel Navarro and Pham Huu Tiep in 2013.[6] Gabriel Navarro and Britta Späth showed that the so-called inductive Alperin—McKay condition for simple groups implied Brauer's Height Zero Conjecture.[7] Lucas Ruhstorfer completed the proof of these conditions for the case  .[8] The case of odd primes was finally settled by Gunter Malle, Gabriel Navarro, A. A. Schaeffer Fry and Pham Huu Tiep using a different reduction theorem.[9]

References

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  1. ^ Brauer, Richard D. (1955). "Number theoretical investigations on groups of finite order". Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955. Science Council of Japan. pp. 55–62.
  2. ^ Brauer, Richard D. (1963). "Representations of finite groups". Lectures in Mathematics. Vol. 1. Wiley. pp. 133–175.
  3. ^ Kessar, Radha; Malle, Gunter (2013). "Quasi-isolated blocks and Brauer's height zero conjecture". Annals of Mathematics. 178: 321–384. doi:10.4007/annals.2013.178.1.6.
  4. ^ Berger, Thomas R.; Knörr, Reinhard (1988). "On Brauer's height 0 conjecture". Nagoya Mathematical Journal. 109: 109–116. doi:10.1017/S0027763000002798.
  5. ^ Gluck, David; Wolf, Thomas R. (1984). "Brauer's height conjecture for p-solvable groups". Transactions of the American Mathematical Society. 282: 137–152. doi:10.2307/1999582.
  6. ^ Navarro, Gabriel; Tiep, Pham Huu (2013). "Characters of relative  -degree over normal subgroups". Annals of Mathematics. 178: 1135–1171. doi:10.4007/annals.2013.178.
  7. ^ Navarro, Gabriel; Späth, Britta (2014). "On Brauer's height zero conjecture". Journal of the European Mathematical Society. 16: 695–747. doi:10.4171/JEMS/444.
  8. ^ Ruhstorfer, Lucas (2022). "The Alperin-McKay conjecture for the prime 2". to appear in Annals of Mathematics.
  9. ^ Malle, Gunter; Navarro, Gabriel; Schaeffer Fry, A. A.; Tiep, Pham Huu (2024). "Brauer's Height Zero Conjecture". Annals of Mathematics. 200: 557–608. doi:10.4007/annals.2024.200.2.4.