Some decomposition numbers for Hecke algebras of general linear groups

1996 ◽  
Vol 119 (3) ◽  
pp. 383-402 ◽  
Author(s):  
Matthew J. Richards
Keyword(s):  
Symmetric Group ◽  
Recent Work ◽  
Hecke Algebras ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Decomposition Numbers ◽  
Weight One

The theorem which is still known as Nakayama's Conjecture shows how the modular characters of the symmetric group Sn can be divided into blocks of various weights, those with the same weight having similar properties. In fact, all blocks of weight one have essentially the same decomposition numbers and these are easy to describe. In recent work, Scopes [16, 17] has shown that there are essentially only finitely many possibilities for the decomposition numbers for blocks of any given weight, and has given a bound for the number. We develop the combinatorics implicit in her work, and so establish an improved bound.

scholarly journals Hecke algebras, finite general linear groups, and Heisenberg categorification

10.4171/qt/37 ◽  
2013 ◽  
Vol 4 (2) ◽  
pp. 125-185 ◽  
Author(s):  
Anthony Licata ◽  
Alistair Savage
Keyword(s):  
Hecke Algebras ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups

A note on decomposition numbers for general linear groups and symmetric groups

1985 ◽  
Vol 97 (1) ◽  
pp. 57-62 ◽  
Author(s):  
S. Donkin
Keyword(s):  
Chevalley Group ◽  
Reductive Group ◽  
Symmetric Groups ◽  
General Linear ◽  
Linear Groups ◽  
Reductive Groups ◽  
Weyl Module ◽  
General Linear Groups ◽  
Group I ◽  
Decomposition Numbers

In [5] James proved theorems on the decomposition numbers, for the general linear groups and symmetric groups, involving the removal of the first row or column from partitions. In [1] we gave different proofs of these theorems based on a result valid for the decomposition numbers of any reductive group. (I am grateful to J. C. Jantzen for pointing out that the Theorem in [1] may also be derived from the universal Chevalley group case, which follows from the proof of 1 ·18 Satz of [6] – the analogue of equation (1) of the proof being obtained by means of the natural isometry (with respect to contra-variant forms) between a certain sum of weight spaces of a Weyl module V(λ) of highest weight λ and the Weyl module corresponding to λ for the Chevalley group determined by the subset of the base involved.) However, we have recently noticed that this result for reductive groups, even when specialized to the case of GLn, gives a substantial generalization of James's Theorems. This generalization, which we give here, is an expression for the decomposition number [λ: μ] for a pair of partitions λ, μ whose diagrams can be simultaneously cut by a horizontal (or vertical) line so as to leave the same number of nodes above the line (or to the left of the line for a vertical cut) in both cases. Cutting between the first and second rows gives James's principal of row removal ([5], theorem 1) and cutting between the first and second column gives his principle of column removal ([5], theorem 2). Another special case of our horizontal result, involving the removal of bottom rows of a pair of partitions, is stated in [7], Satz 8.

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