Blocks Of Symmetric Groups Research Articles

Morita Equivalent Blocks of Symmetric Groups

A well-known result of Scopes states that there are only finitely many Morita equivalence classes of $p$-blocks of symmetric groups with a given weight (or defect). In this note we investigate a lower bound on the number of those Morita equivalence classes.

RoCK blocks, wreath products and KLR algebras

We consider RoCK (or Rouquier) blocks of symmetric groups and Hecke algebras at roots of unity. We prove a conjecture of Turner asserting that a certain idempotent truncation of a RoCK block of weight d of a symmetric group $${\mathfrak {S}}_n$$ defined over a field F of characteristic e is Morita equivalent to the principal block of the wreath product $$\mathfrak S_e \wr \mathfrak S_d$$ . This generalises a theorem of Chuang and Kessar that applies to RoCK blocks with abelian defect groups. Our proof relies crucially on an isomorphism between $$F{\mathfrak {S}}_n$$ and a cyclotomic Khovanov–Lauda–Rouquier algebra, and the Morita equivalence we produce is that of graded algebras. We also prove the analogous result for an Iwahori–Hecke algebra at a root of unity defined over an arbitrary field.

Basic Orders for Defect Two Blocks of ℤ p Σ n

We show how basic orders for defect two blocks of symmetric groups over the ring of p-adic integers can be constructed by purely combinatorial means.

GRADED CARTAN MATRICES OF DEFECT 2 BLOCKS OF SYMMETRIC GROUPS

In this paper we show how the LLT algorithm for computation of crystal decomposition numbers can be used to construct quivers of defect 2 blocks of symmetric groups. We do this by establishing a connection between tight gradings on blocks of symmetric groups and crystal decomposition numbers.

Cubist algebras

We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these ‘Cubist algebras’ satisfy strong homological properties, such as Koszulity and quasi-heredity, reflecting the combinatorics of the tilings. We construct derived equivalences between Cubist algebras associated to local mutations in tilings. We recover as a special case the Rhombal algebras of Michael Peach and make a precise connection to weight 2 blocks of symmetric groups.

Decomposition numbers for weight three blocks of symmetric groups and Iwahori--Hecke algebras

Let F be a field, q a non-zero element of F and H n = H F,q (G n ) the Iwahori-Hecke algebra of the symmetric group G n . If B is a block of H n of e-weight 3 and the characteristic of F is at least 5, we prove that the decomposition numbers for B are all at most 1. In particular, the decomposition numbers for a p-block of G n of defect 3 are all at most 1.

On weight three blocks of symmetric groups in characteristic three

Quarterly Journal of Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Quart. J. Math. 53 (2002) 403–19. http://dx.doi.org/10.1093/qjmath/53.4.403

On Certain Blocks of Schur Algebras

In this paper, what is already known about defect 2 blocks of symmetric groups is used to deduce information about the corresponding blocks of Schur algebras. This information includes Ext-quivers and decomposition numbers, as well as Loewy structures of the Weyl modules, principal indecomposable modules and tilting modules.

Note on Blocks of Symmetric Groups

A conjecture concerning characterization of blocks of representations of symmetric groups by means of Young diagrams, given formerly by one of the writers, was affirmed recently by R. Brauer and G. de B. Robinson jointly. Namely, it was proved in Brauer’s paper, relying on his general, and profound, theory of blocks in finite groups, that the number of p-blocks in a symmetric group of degree n, is exactly equal to the number of p-cores, or p-kernels in the terminology of NI, with n- Ip (l= 0, 1, 2,…) nodes, and that certain statements designated as (A), (B) and (C) would secure that two irreducible representations with a same p-core belong to a same block (whence the conjecture itself when combined with the estimate of the number of blocks). Then these statements (A), (B), (C) were proved actually in Robinson’s paper.