Generalized blocks for symmetric groups

Abstract

Abstract. We study, via character-theoretic methods, an -analogue of the modular representation theory of the symmetric group, for an arbitrary integer ≥ 2. We find that many of the invariants of the usual block theory (ie. when is prime) generalize in a natural fashion to this new context.The study of the modular representation theory of symmetric groups was initiated in the 1940's. One of the first highlights was the proof of the so-called Nakayama conjecture describing the distribution of the irreducible characters into p-blocks in terms of a combinatorial condition on the partitions labelling them. More specifically two irreducible characters are in the same p-block if and only if the partitions labelling them have the same p-core. There is also a comprehensive literature on decomposition numbers, Cartan matrices and other block-theoretic invariants of symmetric groups.The representation theory of symmetric groups has served as a source of inspiration for the study of representations of other classes of groups and algebras. As an example we may refer to the book [9]. Corollary 5.38 in that book presents an analogue of the Nakayama conjecture for Iwahori-Hecke algebras for the symmetric group S n at an -th root of unity. Donkin [4] has presented a direct link between the representation theory of these algebras and an -analogue of the modular representation theory of the symmetric groups. It thus seems a natural problem to study " -blocks" of S n . We attempt to do this here based primarily on the ordinary character theory of symmetric groups and on some very general ideas from the character theory of finite groups. We study analogues of blocks, of the second main theorem on blocks, of decomposition matrices and of Cartan matrices in this context and prove an -analogue of the Nakayama conjecture. We believe that this approach may provide additional insight, eg. concerning the invariant factors of Cartan matrices. For instance we show that these calculations for a given block of weight w may be performed inside the wreath product Z S w . It should be mentioned that Brundan and Kleshchev [3] have recently given a formula for the determinant of the Cartan matrix of an -block for the Hecke algebras. In view of [4] this also is the determinant of the Cartan matrix of an -block of S n . (See Proposition 6.10 for details).The paper is organized as follows: The first two sections present a very general theory of contributions, perfect isometries, sections and blocks, suitable for our purposes. These sections may have independent interest beyond the questions at hand. In section 3 we introduce -sections and -blocks in symmetric groups and prove an analogue of the second main theorem of blocks. Then in section 4 we construct "basic sets", i.e. integral bases for the restrictions of the generalized

Properties of Some Character Tables Related to the Symmetric Groups

“…The knowledge of these determinants is equivalent to the knowledge of the determinants of certain "generalized Cartan matrices" of S n as considered in [9]. In particular we obtain a new proof of a conjecture of Mathas about the Cartan matrix of an Iwahori-Hecke algebra of S n at a primitive rth root of unity which is simpler than the original proof given by Brundan and Kleshchev in [3].…”

“…Let us finally mention that in [9,Section 6] there is an explicit conjecture about the Smith normal form of C. In the case where r is a prime, this is known to be true by the general theory of R. Brauer. One may also ask about the Smith normal form of X RC ; we answer this question in this article in the prime case.…”

“…Corollary 2 Let C be the r-analogue of the Cartan matrix of S n as defined in [9]. Then we have det (C) = r d C .…”

“…This result was first proved in [12], but the proof relied on results of [9] for which the work of Donkin [4] and Brundan and Kleshchev [3] was used in a crucial way. Our proof of Theorem 1 does not use [4] or [3]; it is direct and thus much shorter.…”

“…Let Y = Y CC = (ξ λ (µ)) λ,µ∈C denote the part of the permutation character table of S n with rows and columns indexed by the class p-regular partitions of n. Set X = X RC . As the characters χ λ with λ in the set R of p-regular partitions of n form a basic set for the characters on the p-regular conjugacy classes by [9], we have a decomposition matrix D = D CR with integer entries such that…”