ON LOWER BOUNDS FOR THE DIMENSIONS OF PROJECTIVE MODULES FOR FINITE SIMPLE GROUPS A.E. Zalesski Group rings and Young-Baxter equations Spa, Belgium, June 2017 1
Introduction Let G be a nite group and p a prime. Let F be an algebraically closed eld of char p > 0. Projective indecomposable F G-modules (PIM) are exactly indecomposable direct summands of the regular F G-module. These were introduced by Brauer and Nesbitt in 1940 and remain important objects of study in representation theory of nite groups. However, there are very poor information on their dimensions; an extremal version of the problem can be stated as follows: Problem 1. Given a nite group G and a prime p, determine the minimum dimension of a projective F G-module.
The absolute lower bound for the dimension of a PIM is G p, the p-part of the order of G. However, it could be wrong to expect that this bound attains for every group G. There are two well known cases where this is true: 1) G has a subgroup of index G p, in particular, G is p-solvable and 2) G is a Chevalley group (or more generally, a nite reductive group) in dening characteristic p. In fact, the majority of works in the area concern with the decomposition of the characters of projective modules in terms of ordinary irreducible characters, or with computation of decomposition numbers. The matter is that every PIM corresponds to a projective indecomposable K p G-module, where K p is the ring of integers of a suitable nite extension of Q p, the p-adic number eld. The character of this can be expressed in terms of irreducible characters of G. The coecients of this are called the decomposition numbers.
In particular, the dimension of a PIM can be expressed in terms of irreducible character degrees of G and the decomposition numbers. This hints that Problem 1 can be studied in the framework of ordinary character theory. The diculty is that no way is known to characterise PIMs in term of ordinary characters. However, as the character of a projective K p G- module vanishes at the p-singular elements, one can somehow ignore this diculty as follows. Denition. An ordinary (reducible) character is called quasi-projective (QP) if it vanishes at the p-singular elements. With this, one can replace Problem 1 by the following one which belongs to the ordinary character theory and is expected to be easier: Problem 2. Determine the minimum degree a QP character of G.
Quasi-projective characters A QP-character is called indecomposable if this is not of a sum of proper QP-characters. For groups with cyclic Sylow p-subgroups the indecomposable QP characters are classied in Willems-Z (J. Alg. 2015). In particular, Theorem 1. Let χ be an indecomposable QP character. Then χ = τ + σ, where τ is irreducible, and σ is either 0, or irreducible or the sum of all exceptional characters. In addition, every PIM character is indecomposable as a QP-character. In addition, the indecomposable QP characters have a nice description in terms of the Brauer tree.
Obviously, the restriction of a quasi-projective character χ to a Sylow p-subgroup of G is a multilple of the regular character, so χ(1) = l G p, where G p is the p-part of the order of G. We call l the level of χ. This is meaningful for projective characters as well. So G p is the absolute lower bound for the degree of a QP character. The natural problem is: Problem 3. Determine reducible quasi-projective (and projective) characters of level 1. This problem was addressed in a paper by Pellegrini-Z (2016), where it was solved for simple groups of Lie type G for p to be dening characteristic of G, except for types B 3 and D 4. These two exceptional cases are still open.
For simple groups Problem 3 was studied by Malle-Weigel (2008), Z. (2013), Pellegrini-Z (2016) and Malle-Z (in preparation). Technics developed there can be used for further progress. In addition, it becomes more clear what kind of diculties arise when dealing with QP characters in comparison with projective ones. To illustrate: Problem 4. Let G = G 1 G 2 be a direct product. Describe indecomposable QP characters in terms of those for G 1, G 2. At least, one cannot expect a simple answer available for projective indecomposable characters, which are the products of those for G 1 and G 2.
A less ambitious problem useful for some application is the following: Problem 5. Set G (n) = G G (n times). Suppose that G has no indecomposable QP character of level 1. Is it true that for every m > 0 there exists n such that the levels of all QP characters of G (n) exceed m? For some groups G this is true, and used to prove the following result (Malle-Z, in preparation).
Theorem 2. Let p > 2 be a prime and A n the alternating group. Then the minimum degree of a QP character of A n tends to the innity as n. This also implies a similar result for classical groups with p to be a cross characteristic, where the degree in question grows together with the rank n of the group. In constrast, this is not true for the natural characteristic of a classical group. Theorem 2 is shown to be failed for p = 2, however, the analogue of it for classical groups is valid for p = 2. Almost nothing is known for non-simple groups, except for p-solvable.
I mention the following result on groups of Lie type (Z, J. Alg. 2013): Theorem 3. Let G = P SL n (p m ), n > 4, and χ be a reducible projective character for the prime p. Then χ(1) (n 1) G p. This bound is sharp. The existence of a projective module of dimension n G p is well known, and for q = 2 there exists such a module of dimension (n 1) G 2. In addition, there exists a QP character of the above degree. I expect that this is projective. Theorem 3 is not probably true for QP characters. In the above paper a similar result is proved also for groups E n (p m ), n = 6, 7, 8, where the bound is shown to be n G p.
Malle-Z (in preparation) complete classication of QP characters of degree G p for G simple, except for p = 2 and G = A n. The list is too large to be exposed here. Some bibliography G. Malle and Th. Weigel, Finite groups with minimal 1-PIM, Manuscripta Math. 126 (2008), 315-332. G. Malle and A. Zalesski, In preparation M. Pellegrini and A. Zalesski, On characters of Chevalley groups vanishing at the non-semisimple elements, Intern. J. Algebra and Comput. Math. 26(2016), 789-841. M. Pellegrini and A. Zalesski, Irreducible characters of Chevalley groups constant on non-identity unipotent elements, Rend. Sem. Mat. Univ. Padova, Vol. 136(2016), 3550. W. Willems and A. Zalesski, Quasi-projective and quasiliftable characters, J. Algebra 442(2015), 548-559. A. Zalesski, Low dimensional projective indecomposable modules for Chevalley groups in dening characteristic, J. Algebra 377(2013), 125 156. A. Zalesski, Invariants of maximal tori and unipotent constituents of some quasi-projective characters for nite classical groups, ArXiv:1705.07179v1 [math.gr] 19 May 2017