### Abstract

We study, via character-theoretic methods, an l-analogue of the modular representation theory of the symmetric group, for an arbitrary integer l greater than or equal to 2. We find that many of the invariants of the usual block theory (ie. when l 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 l-th root of unity. Donkin [4] has presented a direct link between the representation theory of these algebras and an l-analogue of the modular representation theory of the symmetric groups. It thus seems a natural problem to study l-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 l-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(l) 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 e-block for the Hecke algebras. In view of [4] this also is the determinant of the Cartan matrix of an l-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 Sect. 3 we introduce e-sections and e-blocks in symmetric groups and prove an analogue of the second main theorem of blocks. Then in Sect. 4 we construct "basic sets", i.e. integral bases for the restrictions of the generalized characters of S-n to l-regular elements. Generalizing ideas of Osima we study in Sect. 5 the equivalence of blocks of a given weight w. A relation between their decomposition matrices is given, showing also that their Cartan matrices have the same invariant factors. More generally, any such block is "perfectly isometric" to the set of irreducible characters of Z(l) S-w over a set of "regular" elements, defined by Osima. In the final section the invariant factors of the Cartan matrices are studied more closely. First the largest invariant factor is determined. In analogy with the prime case, each l-regular conjugacy class should contribute (in a quite subtle way) to the invariant factors and we make a specific conjecture what this contribution should be. Then we confirm (based on [3]) that the determinant of the l-Cartan matrix of S-n is in accordance with a conjecture of Bessenrodt and Olsson [1], and we explain how our conjectured invariant factors predict the determinant.

Original language | English |
---|---|

Pages (from-to) | 513-552 |

Number of pages | 39 |

Journal | Inventiones Mathematicae |

Volume | 151 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2003 |

### Cite this

*Inventiones Mathematicae*,

*151*(3), 513-552. https://doi.org/10.1007/S00222-002-0258-3

**Generalized blocks for symmetric groups.** / Kulshammer, B.; Olsson, J. B.; Robinson, Geoffrey Raymond.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 151, no. 3, pp. 513-552. https://doi.org/10.1007/S00222-002-0258-3

}

TY - JOUR

T1 - Generalized blocks for symmetric groups

AU - Kulshammer, B.

AU - Olsson, J. B.

AU - Robinson, Geoffrey Raymond

PY - 2003/3

Y1 - 2003/3

N2 - We study, via character-theoretic methods, an l-analogue of the modular representation theory of the symmetric group, for an arbitrary integer l greater than or equal to 2. We find that many of the invariants of the usual block theory (ie. when l 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 l-th root of unity. Donkin [4] has presented a direct link between the representation theory of these algebras and an l-analogue of the modular representation theory of the symmetric groups. It thus seems a natural problem to study l-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 l-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(l) 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 e-block for the Hecke algebras. In view of [4] this also is the determinant of the Cartan matrix of an l-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 Sect. 3 we introduce e-sections and e-blocks in symmetric groups and prove an analogue of the second main theorem of blocks. Then in Sect. 4 we construct "basic sets", i.e. integral bases for the restrictions of the generalized characters of S-n to l-regular elements. Generalizing ideas of Osima we study in Sect. 5 the equivalence of blocks of a given weight w. A relation between their decomposition matrices is given, showing also that their Cartan matrices have the same invariant factors. More generally, any such block is "perfectly isometric" to the set of irreducible characters of Z(l) S-w over a set of "regular" elements, defined by Osima. In the final section the invariant factors of the Cartan matrices are studied more closely. First the largest invariant factor is determined. In analogy with the prime case, each l-regular conjugacy class should contribute (in a quite subtle way) to the invariant factors and we make a specific conjecture what this contribution should be. Then we confirm (based on [3]) that the determinant of the l-Cartan matrix of S-n is in accordance with a conjecture of Bessenrodt and Olsson [1], and we explain how our conjectured invariant factors predict the determinant.

AB - We study, via character-theoretic methods, an l-analogue of the modular representation theory of the symmetric group, for an arbitrary integer l greater than or equal to 2. We find that many of the invariants of the usual block theory (ie. when l 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 l-th root of unity. Donkin [4] has presented a direct link between the representation theory of these algebras and an l-analogue of the modular representation theory of the symmetric groups. It thus seems a natural problem to study l-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 l-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(l) 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 e-block for the Hecke algebras. In view of [4] this also is the determinant of the Cartan matrix of an l-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 Sect. 3 we introduce e-sections and e-blocks in symmetric groups and prove an analogue of the second main theorem of blocks. Then in Sect. 4 we construct "basic sets", i.e. integral bases for the restrictions of the generalized characters of S-n to l-regular elements. Generalizing ideas of Osima we study in Sect. 5 the equivalence of blocks of a given weight w. A relation between their decomposition matrices is given, showing also that their Cartan matrices have the same invariant factors. More generally, any such block is "perfectly isometric" to the set of irreducible characters of Z(l) S-w over a set of "regular" elements, defined by Osima. In the final section the invariant factors of the Cartan matrices are studied more closely. First the largest invariant factor is determined. In analogy with the prime case, each l-regular conjugacy class should contribute (in a quite subtle way) to the invariant factors and we make a specific conjecture what this contribution should be. Then we confirm (based on [3]) that the determinant of the l-Cartan matrix of S-n is in accordance with a conjecture of Bessenrodt and Olsson [1], and we explain how our conjectured invariant factors predict the determinant.

U2 - 10.1007/S00222-002-0258-3

DO - 10.1007/S00222-002-0258-3

M3 - Article

VL - 151

SP - 513

EP - 552

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -