arXiv subjects
Publications and source records attributed to Olivier Brunat.
We construct, for every odd prime $p$, a bijection between the $p'$-spin characters $\widetilde S_n$ and those of a Sylow normalizer which, in particular, is equivariant under the Galois action detecting $p$-rationality. An analogous result holds for $\widetilde A_n$. We also obtain a blockwise refinement for height-zero spin characters and their Brauer correspondents.
We study generalised core partitions arising from affine Grassmannian elements in arbitrary Dynkin type. The corresponding notion of size is given by the atomic length in the sense of [CLG22]. In this paper, we first develop the theory for extended affine Weyl groups. In a series of applications, we give some remarkable parametrisations of the solutions of certain Diophantine equations resembling Pell's equation, by refining the results of [BN22] and [Alp14], and generalising them to further types.
The study of modular representation theory of the double covering groups of the symmetric and alternating groups reveals rich and subtle combinatorial and algebraic phenomena involving their irreducible characters and the structure of their p-blocks, where p is an odd prime number. In this paper, we investigate the action of certain Galois automorphisms, those that act on p'-roots of unity by a power of p, on spin characters, with an emphasis on their interaction with perfect isometries and block theory. In particular, we prove that perfect isometries constructed by the first author and J.\,B. Gramain in \cite{BrGr3}, which were used to establish a weaker form of the Kessar--Schaps conjecture, remain preserved under this Galois action whenever certain natural compatibility conditions occur.
This note is concerned with the set of integral solutions of the equation $x^2+3y^2=12n+4$, where $n$ is a positive integer. We will describe a parametrization of this set using the 3-core partitions of n. In particular we construct a crank using the action of a suitable subgroup of the isometric group of the plane that we connect with the unit group of the ring of Eisenstein integers. We also show that the process goes in the reverse direction: from the solutions of the equation and the crank, we can describe the 3-core partitions of n. As a consequence we describe an explicit bijection between $3$-core partitions and ideals of the ring of Eisenstein integers, explaining a result of G. Han and K. Ono obtained using modular forms.
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the decomposition matrices, that is, the existence of a unitriangular basic set. We study several ways to obtain such sets in the general case of a symmetric algebra. We apply our results to the symmetric groups and to their Hecke algebras and thus obtain new ways to label the simple modules for these objects. Finally, we show that these sets do not always exist in the case of the alternating groups by studying two explicit cases in characteristic 3.
Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture holds for the alternating groups when the prime p is odd.
We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.
The Frobenius symbol was first introduced in 1900 by Frobenius as a way to encode an integer partition. In 1941, motivated by the modular representation theory of the symmetric group, Nakayama introduced the idea of a p-core partition, for p prime, using hook removals. In the following decade, Robinson, Littlewood, Staal and Farahat codified the $p$-quotient of such a partition, using variations of the star diagram. Since the 1970s, the convention has been to build up the theory of both core and quotients with the abacus construction, first introduced by G. James. In this paper we return to the earlier point of view. First we show that, for any positive integer t, the t-core and t-quotient of an integer partition can be directly obtained from its Frobenius symbol. The argument also works in the opposite direction: that is, given the Frobenius symbol of a t-core and a $t$-tuple of Frobenius symbols, one can recover the Frobenius symbol of the corresponding partition. One immediate application is the calculation of the Durfee number of the associated partition from the Frobenius symbols of the core and quotient. In 1991, J. Scopes gathered together the $p$-core partitions into families to prove that Donovan's conjecture holds for the symmetric groups at the prime p. We describe, using our methods, the action of the affine Weyl group W_p of type $A$ on Frobenius symbols, and use this to parametrize and compute the explicit number of Scopes families. In particular we enumerate both the infinite and finite Scopes families. Core partitions have also attracted recent interest in number theory. By constructing explicit and combinatorial bijections, we revisit some well-known identities originally obtained using sophisticated methods. We end with a close study of the relationship between certain hooks in the quotient and certain hooks in the associated partition.
In this paper, we prove that, given any integers $d$, $e$, $r$ and $r'$, and a prime $p$ not dividing $de$, any two blocks of the complex reflection groups $G(de,e,r)$ and $G(de,e,r')$ with the same $p$-weight are perfectly isometric.
In this paper, following the methods of http://arxiv.org/abs/1305.7449v2, we show that the double covering groups of the symmetric and alternating groups have p-basic sets for any odd prime p.
This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two p-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for p-blocks of alternating groups (where the blocks must also have the same sign when p is odd), of double covers of alternating and symmetric groups (for p odd, and where we obtain crossover isometries when the blocks have opposite signs),of complex reflection groups G(d,1,n) (for d prime to p), of Weyl groups of type B and D (for p odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks, in a way which should be of independent interest.
Eaton and Moretó proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~$p$. We provide some further evidence for blocks of groups of Lie type considered in cross characteristic.
We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras over a finite field are unitary for the action of the braid groups.
We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.
We give parameterizations of the irreducible representations of finite groups of Lie type in their defining characteristic.
In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group G defined over a finite field with corresponding Frobenius map F and derive the number of F-stable semisimple classes of G with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters to their inertia group in the full automorphism group. As a consequence, we prove that "twisted" and "untwisted" simple groups of type E_6 are "good" in defining characteristic, which is a contribution to the general program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture in representation theory of finite groups.
This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in the defining characteristic case. Using a general argument based on the representation theory of connected reductive groups with disconnected center, we show that the inductive McKay condition holds if the Schur multiplier of the simple group has order 2. As a consequence, the simple groups \Orth_{2m+1}(p^n) and PSp_{2m}(p^n) are "good" for p>2 and the simple groups E_7(p^n) are ``good'' for p>3 in the sense of Isaacs, Malle and Navarro. We also describe the action of the diagonal and field automorphisms on the semisimple and the regular characters.
This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the F-coinvariants of the component group of the center of G has prime order and if p is a good prime for G, then the relative McKay conjecture holds for G at the prime p. In particular, this conjecture is true for G^F in defining characteristic for G a simple and simply-connected group of type B_n, C_n, E_6 and E_7. Our main tools are the theory of Gelfand-Graev characters for connected reductive groups with disconnected center developed by Digne-Lehrer-Michel and the theory of cuspidal Levi subgroups. We also explicitly compute the number of semisimple classes of G^F for any simple algebraic group G.